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

↓

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

\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, \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)}, x\right)}

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r46435809 = x;
        double r46435810 = y;
        double r46435811 = 2.0;
        double r46435812 = z;
        double r46435813 = t;
        double r46435814 = a;
        double r46435815 = r46435813 + r46435814;
        double r46435816 = sqrt(r46435815);
        double r46435817 = r46435812 * r46435816;
        double r46435818 = r46435817 / r46435813;
        double r46435819 = b;
        double r46435820 = c;
        double r46435821 = r46435819 - r46435820;
        double r46435822 = 5.0;
        double r46435823 = 6.0;
        double r46435824 = r46435822 / r46435823;
        double r46435825 = r46435814 + r46435824;
        double r46435826 = 3.0;
        double r46435827 = r46435813 * r46435826;
        double r46435828 = r46435811 / r46435827;
        double r46435829 = r46435825 - r46435828;
        double r46435830 = r46435821 * r46435829;
        double r46435831 = r46435818 - r46435830;
        double r46435832 = r46435811 * r46435831;
        double r46435833 = exp(r46435832);
        double r46435834 = r46435810 * r46435833;
        double r46435835 = r46435809 + r46435834;
        double r46435836 = r46435809 / r46435835;
        return r46435836;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r46435837 = x;
        double r46435838 = y;
        double r46435839 = 2.0;
        double r46435840 = c;
        double r46435841 = b;
        double r46435842 = r46435840 - r46435841;
        double r46435843 = a;
        double r46435844 = 5.0;
        double r46435845 = 6.0;
        double r46435846 = r46435844 / r46435845;
        double r46435847 = r46435843 + r46435846;
        double r46435848 = t;
        double r46435849 = 3.0;
        double r46435850 = r46435848 * r46435849;
        double r46435851 = r46435839 / r46435850;
        double r46435852 = r46435847 - r46435851;
        double r46435853 = z;
        double r46435854 = cbrt(r46435848);
        double r46435855 = r46435854 * r46435854;
        double r46435856 = r46435853 / r46435855;
        double r46435857 = r46435848 + r46435843;
        double r46435858 = sqrt(r46435857);
        double r46435859 = r46435858 / r46435854;
        double r46435860 = r46435856 * r46435859;
        double r46435861 = fma(r46435842, r46435852, r46435860);
        double r46435862 = r46435839 * r46435861;
        double r46435863 = exp(r46435862);
        double r46435864 = fma(r46435838, r46435863, r46435837);
        double r46435865 = r46435837 / r46435864;
        return r46435865;
}

Target

Original	3.9
Target	3.1
Herbie	1.5

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

Derivation

Initial program 3.9
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Simplified2.5
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z \cdot \sqrt{t + a}}{t}\right)}, x\right)}}\]
Using strategy rm
Applied add-cube-cbrt2.5
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\right)}, x\right)}\]
Applied times-frac1.5
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}}\right)}, x\right)}\]
Final simplification1.5
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019173 +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)))))))))))

Error

Target

Derivation

Reproduce