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

↓

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

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

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r20862127 = x;
        double r20862128 = y;
        double r20862129 = 2.0;
        double r20862130 = z;
        double r20862131 = t;
        double r20862132 = a;
        double r20862133 = r20862131 + r20862132;
        double r20862134 = sqrt(r20862133);
        double r20862135 = r20862130 * r20862134;
        double r20862136 = r20862135 / r20862131;
        double r20862137 = b;
        double r20862138 = c;
        double r20862139 = r20862137 - r20862138;
        double r20862140 = 5.0;
        double r20862141 = 6.0;
        double r20862142 = r20862140 / r20862141;
        double r20862143 = r20862132 + r20862142;
        double r20862144 = 3.0;
        double r20862145 = r20862131 * r20862144;
        double r20862146 = r20862129 / r20862145;
        double r20862147 = r20862143 - r20862146;
        double r20862148 = r20862139 * r20862147;
        double r20862149 = r20862136 - r20862148;
        double r20862150 = r20862129 * r20862149;
        double r20862151 = exp(r20862150);
        double r20862152 = r20862128 * r20862151;
        double r20862153 = r20862127 + r20862152;
        double r20862154 = r20862127 / r20862153;
        return r20862154;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r20862155 = x;
        double r20862156 = y;
        double r20862157 = 2.0;
        double r20862158 = c;
        double r20862159 = b;
        double r20862160 = r20862158 - r20862159;
        double r20862161 = 5.0;
        double r20862162 = 6.0;
        double r20862163 = r20862161 / r20862162;
        double r20862164 = t;
        double r20862165 = r20862157 / r20862164;
        double r20862166 = 3.0;
        double r20862167 = r20862165 / r20862166;
        double r20862168 = a;
        double r20862169 = r20862167 - r20862168;
        double r20862170 = r20862163 - r20862169;
        double r20862171 = r20862168 + r20862164;
        double r20862172 = sqrt(r20862171);
        double r20862173 = cbrt(r20862164);
        double r20862174 = r20862173 * r20862173;
        double r20862175 = r20862172 / r20862174;
        double r20862176 = z;
        double r20862177 = r20862173 / r20862176;
        double r20862178 = r20862175 / r20862177;
        double r20862179 = fma(r20862160, r20862170, r20862178);
        double r20862180 = r20862157 * r20862179;
        double r20862181 = exp(r20862180);
        double r20862182 = fma(r20862156, r20862181, r20862155);
        double r20862183 = r20862155 / r20862182;
        return r20862183;
}

Target

Original	4.0
Target	3.3
Herbie	1.6

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

Derivation

Initial program 4.0
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Simplified1.9
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\frac{t}{z}}\right)}, x\right)}}\]
Using strategy rm
Applied *-un-lft-identity1.9
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\frac{t}{\color{blue}{1 \cdot z}}}\right)}, x\right)}\]
Applied add-cube-cbrt1.9
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\frac{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}{1 \cdot z}}\right)}, x\right)}\]
Applied times-frac1.9
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\sqrt{a + t}}{\color{blue}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{1} \cdot \frac{\sqrt[3]{t}}{z}}}\right)}, x\right)}\]
Applied associate-/r*1.6
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \color{blue}{\frac{\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{1}}}{\frac{\sqrt[3]{t}}{z}}}\right)}, x\right)}\]
Simplified1.6
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\color{blue}{\frac{\sqrt{t + a}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}}{\frac{\sqrt[3]{t}}{z}}\right)}, x\right)}\]
Final simplification1.6
\[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\frac{\sqrt{a + t}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{\sqrt[3]{t}}{z}}\right)}, x\right)}\]

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