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-rw-r--r--05/main.c10
-rw-r--r--05/math.h409
-rw-r--r--05/stdc_common.h1
-rw-r--r--05/tokenize.b7
4 files changed, 415 insertions, 12 deletions
diff --git a/05/main.c b/05/main.c
index a77ccf2..5f57418 100644
--- a/05/main.c
+++ b/05/main.c
@@ -1,15 +1,9 @@
#define _STDLIB_DEBUG
+#include <math.h>
#include <stdio.h>
-#include <string.h>
-#include <stdlib.h>
-#include <stddef.h>
-#include <ctype.h>
-#include <locale.h>
int main(int argc, char **argv) {
- setlocale(LC_ALL, "C");
- struct lconv *c = localeconv();
- printf("%s\n",c->negative_sign);
+ printf("%.16g\n", cosh(10));
return 0;
}
diff --git a/05/math.h b/05/math.h
new file mode 100644
index 0000000..7b029ac
--- /dev/null
+++ b/05/math.h
@@ -0,0 +1,409 @@
+#ifndef _MATH_H
+#define _MATH_H
+
+#include <stdc_common.h>
+#define HUGE_VAL _INFINITY // glibc defines HUGE_VAL as infinity (the C standard only requires it to be positive, funnily enough)
+#define _NAN (-(_INFINITY-_INFINITY))
+#define _PI 3.141592653589793
+#define _2PI 6.283185307179586
+#define _HALF_PI 1.5707963267948966
+#define _THREE_HALVES_PI 4.71238898038469
+
+// NOTE: these functions are not IEEE 754-compliant (the C standard doesn't require them to be), but they're pretty good
+
+double frexp(double value, int *exp) {
+ if (value == 0) {
+ *exp = 0;
+ return 0;
+ }
+ unsigned long u = *(unsigned long *)&value, significand;
+ *exp = ((u >> 52) & 0x7ff) - 1022;
+ // replace exponent with 1022
+ u &= 0x800fffffffffffff;
+ u |= 0x3fe0000000000000;
+ return *(double *)&u;
+}
+
+double ldexp(double x, int exp) {
+ int e;
+ double y = frexp(x, &e);
+ // since x = y * 2^e, x * 2^exp = y * 2^(e+exp)
+ exp += e;
+ if (exp < -1022) return 0;
+ if (exp > 1023) return _INFINITY;
+ unsigned long pow2 = (unsigned long)(exp + 1023) << 52;
+ return y * *(double *)&pow2;
+}
+
+double floor(double x) {
+ if (x >= 0.0) {
+ if (x > 1073741824.0 * 1073741824.0)
+ return x; // floats this big must be integers
+ return (unsigned long)x;
+ } else {
+ if (x < -1073741824.0 * 1073741824.0)
+ return x; // floats this big must be integers
+ double i = (long)x;
+ if (x == i) return x;
+ return i - 1.0;
+ }
+}
+
+double ceil(double x) {
+ double f = floor(x);
+ if (x == f) return f;
+ return f + 1.;
+}
+
+double fabs(double x) {
+ // this is better than x >= 0 ? x : -x because it deals with -0 properly
+ unsigned long u = *(unsigned long *)&x;
+ u &= 0x7fffffffffffffff;
+ return *(double *)&u;
+}
+
+double fmod(double x, double y) {
+ if (y == 0.0) {
+ errno = EDOM;
+ return 0.0;
+ }
+ return x - (floor(x / y) * y);
+}
+
+double _sin_taylor(double x) {
+ double i;
+ double term = x;
+ // taylor expansion for sin: x - x³/3! + x⁵/5! - ...
+
+ // https://en.wikipedia.org/wiki/Kahan_summation_algorithm
+ double prev = -1.0;
+ double sum = 0.0;
+ double c = 0.0;
+ for (i = 0.0; i < 100.0 && sum != prev; ++i) {
+ prev = sum;
+ double y = term - c;
+ double t = sum + y;
+ c = (t - sum) - y;
+ sum = t;
+ term *= -(x * x) / ((2.0*i+2.0)*(2.0*i+3.0));
+ }
+ return sum;
+}
+
+double _cos_taylor(double x) {
+ double i;
+ double term = 1.0;
+ // taylor expansion for cos: 1 - x²/2! + x⁴/4! - ...
