Average Error: 4.1 → 1.8
Time: 35.1s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{\mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \frac{\sqrt{t + a}}{\frac{t}{z}}\right) \cdot 2}, x\right)}\]
\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}
↓
\frac{x}{\mathsf{fma}\left(y, e^{\mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \frac{\sqrt{t + a}}{\frac{t}{z}}\right) \cdot 2}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2586904 = x;
        double r2586905 = y;
        double r2586906 = 2.0;
        double r2586907 = z;
        double r2586908 = t;
        double r2586909 = a;
        double r2586910 = r2586908 + r2586909;
        double r2586911 = sqrt(r2586910);
        double r2586912 = r2586907 * r2586911;
        double r2586913 = r2586912 / r2586908;
        double r2586914 = b;
        double r2586915 = c;
        double r2586916 = r2586914 - r2586915;
        double r2586917 = 5.0;
        double r2586918 = 6.0;
        double r2586919 = r2586917 / r2586918;
        double r2586920 = r2586909 + r2586919;
        double r2586921 = 3.0;
        double r2586922 = r2586908 * r2586921;
        double r2586923 = r2586906 / r2586922;
        double r2586924 = r2586920 - r2586923;
        double r2586925 = r2586916 * r2586924;
        double r2586926 = r2586913 - r2586925;
        double r2586927 = r2586906 * r2586926;
        double r2586928 = exp(r2586927);
        double r2586929 = r2586905 * r2586928;
        double r2586930 = r2586904 + r2586929;
        double r2586931 = r2586904 / r2586930;
        return r2586931;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2586932 = x;
        double r2586933 = y;
        double r2586934 = c;
        double r2586935 = b;
        double r2586936 = r2586934 - r2586935;
        double r2586937 = 5.0;
        double r2586938 = 6.0;
        double r2586939 = r2586937 / r2586938;
        double r2586940 = a;
        double r2586941 = 2.0;
        double r2586942 = t;
        double r2586943 = r2586941 / r2586942;
        double r2586944 = 3.0;
        double r2586945 = r2586943 / r2586944;
        double r2586946 = r2586940 - r2586945;
        double r2586947 = r2586939 + r2586946;
        double r2586948 = r2586942 + r2586940;
        double r2586949 = sqrt(r2586948);
        double r2586950 = z;
        double r2586951 = r2586942 / r2586950;
        double r2586952 = r2586949 / r2586951;
        double r2586953 = fma(r2586936, r2586947, r2586952);
        double r2586954 = r2586953 * r2586941;
        double r2586955 = exp(r2586954);
        double r2586956 = fma(r2586933, r2586955, r2586932);
        double r2586957 = r2586932 / r2586956;
        return r2586957;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 4.1

    \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

  2. Simplified1.8

    \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, e^{\mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \frac{\sqrt{a + t}}{\frac{t}{z}}\right) \cdot 2}, x\right)}}\]

  3. Final simplification1.8

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \frac{\sqrt{t + a}}{\frac{t}{z}}\right) \cdot 2}, x\right)}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))