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

↓

\[\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)}, x\right)}\]

\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, \left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)}, x\right)}

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r6895534 = x;
        double r6895535 = y;
        double r6895536 = 2.0;
        double r6895537 = z;
        double r6895538 = t;
        double r6895539 = a;
        double r6895540 = r6895538 + r6895539;
        double r6895541 = sqrt(r6895540);
        double r6895542 = r6895537 * r6895541;
        double r6895543 = r6895542 / r6895538;
        double r6895544 = b;
        double r6895545 = c;
        double r6895546 = r6895544 - r6895545;
        double r6895547 = 5.0;
        double r6895548 = 6.0;
        double r6895549 = r6895547 / r6895548;
        double r6895550 = r6895539 + r6895549;
        double r6895551 = 3.0;
        double r6895552 = r6895538 * r6895551;
        double r6895553 = r6895536 / r6895552;
        double r6895554 = r6895550 - r6895553;
        double r6895555 = r6895546 * r6895554;
        double r6895556 = r6895543 - r6895555;
        double r6895557 = r6895536 * r6895556;
        double r6895558 = exp(r6895557);
        double r6895559 = r6895535 * r6895558;
        double r6895560 = r6895534 + r6895559;
        double r6895561 = r6895534 / r6895560;
        return r6895561;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r6895562 = x;
        double r6895563 = y;
        double r6895564 = 2.0;
        double r6895565 = c;
        double r6895566 = b;
        double r6895567 = r6895565 - r6895566;
        double r6895568 = 5.0;
        double r6895569 = 6.0;
        double r6895570 = r6895568 / r6895569;
        double r6895571 = a;
        double r6895572 = r6895570 + r6895571;
        double r6895573 = t;
        double r6895574 = 3.0;
        double r6895575 = r6895573 * r6895574;
        double r6895576 = r6895564 / r6895575;
        double r6895577 = r6895572 - r6895576;
        double r6895578 = z;
        double r6895579 = cbrt(r6895573);
        double r6895580 = r6895579 * r6895579;
        double r6895581 = r6895578 / r6895580;
        double r6895582 = r6895573 + r6895571;
        double r6895583 = sqrt(r6895582);
        double r6895584 = r6895583 / r6895579;
        double r6895585 = r6895581 * r6895584;
        double r6895586 = fma(r6895567, r6895577, r6895585);
        double r6895587 = r6895564 * r6895586;
        double r6895588 = exp(r6895587);
        double r6895589 = fma(r6895563, r6895588, r6895562);
        double r6895590 = r6895562 / r6895589;
        return r6895590;
}

Target

Original	3.4
Target	2.8
Herbie	1.4

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

Derivation

Initial program 3.4
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
Simplified2.4
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}, \frac{z \cdot \sqrt{a + t}}{t}\right)}, x\right)}}\]
Using strategy rm
Applied add-cube-cbrt2.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}, \frac{z \cdot \sqrt{a + t}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\right)}, x\right)}\]
Applied times-frac1.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}, \color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{a + t}}{\sqrt[3]{t}}}\right)}, x\right)}\]
Final simplification1.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019156 +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