+
+ // https://en.wikipedia.org/wiki/Kahan_summation_algorithm
+ double prev = -1.0;
+ double sum = 0.0;
+ double c = 0.0;
+ for (i = 0.0; i < 100.0 && sum != prev; ++i) {
+ prev = sum;
+ double y = term - c;
+ double t = sum + y;
+ c = (t - sum) - y;
+ sum = t;
+ term *= -(x * x) / ((2.0*i+1.0)*(2.0*i+2.0));
+ }
+ return sum;
+}
+
+double sin(double x) {
+ x = fmod(x, 2.0*_PI);
+ // the Taylor series works best for small inputs. so, provide _sin_taylor with a value in the range [0,π/2]
+ if (x < _HALF_PI)
+ return _sin_taylor(x);
+ if (x < _PI)
+ return _sin_taylor(_PI - x);
+ if (x < _THREE_HALVES_PI)
+ return -_sin_taylor(x - _PI);
+ return -_sin_taylor(_2PI - x);
+}
+
+double cos(double x) {
+ x = fmod(x, 2.0*_PI);
+ // the Taylor series works best for small inputs. so, provide _cos_taylor with a value in the range [0,π/2]
+ if (x < _HALF_PI)
+ return _cos_taylor(x);
+ if (x < _PI)
+ return -_cos_taylor(_PI - x);
+ if (x < _THREE_HALVES_PI)
+ return -_cos_taylor(x - _PI);
+ return _cos_taylor(_2PI - x);
+}
+
+double tan(double x) {
+ return sin(x)/cos(x);
+}
+
+// for sqrt and the inverse trigonometric functions, we use Newton's method
+// https://en.wikipedia.org/wiki/Newton%27s_method
+
+double sqrt(double x) {
+ if (x < 0.0) {
+ errno = EDOM;
+ return _NAN;
+ }
+ if (x == 0.0) return 0.0;
+ if (x == _INFINITY) return _INFINITY;
+ // we want to find the root of: f(t) = t² - x
+ // f'(t) = 2t
+ int exp;
+ double y = frexp(x, &exp);
+ if (exp & 1) {
+ y *= 2;
+ --exp;
+ }
+ // newton's method will be slow for very small or very large numbers.
+ // so we have ensured that
+ // 0.5 ≤ y < 2
+ // and also x = y * 2^exp; sqrt(x) = sqrt(y) * 2^(exp/2) NB: we've ensured that exp is even
+
+ // 7 iterations seems to be more than enough for any number
+ double t = y;
+ t = (y / t + t) * 0.5;
+ t = (y / t + t) * 0.5;
+ t = (y / t + t) * 0.5;
+ t = (y / t + t) * 0.5;
+ t = (y / t + t) * 0.5;
+ t = (y / t + t) * 0.5;
+ t = (y / t + t) * 0.5;
+
+ return ldexp(t, exp>>1);
+
+}
+
+double _acos_newton(double x) {
+ // we want to find the root of: f(t) = cos(t) - x
+ // f'(t) = -sin(t)
+ double t = _HALF_PI - x; // reasonably good first approximation
+ double prev_t = -100.0;
+ int i;
+
+ for (i = 0; i < 100 && prev_t != t; ++i) {
+ prev_t = t;
+ t += (cos(t) - x) / sin(t);
+ }
+ return t;
+}
+
+double _asin_newton(double x) {
+ // we want to find the root of: f(t) = sin(t) - x
+ // f'(t) = cos(t)
+ double t = x; // reasonably good first approximation
+ double prev_t = -100.0;
+ int i;
+
+ for (i = 0; i < 100 && prev_t != t; ++i) {
+ prev_t = t;
+ t += (x - sin(t)) / cos(t);
+ }
+ return t;
+}
+
+double acos(double x) {
+ if (x > 1.0 || x < -1.0) {
+ errno = EDOM;
+ return _NAN;
+ }
+ // Newton's method doesn't work well near -1 and 1, because f(x) / f'(x) is very large.
