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

↓

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

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

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r315201 = x;
        double r315202 = y;
        double r315203 = 2.0;
        double r315204 = z;
        double r315205 = t;
        double r315206 = a;
        double r315207 = r315205 + r315206;
        double r315208 = sqrt(r315207);
        double r315209 = r315204 * r315208;
        double r315210 = r315209 / r315205;
        double r315211 = b;
        double r315212 = c;
        double r315213 = r315211 - r315212;
        double r315214 = 5.0;
        double r315215 = 6.0;
        double r315216 = r315214 / r315215;
        double r315217 = r315206 + r315216;
        double r315218 = 3.0;
        double r315219 = r315205 * r315218;
        double r315220 = r315203 / r315219;
        double r315221 = r315217 - r315220;
        double r315222 = r315213 * r315221;
        double r315223 = r315210 - r315222;
        double r315224 = r315203 * r315223;
        double r315225 = exp(r315224);
        double r315226 = r315202 * r315225;
        double r315227 = r315201 + r315226;
        double r315228 = r315201 / r315227;
        return r315228;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r315229 = x;
        double r315230 = y;
        double r315231 = 2.0;
        double r315232 = exp(r315231);
        double r315233 = t;
        double r315234 = r315231 / r315233;
        double r315235 = 3.0;
        double r315236 = r315234 / r315235;
        double r315237 = a;
        double r315238 = 5.0;
        double r315239 = 6.0;
        double r315240 = r315238 / r315239;
        double r315241 = r315237 + r315240;
        double r315242 = r315236 - r315241;
        double r315243 = b;
        double r315244 = c;
        double r315245 = r315243 - r315244;
        double r315246 = z;
        double r315247 = r315233 + r315237;
        double r315248 = sqrt(r315247);
        double r315249 = r315233 / r315248;
        double r315250 = r315246 / r315249;
        double r315251 = fma(r315242, r315245, r315250);
        double r315252 = pow(r315232, r315251);
        double r315253 = fma(r315230, r315252, r315229);
        double r315254 = r315229 / r315253;
        return r315254;
}

Target

Original	3.8
Target	3.2
Herbie	1.9

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

Derivation

Initial program 3.8
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Simplified2.6
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}}\]
Using strategy rm
Applied associate-/l*1.9
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \color{blue}{\frac{z}{\frac{t}{\sqrt{t + a}}}}\right)\right)}, x\right)}\]
Final simplification1.9
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\frac{t}{\sqrt{t + a}}}\right)\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019305 +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