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

↓

\[\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

\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}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r325408 = x;
        double r325409 = y;
        double r325410 = 2.0;
        double r325411 = z;
        double r325412 = t;
        double r325413 = a;
        double r325414 = r325412 + r325413;
        double r325415 = sqrt(r325414);
        double r325416 = r325411 * r325415;
        double r325417 = r325416 / r325412;
        double r325418 = b;
        double r325419 = c;
        double r325420 = r325418 - r325419;
        double r325421 = 5.0;
        double r325422 = 6.0;
        double r325423 = r325421 / r325422;
        double r325424 = r325413 + r325423;
        double r325425 = 3.0;
        double r325426 = r325412 * r325425;
        double r325427 = r325410 / r325426;
        double r325428 = r325424 - r325427;
        double r325429 = r325420 * r325428;
        double r325430 = r325417 - r325429;
        double r325431 = r325410 * r325430;
        double r325432 = exp(r325431);
        double r325433 = r325409 * r325432;
        double r325434 = r325408 + r325433;
        double r325435 = r325408 / r325434;
        return r325435;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r325436 = x;
        double r325437 = y;
        double r325438 = 2.0;
        double r325439 = z;
        double r325440 = t;
        double r325441 = a;
        double r325442 = r325440 + r325441;
        double r325443 = sqrt(r325442);
        double r325444 = r325439 * r325443;
        double r325445 = 1.0;
        double r325446 = r325445 / r325440;
        double r325447 = b;
        double r325448 = c;
        double r325449 = r325447 - r325448;
        double r325450 = 5.0;
        double r325451 = 6.0;
        double r325452 = r325450 / r325451;
        double r325453 = r325441 + r325452;
        double r325454 = 3.0;
        double r325455 = r325440 * r325454;
        double r325456 = r325438 / r325455;
        double r325457 = r325453 - r325456;
        double r325458 = r325449 * r325457;
        double r325459 = -r325458;
        double r325460 = fma(r325444, r325446, r325459);
        double r325461 = r325438 * r325460;
        double r325462 = exp(r325461);
        double r325463 = r325437 * r325462;
        double r325464 = r325436 + r325463;
        double r325465 = r325436 / r325464;
        return r325465;
}

Target

Original	4.0
Target	3.1
Herbie	2.8

\[\begin{array}{l} \mathbf{if}\;t \lt -2.1183266448915811 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.19658877065154709 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

Initial program 4.0
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Using strategy rm
Applied div-inv4.0
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(z \cdot \sqrt{t + a}\right) \cdot \frac{1}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Applied fma-neg2.8
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}}\]
Final simplification2.8
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

Reproduce

herbie shell --seed 2020062 +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