Average Error: 1.8 → 1.4
Time: 41.5s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\frac{\sqrt[3]{\left(\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{{e}^{\left(\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b\right)}}}}{\sqrt[3]{y}} \cdot \frac{\left(x \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}}{\sqrt[3]{y}}}{\sqrt[3]{y}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}
↓
\frac{\frac{\sqrt[3]{\left(\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{{e}^{\left(\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b\right)}}}}{\sqrt[3]{y}} \cdot \frac{\left(x \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}}{\sqrt[3]{y}}}{\sqrt[3]{y}}
double f(double x, double y, double z, double t, double a, double b) {
        double r1546262 = x;
        double r1546263 = y;
        double r1546264 = z;
        double r1546265 = log(r1546264);
        double r1546266 = r1546263 * r1546265;
        double r1546267 = t;
        double r1546268 = 1.0;
        double r1546269 = r1546267 - r1546268;
        double r1546270 = a;
        double r1546271 = log(r1546270);
        double r1546272 = r1546269 * r1546271;
        double r1546273 = r1546266 + r1546272;
        double r1546274 = b;
        double r1546275 = r1546273 - r1546274;
        double r1546276 = exp(r1546275);
        double r1546277 = r1546262 * r1546276;
        double r1546278 = r1546277 / r1546263;
        return r1546278;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r1546279 = a;
        double r1546280 = log(r1546279);
        double r1546281 = t;
        double r1546282 = 1.0;
        double r1546283 = r1546281 - r1546282;
        double r1546284 = y;
        double r1546285 = z;
        double r1546286 = log(r1546285);
        double r1546287 = r1546284 * r1546286;
        double r1546288 = fma(r1546280, r1546283, r1546287);
        double r1546289 = b;
        double r1546290 = r1546288 - r1546289;
        double r1546291 = exp(r1546290);
        double r1546292 = cbrt(r1546291);
        double r1546293 = r1546292 * r1546292;
        double r1546294 = exp(1.0);
        double r1546295 = pow(r1546294, r1546290);
        double r1546296 = cbrt(r1546295);
        double r1546297 = r1546293 * r1546296;
        double r1546298 = cbrt(r1546297);
        double r1546299 = cbrt(r1546284);
        double r1546300 = r1546298 / r1546299;
        double r1546301 = x;
        double r1546302 = r1546301 * r1546292;
        double r1546303 = r1546302 * r1546292;
        double r1546304 = r1546303 / r1546299;
        double r1546305 = r1546300 * r1546304;
        double r1546306 = r1546305 / r1546299;
        return r1546306;
}

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

Derivation

  1. Initial program 1.8

    \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.8

    \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}} \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}\right) \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}\right)}}{y}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt1.8

    \[\leadsto \frac{x \cdot \left(\left(\sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}} \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}\right) \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}\right)}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\]

  6. Applied associate-/r*1.8

    \[\leadsto \color{blue}{\frac{\frac{x \cdot \left(\left(\sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}} \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}\right) \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}\right)}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt[3]{y}}}\]

  7. Simplified1.4

    \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}}{\sqrt[3]{y}}}}{\sqrt[3]{y}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt1.4

    \[\leadsto \frac{\frac{\left(x \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}}}}{\sqrt[3]{y}}}{\sqrt[3]{y}}\]
  10. Using strategy rm

  11. Applied *-un-lft-identity1.4

    \[\leadsto \frac{\frac{\left(x \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{\left(\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{e^{\color{blue}{1 \cdot \left(\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b\right)}}}}}{\sqrt[3]{y}}}{\sqrt[3]{y}}\]

  12. Applied exp-prod1.4

    \[\leadsto \frac{\frac{\left(x \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{\left(\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{\color{blue}{{\left(e^{1}\right)}^{\left(\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b\right)}}}}}{\sqrt[3]{y}}}{\sqrt[3]{y}}\]

  13. Simplified1.4

    \[\leadsto \frac{\frac{\left(x \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{\left(\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{{\color{blue}{e}}^{\left(\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b\right)}}}}{\sqrt[3]{y}}}{\sqrt[3]{y}}\]

  14. Final simplification1.4

    \[\leadsto \frac{\frac{\sqrt[3]{\left(\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{{e}^{\left(\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b\right)}}}}{\sqrt[3]{y}} \cdot \frac{\left(x \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, y \cdot \log z\right) - b}}}{\sqrt[3]{y}}}{\sqrt[3]{y}}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))