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

↓

\[\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

\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}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2073 = x;
        double r2074 = y;
        double r2075 = 2.0;
        double r2076 = z;
        double r2077 = t;
        double r2078 = a;
        double r2079 = r2077 + r2078;
        double r2080 = sqrt(r2079);
        double r2081 = r2076 * r2080;
        double r2082 = r2081 / r2077;
        double r2083 = b;
        double r2084 = c;
        double r2085 = r2083 - r2084;
        double r2086 = 5.0;
        double r2087 = 6.0;
        double r2088 = r2086 / r2087;
        double r2089 = r2078 + r2088;
        double r2090 = 3.0;
        double r2091 = r2077 * r2090;
        double r2092 = r2075 / r2091;
        double r2093 = r2089 - r2092;
        double r2094 = r2085 * r2093;
        double r2095 = r2082 - r2094;
        double r2096 = r2075 * r2095;
        double r2097 = exp(r2096);
        double r2098 = r2074 * r2097;
        double r2099 = r2073 + r2098;
        double r2100 = r2073 / r2099;
        return r2100;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2101 = x;
        double r2102 = y;
        double r2103 = 2.0;
        double r2104 = z;
        double r2105 = t;
        double r2106 = a;
        double r2107 = r2105 + r2106;
        double r2108 = sqrt(r2107);
        double r2109 = r2104 * r2108;
        double r2110 = 1.0;
        double r2111 = r2110 / r2105;
        double r2112 = b;
        double r2113 = c;
        double r2114 = r2112 - r2113;
        double r2115 = 5.0;
        double r2116 = 6.0;
        double r2117 = r2115 / r2116;
        double r2118 = r2106 + r2117;
        double r2119 = 3.0;
        double r2120 = r2105 * r2119;
        double r2121 = r2103 / r2120;
        double r2122 = r2118 - r2121;
        double r2123 = r2114 * r2122;
        double r2124 = -r2123;
        double r2125 = fma(r2109, r2111, r2124);
        double r2126 = r2103 * r2125;
        double r2127 = exp(r2126);
        double r2128 = r2102 * r2127;
        double r2129 = r2101 + r2128;
        double r2130 = r2101 / r2129;
        return r2130;
}

Target

Original	3.5
Target	2.9
Herbie	2.3

\[\begin{array}{l} \mathbf{if}\;t \lt -2.1183266448915811 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.19658877065154709 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

Initial program 3.5
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Using strategy rm
Applied div-inv3.5
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(z \cdot \sqrt{t + a}\right) \cdot \frac{1}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
Applied fma-neg2.3
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}}\]
Final simplification2.3
\[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

Reproduce

herbie shell --seed 2020025 +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)))))))))))

Error

Target

Derivation

Reproduce