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

↓

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

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

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r347282 = x;
        double r347283 = y;
        double r347284 = 2.0;
        double r347285 = z;
        double r347286 = t;
        double r347287 = a;
        double r347288 = r347286 + r347287;
        double r347289 = sqrt(r347288);
        double r347290 = r347285 * r347289;
        double r347291 = r347290 / r347286;
        double r347292 = b;
        double r347293 = c;
        double r347294 = r347292 - r347293;
        double r347295 = 5.0;
        double r347296 = 6.0;
        double r347297 = r347295 / r347296;
        double r347298 = r347287 + r347297;
        double r347299 = 3.0;
        double r347300 = r347286 * r347299;
        double r347301 = r347284 / r347300;
        double r347302 = r347298 - r347301;
        double r347303 = r347294 * r347302;
        double r347304 = r347291 - r347303;
        double r347305 = r347284 * r347304;
        double r347306 = exp(r347305);
        double r347307 = r347283 * r347306;
        double r347308 = r347282 + r347307;
        double r347309 = r347282 / r347308;
        return r347309;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r347310 = x;
        double r347311 = y;
        double r347312 = 2.0;
        double r347313 = exp(r347312);
        double r347314 = t;
        double r347315 = r347312 / r347314;
        double r347316 = 3.0;
        double r347317 = r347315 / r347316;
        double r347318 = a;
        double r347319 = 5.0;
        double r347320 = 6.0;
        double r347321 = r347319 / r347320;
        double r347322 = r347318 + r347321;
        double r347323 = r347317 - r347322;
        double r347324 = b;
        double r347325 = c;
        double r347326 = r347324 - r347325;
        double r347327 = z;
        double r347328 = cbrt(r347314);
        double r347329 = r347328 * r347328;
        double r347330 = r347327 / r347329;
        double r347331 = r347314 + r347318;
        double r347332 = sqrt(r347331);
        double r347333 = r347332 / r347328;
        double r347334 = r347330 * r347333;
        double r347335 = fma(r347323, r347326, r347334);
        double r347336 = pow(r347313, r347335);
        double r347337 = fma(r347311, r347336, r347310);
        double r347338 = r347310 / r347337;
        return r347338;
}

Target

Original	3.7
Target	2.8
Herbie	1.3

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

Derivation

Initial program 3.7
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Simplified2.4
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}}\]
Using strategy rm
Applied add-cube-cbrt2.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\right)\right)}, x\right)}\]
Applied times-frac1.3
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}}\right)\right)}, x\right)}\]
Final simplification1.3
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019303 +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 -2.1183266448915811e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.83333333333333337 c)) (* a b))))))) (if (< t 5.19658877065154709e-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