Average Error: 4.0 → 1.4
Time: 21.6s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{\mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \left(\sqrt{t + a} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right) \cdot 2}, x\right)}\]
\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), \left(\sqrt{t + a} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right) \cdot 2}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r16579851 = x;
        double r16579852 = y;
        double r16579853 = 2.0;
        double r16579854 = z;
        double r16579855 = t;
        double r16579856 = a;
        double r16579857 = r16579855 + r16579856;
        double r16579858 = sqrt(r16579857);
        double r16579859 = r16579854 * r16579858;
        double r16579860 = r16579859 / r16579855;
        double r16579861 = b;
        double r16579862 = c;
        double r16579863 = r16579861 - r16579862;
        double r16579864 = 5.0;
        double r16579865 = 6.0;
        double r16579866 = r16579864 / r16579865;
        double r16579867 = r16579856 + r16579866;
        double r16579868 = 3.0;
        double r16579869 = r16579855 * r16579868;
        double r16579870 = r16579853 / r16579869;
        double r16579871 = r16579867 - r16579870;
        double r16579872 = r16579863 * r16579871;
        double r16579873 = r16579860 - r16579872;
        double r16579874 = r16579853 * r16579873;
        double r16579875 = exp(r16579874);
        double r16579876 = r16579852 * r16579875;
        double r16579877 = r16579851 + r16579876;
        double r16579878 = r16579851 / r16579877;
        return r16579878;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r16579879 = x;
        double r16579880 = y;
        double r16579881 = c;
        double r16579882 = b;
        double r16579883 = r16579881 - r16579882;
        double r16579884 = 5.0;
        double r16579885 = 6.0;
        double r16579886 = r16579884 / r16579885;
        double r16579887 = a;
        double r16579888 = 2.0;
        double r16579889 = t;
        double r16579890 = r16579888 / r16579889;
        double r16579891 = 3.0;
        double r16579892 = r16579890 / r16579891;
        double r16579893 = r16579887 - r16579892;
        double r16579894 = r16579886 + r16579893;
        double r16579895 = r16579889 + r16579887;
        double r16579896 = sqrt(r16579895);
        double r16579897 = z;
        double r16579898 = cbrt(r16579897);
        double r16579899 = cbrt(r16579889);
        double r16579900 = r16579898 / r16579899;
        double r16579901 = r16579900 * r16579900;
        double r16579902 = r16579896 * r16579901;
        double r16579903 = r16579902 * r16579900;
        double r16579904 = fma(r16579883, r16579894, r16579903);
        double r16579905 = r16579904 * r16579888;
        double r16579906 = exp(r16579905);
        double r16579907 = fma(r16579880, r16579906, r16579879);
        double r16579908 = r16579879 / r16579907;
        return r16579908;
}

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

Target

Original4.0
Target3.3
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581057561884576920117070548 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333333703407674875052180141211 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547088010424937268931048836 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

  1. Initial program 4.0

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

  2. Simplified1.8

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

  4. Applied add-cube-cbrt1.8

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

  5. Applied add-cube-cbrt1.8

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

  6. Applied times-frac1.8

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

  7. Applied associate-*r*1.4

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

  8. Simplified1.4

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

  9. Final simplification1.4

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2.0 (/ (- (* (* z (sqrt (+ t a))) (* (* 3.0 t) (- a (/ 5.0 6.0)))) (* (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0) (* (- a (/ 5.0 6.0)) (* (- b c) t)))) (* (* (* t t) 3.0) (- a (/ 5.0 6.0))))))))) (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))

  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))