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

↓

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

\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 \left(\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r354746 = x;
        double r354747 = y;
        double r354748 = 2.0;
        double r354749 = z;
        double r354750 = t;
        double r354751 = a;
        double r354752 = r354750 + r354751;
        double r354753 = sqrt(r354752);
        double r354754 = r354749 * r354753;
        double r354755 = r354754 / r354750;
        double r354756 = b;
        double r354757 = c;
        double r354758 = r354756 - r354757;
        double r354759 = 5.0;
        double r354760 = 6.0;
        double r354761 = r354759 / r354760;
        double r354762 = r354751 + r354761;
        double r354763 = 3.0;
        double r354764 = r354750 * r354763;
        double r354765 = r354748 / r354764;
        double r354766 = r354762 - r354765;
        double r354767 = r354758 * r354766;
        double r354768 = r354755 - r354767;
        double r354769 = r354748 * r354768;
        double r354770 = exp(r354769);
        double r354771 = r354747 * r354770;
        double r354772 = r354746 + r354771;
        double r354773 = r354746 / r354772;
        return r354773;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r354774 = x;
        double r354775 = y;
        double r354776 = 2.0;
        double r354777 = z;
        double r354778 = t;
        double r354779 = a;
        double r354780 = r354778 + r354779;
        double r354781 = sqrt(r354780);
        double r354782 = r354777 * r354781;
        double r354783 = 1.0;
        double r354784 = r354783 / r354778;
        double r354785 = 5.0;
        double r354786 = 6.0;
        double r354787 = r354785 / r354786;
        double r354788 = r354779 + r354787;
        double r354789 = 3.0;
        double r354790 = r354778 * r354789;
        double r354791 = r354776 / r354790;
        double r354792 = r354788 - r354791;
        double r354793 = b;
        double r354794 = c;
        double r354795 = r354793 - r354794;
        double r354796 = r354792 * r354795;
        double r354797 = -r354796;
        double r354798 = fma(r354782, r354784, r354797);
        double r354799 = -r354795;
        double r354800 = r354799 + r354795;
        double r354801 = r354792 * r354800;
        double r354802 = r354798 + r354801;
        double r354803 = r354776 * r354802;
        double r354804 = exp(r354803);
        double r354805 = r354775 * r354804;
        double r354806 = r354774 + r354805;
        double r354807 = r354774 / r354806;
        return r354807;
}

Target

Original	4.0
Target	3.1
Herbie	2.6

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

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 div-inv4.0
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(z \cdot \sqrt{t + a}\right) \cdot \frac{1}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Applied prod-diff22.3
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \mathsf{fma}\left(-\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right), b - c, \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right)\right)}}}\]
Simplified2.6
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \color{blue}{\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)}\right)}}\]
Final simplification2.6
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\]

Reproduce

herbie shell --seed 2020024 +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