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

↓

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

\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 r425164 = x;
        double r425165 = y;
        double r425166 = 2.0;
        double r425167 = z;
        double r425168 = t;
        double r425169 = a;
        double r425170 = r425168 + r425169;
        double r425171 = sqrt(r425170);
        double r425172 = r425167 * r425171;
        double r425173 = r425172 / r425168;
        double r425174 = b;
        double r425175 = c;
        double r425176 = r425174 - r425175;
        double r425177 = 5.0;
        double r425178 = 6.0;
        double r425179 = r425177 / r425178;
        double r425180 = r425169 + r425179;
        double r425181 = 3.0;
        double r425182 = r425168 * r425181;
        double r425183 = r425166 / r425182;
        double r425184 = r425180 - r425183;
        double r425185 = r425176 * r425184;
        double r425186 = r425173 - r425185;
        double r425187 = r425166 * r425186;
        double r425188 = exp(r425187);
        double r425189 = r425165 * r425188;
        double r425190 = r425164 + r425189;
        double r425191 = r425164 / r425190;
        return r425191;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r425192 = x;
        double r425193 = y;
        double r425194 = 2.0;
        double r425195 = z;
        double r425196 = t;
        double r425197 = a;
        double r425198 = r425196 + r425197;
        double r425199 = sqrt(r425198);
        double r425200 = r425195 * r425199;
        double r425201 = 1.0;
        double r425202 = r425201 / r425196;
        double r425203 = b;
        double r425204 = c;
        double r425205 = r425203 - r425204;
        double r425206 = 5.0;
        double r425207 = 6.0;
        double r425208 = r425206 / r425207;
        double r425209 = r425197 + r425208;
        double r425210 = 3.0;
        double r425211 = r425196 * r425210;
        double r425212 = r425194 / r425211;
        double r425213 = r425209 - r425212;
        double r425214 = r425205 * r425213;
        double r425215 = -r425214;
        double r425216 = fma(r425200, r425202, r425215);
        double r425217 = r425194 * r425216;
        double r425218 = exp(r425217);
        double r425219 = r425193 * r425218;
        double r425220 = r425192 + r425219;
        double r425221 = r425192 / r425220;
        return r425221;
}

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