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

↓

\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{0.6666666666666666296592325124947819858789}{t} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}\]

\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{0.6666666666666666296592325124947819858789}{t} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r507115 = x;
        double r507116 = y;
        double r507117 = 2.0;
        double r507118 = z;
        double r507119 = t;
        double r507120 = a;
        double r507121 = r507119 + r507120;
        double r507122 = sqrt(r507121);
        double r507123 = r507118 * r507122;
        double r507124 = r507123 / r507119;
        double r507125 = b;
        double r507126 = c;
        double r507127 = r507125 - r507126;
        double r507128 = 5.0;
        double r507129 = 6.0;
        double r507130 = r507128 / r507129;
        double r507131 = r507120 + r507130;
        double r507132 = 3.0;
        double r507133 = r507119 * r507132;
        double r507134 = r507117 / r507133;
        double r507135 = r507131 - r507134;
        double r507136 = r507127 * r507135;
        double r507137 = r507124 - r507136;
        double r507138 = r507117 * r507137;
        double r507139 = exp(r507138);
        double r507140 = r507116 * r507139;
        double r507141 = r507115 + r507140;
        double r507142 = r507115 / r507141;
        return r507142;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r507143 = x;
        double r507144 = y;
        double r507145 = 2.0;
        double r507146 = exp(r507145);
        double r507147 = 0.6666666666666666;
        double r507148 = t;
        double r507149 = r507147 / r507148;
        double r507150 = a;
        double r507151 = 5.0;
        double r507152 = 6.0;
        double r507153 = r507151 / r507152;
        double r507154 = r507150 + r507153;
        double r507155 = r507149 - r507154;
        double r507156 = b;
        double r507157 = c;
        double r507158 = r507156 - r507157;
        double r507159 = z;
        double r507160 = r507148 + r507150;
        double r507161 = sqrt(r507160);
        double r507162 = r507159 * r507161;
        double r507163 = r507162 / r507148;
        double r507164 = fma(r507155, r507158, r507163);
        double r507165 = pow(r507146, r507164);
        double r507166 = fma(r507144, r507165, r507143);
        double r507167 = r507143 / r507166;
        return r507167;
}

Target

Original	4.1
Target	3.1
Herbie	2.8

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

Derivation

Initial program 4.1
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Simplified2.8
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}}\]
Taylor expanded around 0 2.8
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\color{blue}{\frac{0.6666666666666666296592325124947819858789}{t}} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}\]
Final simplification2.8
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{0.6666666666666666296592325124947819858789}{t} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019351 +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)))))))))))

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Target

Derivation

Reproduce