Average Error: 3.8 → 1.3
Time: 22.3s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, c - b, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
\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(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, c - b, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r77057 = x;
        double r77058 = y;
        double r77059 = 2.0;
        double r77060 = z;
        double r77061 = t;
        double r77062 = a;
        double r77063 = r77061 + r77062;
        double r77064 = sqrt(r77063);
        double r77065 = r77060 * r77064;
        double r77066 = r77065 / r77061;
        double r77067 = b;
        double r77068 = c;
        double r77069 = r77067 - r77068;
        double r77070 = 5.0;
        double r77071 = 6.0;
        double r77072 = r77070 / r77071;
        double r77073 = r77062 + r77072;
        double r77074 = 3.0;
        double r77075 = r77061 * r77074;
        double r77076 = r77059 / r77075;
        double r77077 = r77073 - r77076;
        double r77078 = r77069 * r77077;
        double r77079 = r77066 - r77078;
        double r77080 = r77059 * r77079;
        double r77081 = exp(r77080);
        double r77082 = r77058 * r77081;
        double r77083 = r77057 + r77082;
        double r77084 = r77057 / r77083;
        return r77084;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r77085 = x;
        double r77086 = y;
        double r77087 = 2.0;
        double r77088 = exp(r77087);
        double r77089 = a;
        double r77090 = 5.0;
        double r77091 = 6.0;
        double r77092 = r77090 / r77091;
        double r77093 = r77089 + r77092;
        double r77094 = t;
        double r77095 = 3.0;
        double r77096 = r77094 * r77095;
        double r77097 = r77087 / r77096;
        double r77098 = r77093 - r77097;
        double r77099 = c;
        double r77100 = b;
        double r77101 = r77099 - r77100;
        double r77102 = z;
        double r77103 = cbrt(r77094);
        double r77104 = r77103 * r77103;
        double r77105 = r77102 / r77104;
        double r77106 = r77094 + r77089;
        double r77107 = sqrt(r77106);
        double r77108 = r77107 / r77103;
        double r77109 = r77105 * r77108;
        double r77110 = fma(r77098, r77101, r77109);
        double r77111 = pow(r77088, r77110);
        double r77112 = fma(r77086, r77111, r77085);
        double r77113 = r77085 / r77112;
        return r77113;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 3.8

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

  2. Simplified2.5

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

  4. Applied add-cube-cbrt2.5

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

  5. Applied times-frac1.3

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

  6. Final simplification1.3

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))