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

↓

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

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

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r16662795 = x;
        double r16662796 = y;
        double r16662797 = 2.0;
        double r16662798 = z;
        double r16662799 = t;
        double r16662800 = a;
        double r16662801 = r16662799 + r16662800;
        double r16662802 = sqrt(r16662801);
        double r16662803 = r16662798 * r16662802;
        double r16662804 = r16662803 / r16662799;
        double r16662805 = b;
        double r16662806 = c;
        double r16662807 = r16662805 - r16662806;
        double r16662808 = 5.0;
        double r16662809 = 6.0;
        double r16662810 = r16662808 / r16662809;
        double r16662811 = r16662800 + r16662810;
        double r16662812 = 3.0;
        double r16662813 = r16662799 * r16662812;
        double r16662814 = r16662797 / r16662813;
        double r16662815 = r16662811 - r16662814;
        double r16662816 = r16662807 * r16662815;
        double r16662817 = r16662804 - r16662816;
        double r16662818 = r16662797 * r16662817;
        double r16662819 = exp(r16662818);
        double r16662820 = r16662796 * r16662819;
        double r16662821 = r16662795 + r16662820;
        double r16662822 = r16662795 / r16662821;
        return r16662822;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r16662823 = x;
        double r16662824 = y;
        double r16662825 = 2.0;
        double r16662826 = c;
        double r16662827 = b;
        double r16662828 = r16662826 - r16662827;
        double r16662829 = 5.0;
        double r16662830 = 6.0;
        double r16662831 = r16662829 / r16662830;
        double r16662832 = a;
        double r16662833 = t;
        double r16662834 = r16662825 / r16662833;
        double r16662835 = 3.0;
        double r16662836 = r16662834 / r16662835;
        double r16662837 = r16662832 - r16662836;
        double r16662838 = r16662831 + r16662837;
        double r16662839 = r16662833 + r16662832;
        double r16662840 = sqrt(r16662839);
        double r16662841 = z;
        double r16662842 = r16662841 / r16662833;
        double r16662843 = r16662840 * r16662842;
        double r16662844 = fma(r16662828, r16662838, r16662843);
        double r16662845 = r16662825 * r16662844;
        double r16662846 = exp(r16662845);
        double r16662847 = fma(r16662824, r16662846, r16662823);
        double r16662848 = r16662823 / r16662847;
        return r16662848;
}

Target

Original	3.9
Target	2.9
Herbie	1.8

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

Derivation

Initial program 3.9
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Simplified1.8
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \sqrt{a + t} \cdot \frac{z}{t}\right)}, x\right)}}\]
Final simplification1.8
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \sqrt{t + a} \cdot \frac{z}{t}\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019192 +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