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

↓

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

\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, z \cdot \frac{\sqrt{t + a}}{t}\right)\right)}, x\right)}

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r208380 = x;
        double r208381 = y;
        double r208382 = 2.0;
        double r208383 = z;
        double r208384 = t;
        double r208385 = a;
        double r208386 = r208384 + r208385;
        double r208387 = sqrt(r208386);
        double r208388 = r208383 * r208387;
        double r208389 = r208388 / r208384;
        double r208390 = b;
        double r208391 = c;
        double r208392 = r208390 - r208391;
        double r208393 = 5.0;
        double r208394 = 6.0;
        double r208395 = r208393 / r208394;
        double r208396 = r208385 + r208395;
        double r208397 = 3.0;
        double r208398 = r208384 * r208397;
        double r208399 = r208382 / r208398;
        double r208400 = r208396 - r208399;
        double r208401 = r208392 * r208400;
        double r208402 = r208389 - r208401;
        double r208403 = r208382 * r208402;
        double r208404 = exp(r208403);
        double r208405 = r208381 * r208404;
        double r208406 = r208380 + r208405;
        double r208407 = r208380 / r208406;
        return r208407;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r208408 = x;
        double r208409 = y;
        double r208410 = 2.0;
        double r208411 = exp(r208410);
        double r208412 = t;
        double r208413 = r208410 / r208412;
        double r208414 = 3.0;
        double r208415 = r208413 / r208414;
        double r208416 = a;
        double r208417 = 5.0;
        double r208418 = 6.0;
        double r208419 = r208417 / r208418;
        double r208420 = r208416 + r208419;
        double r208421 = r208415 - r208420;
        double r208422 = b;
        double r208423 = c;
        double r208424 = r208422 - r208423;
        double r208425 = z;
        double r208426 = r208412 + r208416;
        double r208427 = sqrt(r208426);
        double r208428 = r208427 / r208412;
        double r208429 = r208425 * r208428;
        double r208430 = fma(r208421, r208424, r208429);
        double r208431 = pow(r208411, r208430);
        double r208432 = fma(r208409, r208431, r208408);
        double r208433 = r208408 / r208432;
        return r208433;
}

Target

Original	4.2
Target	3.1
Herbie	2.1

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

Derivation

Initial program 4.2
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Simplified2.7
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}}\]
Using strategy rm
Applied *-un-lft-identity2.7
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{\color{blue}{1 \cdot t}}\right)\right)}, x\right)}\]
Applied times-frac2.1
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \color{blue}{\frac{z}{1} \cdot \frac{\sqrt{t + a}}{t}}\right)\right)}, x\right)}\]
Simplified2.1
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \color{blue}{z} \cdot \frac{\sqrt{t + a}}{t}\right)\right)}, x\right)}\]
Final simplification2.1
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, z \cdot \frac{\sqrt{t + a}}{t}\right)\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019326 +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 -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-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