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

↓

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

\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(z \cdot \sqrt{t + a}, \frac{1}{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 r382698 = x;
        double r382699 = y;
        double r382700 = 2.0;
        double r382701 = z;
        double r382702 = t;
        double r382703 = a;
        double r382704 = r382702 + r382703;
        double r382705 = sqrt(r382704);
        double r382706 = r382701 * r382705;
        double r382707 = r382706 / r382702;
        double r382708 = b;
        double r382709 = c;
        double r382710 = r382708 - r382709;
        double r382711 = 5.0;
        double r382712 = 6.0;
        double r382713 = r382711 / r382712;
        double r382714 = r382703 + r382713;
        double r382715 = 3.0;
        double r382716 = r382702 * r382715;
        double r382717 = r382700 / r382716;
        double r382718 = r382714 - r382717;
        double r382719 = r382710 * r382718;
        double r382720 = r382707 - r382719;
        double r382721 = r382700 * r382720;
        double r382722 = exp(r382721);
        double r382723 = r382699 * r382722;
        double r382724 = r382698 + r382723;
        double r382725 = r382698 / r382724;
        return r382725;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r382726 = x;
        double r382727 = y;
        double r382728 = 2.0;
        double r382729 = z;
        double r382730 = t;
        double r382731 = a;
        double r382732 = r382730 + r382731;
        double r382733 = sqrt(r382732);
        double r382734 = r382729 * r382733;
        double r382735 = 1.0;
        double r382736 = r382735 / r382730;
        double r382737 = b;
        double r382738 = c;
        double r382739 = r382737 - r382738;
        double r382740 = 5.0;
        double r382741 = 6.0;
        double r382742 = r382740 / r382741;
        double r382743 = r382731 + r382742;
        double r382744 = 3.0;
        double r382745 = r382730 * r382744;
        double r382746 = r382728 / r382745;
        double r382747 = r382743 - r382746;
        double r382748 = r382739 * r382747;
        double r382749 = -r382748;
        double r382750 = fma(r382734, r382736, r382749);
        double r382751 = r382728 * r382750;
        double r382752 = exp(r382751);
        double r382753 = r382727 * r382752;
        double r382754 = r382726 + r382753;
        double r382755 = r382726 / r382754;
        return r382755;
}

Target

Original	3.6
Target	3.2
Herbie	2.4

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

Derivation

Initial program 3.6
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Using strategy rm
Applied div-inv3.6
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(z \cdot \sqrt{t + a}\right) \cdot \frac{1}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Applied fma-neg2.4
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}}\]
Final simplification2.4
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

Reproduce

herbie shell --seed 2020036 +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