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

↓

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

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

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r19596505 = x;
        double r19596506 = y;
        double r19596507 = 2.0;
        double r19596508 = z;
        double r19596509 = t;
        double r19596510 = a;
        double r19596511 = r19596509 + r19596510;
        double r19596512 = sqrt(r19596511);
        double r19596513 = r19596508 * r19596512;
        double r19596514 = r19596513 / r19596509;
        double r19596515 = b;
        double r19596516 = c;
        double r19596517 = r19596515 - r19596516;
        double r19596518 = 5.0;
        double r19596519 = 6.0;
        double r19596520 = r19596518 / r19596519;
        double r19596521 = r19596510 + r19596520;
        double r19596522 = 3.0;
        double r19596523 = r19596509 * r19596522;
        double r19596524 = r19596507 / r19596523;
        double r19596525 = r19596521 - r19596524;
        double r19596526 = r19596517 * r19596525;
        double r19596527 = r19596514 - r19596526;
        double r19596528 = r19596507 * r19596527;
        double r19596529 = exp(r19596528);
        double r19596530 = r19596506 * r19596529;
        double r19596531 = r19596505 + r19596530;
        double r19596532 = r19596505 / r19596531;
        return r19596532;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r19596533 = x;
        double r19596534 = y;
        double r19596535 = 2.0;
        double r19596536 = c;
        double r19596537 = b;
        double r19596538 = r19596536 - r19596537;
        double r19596539 = 5.0;
        double r19596540 = 6.0;
        double r19596541 = r19596539 / r19596540;
        double r19596542 = t;
        double r19596543 = r19596535 / r19596542;
        double r19596544 = 3.0;
        double r19596545 = r19596543 / r19596544;
        double r19596546 = a;
        double r19596547 = r19596545 - r19596546;
        double r19596548 = r19596541 - r19596547;
        double r19596549 = r19596546 + r19596542;
        double r19596550 = sqrt(r19596549);
        double r19596551 = z;
        double r19596552 = cbrt(r19596551);
        double r19596553 = r19596542 / r19596552;
        double r19596554 = r19596550 / r19596553;
        double r19596555 = r19596552 * r19596552;
        double r19596556 = r19596554 * r19596555;
        double r19596557 = fma(r19596538, r19596548, r19596556);
        double r19596558 = r19596535 * r19596557;
        double r19596559 = exp(r19596558);
        double r19596560 = fma(r19596534, r19596559, r19596533);
        double r19596561 = r19596533 / r19596560;
        return r19596561;
}

Target

Original	3.6
Target	2.6
Herbie	1.4

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

Derivation

Initial program 3.6
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Simplified1.5
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\frac{t}{z}}\right)}, x\right)}}\]
Using strategy rm
Applied add-cube-cbrt1.5
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\frac{t}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}}\right)}, x\right)}\]
Applied *-un-lft-identity1.5
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\frac{\color{blue}{1 \cdot t}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\right)}, x\right)}\]
Applied times-frac1.5
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\color{blue}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{t}{\sqrt[3]{z}}}}\right)}, x\right)}\]
Applied *-un-lft-identity1.5
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{\color{blue}{1 \cdot \left(a + t\right)}}}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{t}{\sqrt[3]{z}}}\right)}, x\right)}\]
Applied sqrt-prod1.5
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\color{blue}{\sqrt{1} \cdot \sqrt{a + t}}}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{t}{\sqrt[3]{z}}}\right)}, x\right)}\]
Applied times-frac1.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \color{blue}{\frac{\sqrt{1}}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{\sqrt{a + t}}{\frac{t}{\sqrt[3]{z}}}}\right)}, x\right)}\]
Simplified1.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)} \cdot \frac{\sqrt{a + t}}{\frac{t}{\sqrt[3]{z}}}\right)}, x\right)}\]
Final simplification1.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\frac{t}{\sqrt[3]{z}}} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019170 +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