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

↓

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

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

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r369616 = x;
        double r369617 = y;
        double r369618 = 2.0;
        double r369619 = z;
        double r369620 = t;
        double r369621 = a;
        double r369622 = r369620 + r369621;
        double r369623 = sqrt(r369622);
        double r369624 = r369619 * r369623;
        double r369625 = r369624 / r369620;
        double r369626 = b;
        double r369627 = c;
        double r369628 = r369626 - r369627;
        double r369629 = 5.0;
        double r369630 = 6.0;
        double r369631 = r369629 / r369630;
        double r369632 = r369621 + r369631;
        double r369633 = 3.0;
        double r369634 = r369620 * r369633;
        double r369635 = r369618 / r369634;
        double r369636 = r369632 - r369635;
        double r369637 = r369628 * r369636;
        double r369638 = r369625 - r369637;
        double r369639 = r369618 * r369638;
        double r369640 = exp(r369639);
        double r369641 = r369617 * r369640;
        double r369642 = r369616 + r369641;
        double r369643 = r369616 / r369642;
        return r369643;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r369644 = x;
        double r369645 = y;
        double r369646 = 2.0;
        double r369647 = z;
        double r369648 = t;
        double r369649 = cbrt(r369648);
        double r369650 = r369649 * r369649;
        double r369651 = r369647 / r369650;
        double r369652 = a;
        double r369653 = r369648 + r369652;
        double r369654 = sqrt(r369653);
        double r369655 = r369654 / r369649;
        double r369656 = b;
        double r369657 = c;
        double r369658 = r369656 - r369657;
        double r369659 = 5.0;
        double r369660 = 6.0;
        double r369661 = r369659 / r369660;
        double r369662 = r369652 + r369661;
        double r369663 = 3.0;
        double r369664 = r369648 * r369663;
        double r369665 = r369646 / r369664;
        double r369666 = r369662 - r369665;
        double r369667 = r369658 * r369666;
        double r369668 = -r369667;
        double r369669 = fma(r369651, r369655, r369668);
        double r369670 = r369646 * r369669;
        double r369671 = exp(r369670);
        double r369672 = r369645 * r369671;
        double r369673 = r369644 + r369672;
        double r369674 = r369644 / r369673;
        return r369674;
}

Target

Original	4.0
Target	3.1
Herbie	1.7

\[\begin{array}{l} \mathbf{if}\;t \lt -2.1183266448915811 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.19658877065154709 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

Initial program 4.0
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Using strategy rm
Applied add-cube-cbrt4.0
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Applied times-frac2.7
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Applied fma-neg1.7
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}}\]
Final simplification1.7
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2 (/ (- (* (* z (sqrt (+ t a))) (* (* 3 t) (- a (/ 5 6)))) (* (- (* (+ (/ 5 6) a) (* 3 t)) 2) (* (- a (/ 5 6)) (* (- b c) t)))) (* (* (* t t) 3) (- a (/ 5 6))))))))) (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3))))))))))))

  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))

Error

Target

Derivation

Reproduce