Average Error: 3.9 → 1.6
Time: 30.9s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}{\frac{t}{\sqrt[3]{z}}}\right)}, x\right)}\]
\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, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}{\frac{t}{\sqrt[3]{z}}}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2465853 = x;
        double r2465854 = y;
        double r2465855 = 2.0;
        double r2465856 = z;
        double r2465857 = t;
        double r2465858 = a;
        double r2465859 = r2465857 + r2465858;
        double r2465860 = sqrt(r2465859);
        double r2465861 = r2465856 * r2465860;
        double r2465862 = r2465861 / r2465857;
        double r2465863 = b;
        double r2465864 = c;
        double r2465865 = r2465863 - r2465864;
        double r2465866 = 5.0;
        double r2465867 = 6.0;
        double r2465868 = r2465866 / r2465867;
        double r2465869 = r2465858 + r2465868;
        double r2465870 = 3.0;
        double r2465871 = r2465857 * r2465870;
        double r2465872 = r2465855 / r2465871;
        double r2465873 = r2465869 - r2465872;
        double r2465874 = r2465865 * r2465873;
        double r2465875 = r2465862 - r2465874;
        double r2465876 = r2465855 * r2465875;
        double r2465877 = exp(r2465876);
        double r2465878 = r2465854 * r2465877;
        double r2465879 = r2465853 + r2465878;
        double r2465880 = r2465853 / r2465879;
        return r2465880;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2465881 = x;
        double r2465882 = y;
        double r2465883 = 2.0;
        double r2465884 = c;
        double r2465885 = b;
        double r2465886 = r2465884 - r2465885;
        double r2465887 = 5.0;
        double r2465888 = 6.0;
        double r2465889 = r2465887 / r2465888;
        double r2465890 = t;
        double r2465891 = r2465883 / r2465890;
        double r2465892 = 3.0;
        double r2465893 = r2465891 / r2465892;
        double r2465894 = a;
        double r2465895 = r2465893 - r2465894;
        double r2465896 = r2465889 - r2465895;
        double r2465897 = r2465894 + r2465890;
        double r2465898 = sqrt(r2465897);
        double r2465899 = z;
        double r2465900 = cbrt(r2465899);
        double r2465901 = r2465900 * r2465900;
        double r2465902 = r2465898 * r2465901;
        double r2465903 = r2465890 / r2465900;
        double r2465904 = r2465902 / r2465903;
        double r2465905 = fma(r2465886, r2465896, r2465904);
        double r2465906 = r2465883 * r2465905;
        double r2465907 = exp(r2465906);
        double r2465908 = fma(r2465882, r2465907, r2465881);
        double r2465909 = r2465881 / r2465908;
        return r2465909;
}

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 3.9

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

  2. Simplified1.6

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

  4. Applied add-cube-cbrt1.6

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

  5. Applied *-un-lft-identity1.6

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\frac{\color{blue}{1 \cdot t}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\right)}, x\right)}\]

  6. Applied times-frac1.6

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\color{blue}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{t}{\sqrt[3]{z}}}}\right)}, x\right)}\]

  7. Applied associate-/r*1.6

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \color{blue}{\frac{\frac{\sqrt{a + t}}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}}{\frac{t}{\sqrt[3]{z}}}}\right)}, x\right)}\]

  8. Simplified1.6

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

  9. Final simplification1.6

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

Reproduce

herbie shell --seed 2019171 +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)))))))))))