Average Error: 3.9 → 1.4
Time: 23.0s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \frac{z}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}, x\right)}\]
\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, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \frac{z}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2320147 = x;
        double r2320148 = y;
        double r2320149 = 2.0;
        double r2320150 = z;
        double r2320151 = t;
        double r2320152 = a;
        double r2320153 = r2320151 + r2320152;
        double r2320154 = sqrt(r2320153);
        double r2320155 = r2320150 * r2320154;
        double r2320156 = r2320155 / r2320151;
        double r2320157 = b;
        double r2320158 = c;
        double r2320159 = r2320157 - r2320158;
        double r2320160 = 5.0;
        double r2320161 = 6.0;
        double r2320162 = r2320160 / r2320161;
        double r2320163 = r2320152 + r2320162;
        double r2320164 = 3.0;
        double r2320165 = r2320151 * r2320164;
        double r2320166 = r2320149 / r2320165;
        double r2320167 = r2320163 - r2320166;
        double r2320168 = r2320159 * r2320167;
        double r2320169 = r2320156 - r2320168;
        double r2320170 = r2320149 * r2320169;
        double r2320171 = exp(r2320170);
        double r2320172 = r2320148 * r2320171;
        double r2320173 = r2320147 + r2320172;
        double r2320174 = r2320147 / r2320173;
        return r2320174;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2320175 = x;
        double r2320176 = y;
        double r2320177 = 2.0;
        double r2320178 = c;
        double r2320179 = b;
        double r2320180 = r2320178 - r2320179;
        double r2320181 = 5.0;
        double r2320182 = 6.0;
        double r2320183 = r2320181 / r2320182;
        double r2320184 = a;
        double r2320185 = t;
        double r2320186 = r2320177 / r2320185;
        double r2320187 = 3.0;
        double r2320188 = r2320186 / r2320187;
        double r2320189 = r2320184 - r2320188;
        double r2320190 = r2320183 + r2320189;
        double r2320191 = z;
        double r2320192 = cbrt(r2320185);
        double r2320193 = r2320191 / r2320192;
        double r2320194 = r2320185 + r2320184;
        double r2320195 = sqrt(r2320194);
        double r2320196 = r2320192 * r2320192;
        double r2320197 = r2320195 / r2320196;
        double r2320198 = r2320193 * r2320197;
        double r2320199 = fma(r2320180, r2320190, r2320198);
        double r2320200 = r2320177 * r2320199;
        double r2320201 = exp(r2320200);
        double r2320202 = fma(r2320176, r2320201, r2320175);
        double r2320203 = r2320175 / r2320202;
        return r2320203;
}

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 3.9

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

  2. Simplified1.6

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

  4. Applied add-cube-cbrt1.6

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

  5. Applied *-un-lft-identity1.6

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

  6. Applied times-frac1.6

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

  7. Applied associate-*r*1.4

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

  8. Simplified1.4

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

  9. Final simplification1.4

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

Reproduce

herbie shell --seed 2019174 +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)))))))))))