\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

↓

\[\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{t} \cdot z\right)}, x\right)}\]

\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}

↓

\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{t} \cdot z\right)}, x\right)}

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r15108682 = x;
        double r15108683 = y;
        double r15108684 = 2.0;
        double r15108685 = z;
        double r15108686 = t;
        double r15108687 = a;
        double r15108688 = r15108686 + r15108687;
        double r15108689 = sqrt(r15108688);
        double r15108690 = r15108685 * r15108689;
        double r15108691 = r15108690 / r15108686;
        double r15108692 = b;
        double r15108693 = c;
        double r15108694 = r15108692 - r15108693;
        double r15108695 = 5.0;
        double r15108696 = 6.0;
        double r15108697 = r15108695 / r15108696;
        double r15108698 = r15108687 + r15108697;
        double r15108699 = 3.0;
        double r15108700 = r15108686 * r15108699;
        double r15108701 = r15108684 / r15108700;
        double r15108702 = r15108698 - r15108701;
        double r15108703 = r15108694 * r15108702;
        double r15108704 = r15108691 - r15108703;
        double r15108705 = r15108684 * r15108704;
        double r15108706 = exp(r15108705);
        double r15108707 = r15108683 * r15108706;
        double r15108708 = r15108682 + r15108707;
        double r15108709 = r15108682 / r15108708;
        return r15108709;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r15108710 = x;
        double r15108711 = y;
        double r15108712 = 2.0;
        double r15108713 = c;
        double r15108714 = b;
        double r15108715 = r15108713 - r15108714;
        double r15108716 = 5.0;
        double r15108717 = 6.0;
        double r15108718 = r15108716 / r15108717;
        double r15108719 = t;
        double r15108720 = r15108712 / r15108719;
        double r15108721 = 3.0;
        double r15108722 = r15108720 / r15108721;
        double r15108723 = a;
        double r15108724 = r15108722 - r15108723;
        double r15108725 = r15108718 - r15108724;
        double r15108726 = r15108723 + r15108719;
        double r15108727 = sqrt(r15108726);
        double r15108728 = r15108727 / r15108719;
        double r15108729 = z;
        double r15108730 = r15108728 * r15108729;
        double r15108731 = fma(r15108715, r15108725, r15108730);
        double r15108732 = r15108712 * r15108731;
        double r15108733 = exp(r15108732);
        double r15108734 = fma(r15108711, r15108733, r15108710);
        double r15108735 = r15108710 / r15108734;
        return r15108735;
}

Target

Original	4.0
Target	3.2
Herbie	2.4

\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) - \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(3.0 \cdot t\right) - 2.0\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array}\]

Derivation

Initial program 4.0
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
Simplified2.0
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{\frac{t}{z}}\right)}, x\right)}}\]
Using strategy rm
Applied associate-/r/2.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \color{blue}{\frac{\sqrt{a + t}}{t} \cdot z}\right)}, x\right)}\]
Final simplification2.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{t} \cdot z\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019164 +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