Average Error: 4.0 → 1.5
Time: 19.3s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
\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(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \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 r55112 = x;
        double r55113 = y;
        double r55114 = 2.0;
        double r55115 = z;
        double r55116 = t;
        double r55117 = a;
        double r55118 = r55116 + r55117;
        double r55119 = sqrt(r55118);
        double r55120 = r55115 * r55119;
        double r55121 = r55120 / r55116;
        double r55122 = b;
        double r55123 = c;
        double r55124 = r55122 - r55123;
        double r55125 = 5.0;
        double r55126 = 6.0;
        double r55127 = r55125 / r55126;
        double r55128 = r55117 + r55127;
        double r55129 = 3.0;
        double r55130 = r55116 * r55129;
        double r55131 = r55114 / r55130;
        double r55132 = r55128 - r55131;
        double r55133 = r55124 * r55132;
        double r55134 = r55121 - r55133;
        double r55135 = r55114 * r55134;
        double r55136 = exp(r55135);
        double r55137 = r55113 * r55136;
        double r55138 = r55112 + r55137;
        double r55139 = r55112 / r55138;
        return r55139;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r55140 = x;
        double r55141 = y;
        double r55142 = 2.0;
        double r55143 = exp(r55142);
        double r55144 = t;
        double r55145 = r55142 / r55144;
        double r55146 = 3.0;
        double r55147 = r55145 / r55146;
        double r55148 = a;
        double r55149 = 5.0;
        double r55150 = 6.0;
        double r55151 = r55149 / r55150;
        double r55152 = r55148 + r55151;
        double r55153 = r55147 - r55152;
        double r55154 = b;
        double r55155 = c;
        double r55156 = r55154 - r55155;
        double r55157 = z;
        double r55158 = cbrt(r55144);
        double r55159 = r55158 * r55158;
        double r55160 = r55157 / r55159;
        double r55161 = r55144 + r55148;
        double r55162 = sqrt(r55161);
        double r55163 = r55162 / r55158;
        double r55164 = r55160 * r55163;
        double r55165 = fma(r55153, r55156, r55164);
        double r55166 = pow(r55143, r55165);
        double r55167 = fma(r55141, r55166, r55140);
        double r55168 = r55140 / r55167;
        return r55168;
}

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 4.0

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

  2. Simplified2.7

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

  4. Applied add-cube-cbrt2.7

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

  5. Applied times-frac1.5

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

  6. Final simplification1.5

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))