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

↓

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

\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{\frac{2}{t}}{3} - \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 r284103 = x;
        double r284104 = y;
        double r284105 = 2.0;
        double r284106 = z;
        double r284107 = t;
        double r284108 = a;
        double r284109 = r284107 + r284108;
        double r284110 = sqrt(r284109);
        double r284111 = r284106 * r284110;
        double r284112 = r284111 / r284107;
        double r284113 = b;
        double r284114 = c;
        double r284115 = r284113 - r284114;
        double r284116 = 5.0;
        double r284117 = 6.0;
        double r284118 = r284116 / r284117;
        double r284119 = r284108 + r284118;
        double r284120 = 3.0;
        double r284121 = r284107 * r284120;
        double r284122 = r284105 / r284121;
        double r284123 = r284119 - r284122;
        double r284124 = r284115 * r284123;
        double r284125 = r284112 - r284124;
        double r284126 = r284105 * r284125;
        double r284127 = exp(r284126);
        double r284128 = r284104 * r284127;
        double r284129 = r284103 + r284128;
        double r284130 = r284103 / r284129;
        return r284130;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r284131 = x;
        double r284132 = y;
        double r284133 = 2.0;
        double r284134 = exp(r284133);
        double r284135 = t;
        double r284136 = r284133 / r284135;
        double r284137 = 3.0;
        double r284138 = r284136 / r284137;
        double r284139 = a;
        double r284140 = 5.0;
        double r284141 = 6.0;
        double r284142 = r284140 / r284141;
        double r284143 = r284139 + r284142;
        double r284144 = r284138 - r284143;
        double r284145 = b;
        double r284146 = c;
        double r284147 = r284145 - r284146;
        double r284148 = z;
        double r284149 = r284135 + r284139;
        double r284150 = sqrt(r284149);
        double r284151 = r284148 * r284150;
        double r284152 = r284151 / r284135;
        double r284153 = fma(r284144, r284147, r284152);
        double r284154 = pow(r284134, r284153);
        double r284155 = fma(r284132, r284154, r284131);
        double r284156 = r284131 / r284155;
        return r284156;
}

Target

Original	3.7
Target	2.9
Herbie	2.5

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

Derivation

Initial program 3.7
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Simplified2.5
\[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}}\]
Final simplification2.5
\[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019306 +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