Average Error: 4.0 → 1.6
Time: 28.4s
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\frac{\sqrt{a + t}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{\sqrt[3]{t}}{z}}\right)}, x\right)}\]
\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}
↓
\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\frac{\sqrt{a + t}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{\sqrt[3]{t}}{z}}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2351854 = x;
        double r2351855 = y;
        double r2351856 = 2.0;
        double r2351857 = z;
        double r2351858 = t;
        double r2351859 = a;
        double r2351860 = r2351858 + r2351859;
        double r2351861 = sqrt(r2351860);
        double r2351862 = r2351857 * r2351861;
        double r2351863 = r2351862 / r2351858;
        double r2351864 = b;
        double r2351865 = c;
        double r2351866 = r2351864 - r2351865;
        double r2351867 = 5.0;
        double r2351868 = 6.0;
        double r2351869 = r2351867 / r2351868;
        double r2351870 = r2351859 + r2351869;
        double r2351871 = 3.0;
        double r2351872 = r2351858 * r2351871;
        double r2351873 = r2351856 / r2351872;
        double r2351874 = r2351870 - r2351873;
        double r2351875 = r2351866 * r2351874;
        double r2351876 = r2351863 - r2351875;
        double r2351877 = r2351856 * r2351876;
        double r2351878 = exp(r2351877);
        double r2351879 = r2351855 * r2351878;
        double r2351880 = r2351854 + r2351879;
        double r2351881 = r2351854 / r2351880;
        return r2351881;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2351882 = x;
        double r2351883 = y;
        double r2351884 = 2.0;
        double r2351885 = c;
        double r2351886 = b;
        double r2351887 = r2351885 - r2351886;
        double r2351888 = 5.0;
        double r2351889 = 6.0;
        double r2351890 = r2351888 / r2351889;
        double r2351891 = t;
        double r2351892 = r2351884 / r2351891;
        double r2351893 = 3.0;
        double r2351894 = r2351892 / r2351893;
        double r2351895 = a;
        double r2351896 = r2351894 - r2351895;
        double r2351897 = r2351890 - r2351896;
        double r2351898 = r2351895 + r2351891;
        double r2351899 = sqrt(r2351898);
        double r2351900 = cbrt(r2351891);
        double r2351901 = r2351900 * r2351900;
        double r2351902 = r2351899 / r2351901;
        double r2351903 = z;
        double r2351904 = r2351900 / r2351903;
        double r2351905 = r2351902 / r2351904;
        double r2351906 = fma(r2351887, r2351897, r2351905);
        double r2351907 = r2351884 * r2351906;
        double r2351908 = exp(r2351907);
        double r2351909 = fma(r2351883, r2351908, r2351882);
        double r2351910 = r2351882 / r2351909;
        return r2351910;
}

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 4.0

    \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

  2. Simplified1.9

    \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{\frac{t}{z}}\right)}, x\right)}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity1.9

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{\frac{t}{\color{blue}{1 \cdot z}}}\right)}, x\right)}\]

  5. Applied add-cube-cbrt1.9

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{\frac{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}{1 \cdot z}}\right)}, x\right)}\]

  6. Applied times-frac1.9

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{\color{blue}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{1} \cdot \frac{\sqrt[3]{t}}{z}}}\right)}, x\right)}\]

  7. Applied associate-/r*1.6

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \color{blue}{\frac{\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{1}}}{\frac{\sqrt[3]{t}}{z}}}\right)}, x\right)}\]

  8. Simplified1.6

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\color{blue}{\frac{\sqrt{t + a}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}}{\frac{\sqrt[3]{t}}{z}}\right)}, x\right)}\]

  9. Final simplification1.6

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\frac{\sqrt{a + t}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{\sqrt[3]{t}}{z}}\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019168 +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)))))))))))