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

↓

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

\frac{x}{x + y \cdot e^{2 \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(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, c - b, \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 r25054276 = x;
        double r25054277 = y;
        double r25054278 = 2.0;
        double r25054279 = z;
        double r25054280 = t;
        double r25054281 = a;
        double r25054282 = r25054280 + r25054281;
        double r25054283 = sqrt(r25054282);
        double r25054284 = r25054279 * r25054283;
        double r25054285 = r25054284 / r25054280;
        double r25054286 = b;
        double r25054287 = c;
        double r25054288 = r25054286 - r25054287;
        double r25054289 = 5.0;
        double r25054290 = 6.0;
        double r25054291 = r25054289 / r25054290;
        double r25054292 = r25054281 + r25054291;
        double r25054293 = 3.0;
        double r25054294 = r25054280 * r25054293;
        double r25054295 = r25054278 / r25054294;
        double r25054296 = r25054292 - r25054295;
        double r25054297 = r25054288 * r25054296;
        double r25054298 = r25054285 - r25054297;
        double r25054299 = r25054278 * r25054298;
        double r25054300 = exp(r25054299);
        double r25054301 = r25054277 * r25054300;
        double r25054302 = r25054276 + r25054301;
        double r25054303 = r25054276 / r25054302;
        return r25054303;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r25054304 = x;
        double r25054305 = y;
        double r25054306 = 2.0;
        double r25054307 = a;
        double r25054308 = 5.0;
        double r25054309 = 6.0;
        double r25054310 = r25054308 / r25054309;
        double r25054311 = r25054307 + r25054310;
        double r25054312 = t;
        double r25054313 = 3.0;
        double r25054314 = r25054312 * r25054313;
        double r25054315 = r25054306 / r25054314;
        double r25054316 = r25054311 - r25054315;
        double r25054317 = c;
        double r25054318 = b;
        double r25054319 = r25054317 - r25054318;
        double r25054320 = z;
        double r25054321 = cbrt(r25054312);
        double r25054322 = r25054321 * r25054321;
        double r25054323 = r25054320 / r25054322;
        double r25054324 = r25054312 + r25054307;
        double r25054325 = sqrt(r25054324);
        double r25054326 = r25054325 / r25054321;
        double r25054327 = r25054323 * r25054326;
        double r25054328 = fma(r25054316, r25054319, r25054327);
        double r25054329 = r25054306 * r25054328;
        double r25054330 = exp(r25054329);
        double r25054331 = fma(r25054305, r25054330, r25054304);
        double r25054332 = r25054304 / r25054331;
        return r25054332;
}

Target

Original	3.9
Target	2.9
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 3.9
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Simplified2.6
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, c - b, \frac{z \cdot \sqrt{t + a}}{t}\right)}, x\right)}}\]
Using strategy rm
Applied add-cube-cbrt2.6
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, c - b, \frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\right)}, x\right)}\]
Applied times-frac1.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, c - b, \color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}}\right)}, x\right)}\]
Final simplification1.4
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, c - b, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019174 +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