\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

↓

\[\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}{\frac{t}{\sqrt[3]{z}}}\right)}, x\right)}\]

\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}

↓

\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}{\frac{t}{\sqrt[3]{z}}}\right)}, x\right)}

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r17506225 = x;
        double r17506226 = y;
        double r17506227 = 2.0;
        double r17506228 = z;
        double r17506229 = t;
        double r17506230 = a;
        double r17506231 = r17506229 + r17506230;
        double r17506232 = sqrt(r17506231);
        double r17506233 = r17506228 * r17506232;
        double r17506234 = r17506233 / r17506229;
        double r17506235 = b;
        double r17506236 = c;
        double r17506237 = r17506235 - r17506236;
        double r17506238 = 5.0;
        double r17506239 = 6.0;
        double r17506240 = r17506238 / r17506239;
        double r17506241 = r17506230 + r17506240;
        double r17506242 = 3.0;
        double r17506243 = r17506229 * r17506242;
        double r17506244 = r17506227 / r17506243;
        double r17506245 = r17506241 - r17506244;
        double r17506246 = r17506237 * r17506245;
        double r17506247 = r17506234 - r17506246;
        double r17506248 = r17506227 * r17506247;
        double r17506249 = exp(r17506248);
        double r17506250 = r17506226 * r17506249;
        double r17506251 = r17506225 + r17506250;
        double r17506252 = r17506225 / r17506251;
        return r17506252;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r17506253 = x;
        double r17506254 = y;
        double r17506255 = 2.0;
        double r17506256 = c;
        double r17506257 = b;
        double r17506258 = r17506256 - r17506257;
        double r17506259 = 5.0;
        double r17506260 = 6.0;
        double r17506261 = r17506259 / r17506260;
        double r17506262 = t;
        double r17506263 = r17506255 / r17506262;
        double r17506264 = 3.0;
        double r17506265 = r17506263 / r17506264;
        double r17506266 = a;
        double r17506267 = r17506265 - r17506266;
        double r17506268 = r17506261 - r17506267;
        double r17506269 = r17506266 + r17506262;
        double r17506270 = sqrt(r17506269);
        double r17506271 = z;
        double r17506272 = cbrt(r17506271);
        double r17506273 = r17506272 * r17506272;
        double r17506274 = r17506270 * r17506273;
        double r17506275 = r17506262 / r17506272;
        double r17506276 = r17506274 / r17506275;
        double r17506277 = fma(r17506258, r17506268, r17506276);
        double r17506278 = r17506255 * r17506277;
        double r17506279 = exp(r17506278);
        double r17506280 = fma(r17506254, r17506279, r17506253);
        double r17506281 = r17506253 / r17506280;
        return r17506281;
}

Target

Original	4.0
Target	3.2
Herbie	1.9

\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) - \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(3.0 \cdot t\right) - 2.0\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array}\]

Derivation

Initial program 4.0
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
Simplified1.9
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{\frac{t}{z}}\right)}, x\right)}}\]
Using strategy rm
Applied add-cube-cbrt1.9
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{\frac{t}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}}\right)}, x\right)}\]
Applied *-un-lft-identity1.9
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{\frac{\color{blue}{1 \cdot t}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\right)}, x\right)}\]
Applied times-frac1.9
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{\color{blue}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{t}{\sqrt[3]{z}}}}\right)}, x\right)}\]
Applied associate-/r*1.9
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \color{blue}{\frac{\frac{\sqrt{a + t}}{\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}}{\frac{t}{\sqrt[3]{z}}}}\right)}, x\right)}\]
Simplified1.9
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt{t + a}}}{\frac{t}{\sqrt[3]{z}}}\right)}, x\right)}\]
Final simplification1.9
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}{\frac{t}{\sqrt[3]{z}}}\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019168 +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