Average Error: 3.8 → 1.4
Time: 30.2s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
\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, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r77948 = x;
        double r77949 = y;
        double r77950 = 2.0;
        double r77951 = z;
        double r77952 = t;
        double r77953 = a;
        double r77954 = r77952 + r77953;
        double r77955 = sqrt(r77954);
        double r77956 = r77951 * r77955;
        double r77957 = r77956 / r77952;
        double r77958 = b;
        double r77959 = c;
        double r77960 = r77958 - r77959;
        double r77961 = 5.0;
        double r77962 = 6.0;
        double r77963 = r77961 / r77962;
        double r77964 = r77953 + r77963;
        double r77965 = 3.0;
        double r77966 = r77952 * r77965;
        double r77967 = r77950 / r77966;
        double r77968 = r77964 - r77967;
        double r77969 = r77960 * r77968;
        double r77970 = r77957 - r77969;
        double r77971 = r77950 * r77970;
        double r77972 = exp(r77971);
        double r77973 = r77949 * r77972;
        double r77974 = r77948 + r77973;
        double r77975 = r77948 / r77974;
        return r77975;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r77976 = x;
        double r77977 = y;
        double r77978 = 2.0;
        double r77979 = exp(r77978);
        double r77980 = t;
        double r77981 = r77978 / r77980;
        double r77982 = 3.0;
        double r77983 = r77981 / r77982;
        double r77984 = a;
        double r77985 = 5.0;
        double r77986 = 6.0;
        double r77987 = r77985 / r77986;
        double r77988 = r77984 + r77987;
        double r77989 = r77983 - r77988;
        double r77990 = b;
        double r77991 = c;
        double r77992 = r77990 - r77991;
        double r77993 = z;
        double r77994 = cbrt(r77980);
        double r77995 = r77994 * r77994;
        double r77996 = r77993 / r77995;
        double r77997 = r77980 + r77984;
        double r77998 = sqrt(r77997);
        double r77999 = r77998 / r77994;
        double r78000 = r77996 * r77999;
        double r78001 = fma(r77989, r77992, r78000);
        double r78002 = pow(r77979, r78001);
        double r78003 = fma(r77977, r78002, r77976);
        double r78004 = r77976 / r78003;
        return r78004;
}

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 3.8

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

  2. Simplified2.5

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

  4. Applied add-cube-cbrt2.5

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

  5. Applied times-frac1.4

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

  6. Final simplification1.4

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))