+ if (x > 0.8)
+ return _asin_newton(sqrt(1-x*x));
+ if (x < -0.8)
+ return _PI-_asin_newton(sqrt(1-x*x));
+
+ return _acos_newton(x);
+}
+
+double asin(double x) {
+ if (x > 1.0 || x < -1.0) {
+ errno = EDOM;
+ return _NAN;
+ }
+ // Newton's method doesn't work well near -1 and 1, because f(x) / f'(x) is very large.
+ if (x > 0.8)
+ return _acos_newton(sqrt(1.0-x*x));
+ if (x < -0.8)
+ return -_acos_newton(sqrt(1.0-x*x));
+
+ return _asin_newton(x);
+}
+
+double atan(double x) {
+ // the formula below breaks for really large inputs; tan(10^20) as a double is indistinguishable from pi/2 anyways
+ if (x > 1e20) return _HALF_PI;
+ if (x < -1e20) return -_HALF_PI;
+
+ // we can use a nice trigonometric identity here
+ return asin(x / sqrt(1+x*x));
+}
+
+double atan2(double y, double x) {
+ if (x == 0.0) {
+ if (y > 0.0) return _HALF_PI;
+ if (y < 0.0) return -_HALF_PI;
+ return 0.0; // this is what IEEE 754 does
+ }
+
+ double a = atan(y/x);
+ if (x > 0.0) {
+ return a;
+ } else if (y > 0.0) {
+ return a + _PI;
+ } else {
+ return a - _PI;
+ }
+}
+
+double _exp_taylor(double x) {
+ double i;
+ double term = 1.0;
+ // taylor expansion for exp: 1 + x/1! + x²/2! + ...
+
+ // https://en.wikipedia.org/wiki/Kahan_summation_algorithm
+ double prev = -1.0;
+ double sum = 0.0;
+ double c = 0.0;
+ for (i = 1.0; i < 100.0 && sum != prev; ++i) {
+ prev = sum;
+ double y = term - c;
+ double t = sum + y;
+ c = (t - sum) - y;
+ sum = t;
+ term *= x / i;
+ }
+ return sum;
+}
+
+double exp(double x) {
+ if (x > 709.782712893384) {
+ errno = ERANGE;
+ return _INFINITY;
+ }
+ if (x == 0.0) return 1;
+ if (x < -744.4400719213812)
+ return 0;
+ int i, e;
+ double y = frexp(x, &e);
+ if (e < 1.0) return _exp_taylor(x);
+ // the taylor series doesn't work well for large x (positive or negative),
+ // so we use the fact that exp(y*2^e) = exp(y)^(2^e)
+ double value = _exp_taylor(y);
+ for (i = 0; i < e; ++i)
+ value *= value;
+ return value;
+}
+
+#define _LOG2 0.6931471805599453
+
+double log(double x) {
+ if (x < 0.0) {
+ errno = EDOM;
+ return _NAN;
+ }
+ if (x == 0.0) return -_INFINITY;
+ if (x == 1.0) return 0.0;
+ int e;
+ double sum;
+ double a = frexp(x, &e);
+ // since x = a * 2^e, log(x) = log(a) + log(2^e) = log(a) + e log(2)
+ sum = e * _LOG2;
+ // now that a is in [1/2,1), the series log(a) = (a-1) - (a-1)²/2 + (a-1)³/3 - ... converges quickly
+
+ a -= 1;
+ // https://en.wikipedia.org/wiki/Kahan_summation_algorithm
+ double prev = HUGE_VAL;
+ double c = 0;
+ double term = a;
+ double i;
+ for (i = 1.0; i < 100.0 && sum != prev; ++i) {
+ prev = sum;
+ double y = term / i - c;
+ double t = sum + y;
+ c = (t - sum) - y;
+ sum = t;
+ term *= -a;
+ }
+ return sum;
+}
+
+#define _INVLOG10 0.43429448190325176 // = 1/log(10)
+double log10(double x) {
+ return log(x) * _INVLOG10;
+}
+
+double modf(double value, double *iptr) {
+ double m = fmod(value, 1.0);
+ if (value >= 0.0) {
+ *iptr = value - m;
+ return m;
+ } else if (m == 0.0) {
+ *iptr = value;
+ return 0.0;
+ } else {
+ *iptr = value - m + 1.0;
+ return m - 1.0;
+ }
+}
+
+// double raised to the power of an integer
+double _dpowi(double x, unsigned long y) {
+ double result = 1.0;
+ if (y & 1) {
+ --y;
+ result *= x;
+ }
+ if (y > 0) {
+ double p = _dpowi(x, y >> 1);
+ result *= p * p;
+ }
+ return result;
+}
+
+double pow(double x, double y) {
+ if (x > 0.0) {
+ return exp(y * log(x));
+ } else if (x < 0.0) {
+ if (fmod(y, 1.0) != 0) {
+ errno = EDOM;
+ return _NAN;
+ }
+ if (y > 1.6602069666338597e+19)
+ return x < -1. ? -_INFINITY : 0.;
+ if (y < -1.6602069666338597e+19)
+ return x < -1. ? 0. : -_INFINITY;
+ return _dpowi(x, (unsigned long)y);
+ } else {
+ if (y < 0) {
+ errno = EDOM;
+ return _NAN;
+ }
+ if (y > 0) {
+ // 0^x = 0 for x>0
+ return 0.;
+ }
+ // 0^0 = 1
+ return 1.;
+ }
+}
+
+double cosh(double x) {
+ double e = exp(x);
+ return (e + 1./e) * 0.5;
+}
+
+double sinh(double x) {
+ double e = exp(x);
+ return (e - 1./e) * 0.5;
+}
+
+double tanh(double x) {
+ if (x > 20.0) return 1.;
+ if (x < -20.0) return -1.;
+ double e = exp(x);
+ double f = 1./e;
+ return (e - f) / (e + f);
+}
+#endif // _MATH_H
diff --git a/05/stdc_common.h b/05/stdc_common.h
index 9e5d7a6..af30b06 100644
--- a/05/stdc_common.h
+++ b/05/stdc_common.h
@@ -424,6 +424,7 @@ double strtod(const char *nptr, char **endptr) {
sum = t;
}
+ if (sum == _INFINITY) errno = ERANGE;
if (endptr) *endptr = nptr;
return sum * sign;
}
diff --git a/05/tokenize.b b/05/tokenize.b
index 809c9ff..56838c5 100644
--- a/05/tokenize.b
+++ b/05/tokenize.b
@@ -287,6 +287,7 @@ function tokenize
fraction = strtoi(&in, 10)
; e.g. to turn 35 into .35, multiply by 10^-2
pow10 = p - in
+ ;putnln_signed(pow10)
if pow10 < -400 goto bad_float
:float_no_fraction
; construct the number integer + fraction*10^pow10
@@ -303,16 +304,13 @@ function tokenize
exponent = new_exponent
significand += right_shift(integer, exponent)
:float_no_integer
+
if *1in == 'e goto float_exponent
if *1in == 'E goto float_exponent
:float_have_significand_and_exponent
if significand == 0 goto float_zero
normalize_float(&significand, &exponent)
- ; putn(significand)
- ; putc(32)
- ; putn_signed(exponent)
- ; putc(10)
; make number round to the nearest representable float roughly (this is what gcc does)
; this fails for 5e-100 probably because of imprecision, but mostly works
significand += 15
@@ -321,6 +319,7 @@ function tokenize
exponent += 5
exponent += 52 ; 1001010111... => 1.001010111...
n = leftmost_1bit(significand)
+ exponent += n - 52 ; in most cases, this is 0, but sometimes it turns out to be 1.
b = 1 < n
significand &= ~b
data = significand