Average Error: 2.1 → 1.0
Time: 59.9s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\left(x \cdot \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\sqrt[3]{y}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}
↓
\left(x \cdot \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\sqrt[3]{y}}
double f(double x, double y, double z, double t, double a, double b) {
        double r2727347 = x;
        double r2727348 = y;
        double r2727349 = z;
        double r2727350 = log(r2727349);
        double r2727351 = r2727348 * r2727350;
        double r2727352 = t;
        double r2727353 = 1.0;
        double r2727354 = r2727352 - r2727353;
        double r2727355 = a;
        double r2727356 = log(r2727355);
        double r2727357 = r2727354 * r2727356;
        double r2727358 = r2727351 + r2727357;
        double r2727359 = b;
        double r2727360 = r2727358 - r2727359;
        double r2727361 = exp(r2727360);
        double r2727362 = r2727347 * r2727361;
        double r2727363 = r2727362 / r2727348;
        return r2727363;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r2727364 = x;
        double r2727365 = t;
        double r2727366 = 1.0;
        double r2727367 = r2727365 - r2727366;
        double r2727368 = a;
        double r2727369 = log(r2727368);
        double r2727370 = z;
        double r2727371 = log(r2727370);
        double r2727372 = y;
        double r2727373 = r2727371 * r2727372;
        double r2727374 = fma(r2727367, r2727369, r2727373);
        double r2727375 = b;
        double r2727376 = r2727374 - r2727375;
        double r2727377 = exp(r2727376);
        double r2727378 = sqrt(r2727377);
        double r2727379 = cbrt(r2727372);
        double r2727380 = r2727379 * r2727379;
        double r2727381 = cbrt(r2727380);
        double r2727382 = cbrt(r2727381);
        double r2727383 = r2727382 * r2727382;
        double r2727384 = r2727382 * r2727383;
        double r2727385 = cbrt(r2727379);
        double r2727386 = r2727384 * r2727385;
        double r2727387 = r2727386 * r2727379;
        double r2727388 = r2727378 / r2727387;
        double r2727389 = r2727364 * r2727388;
        double r2727390 = r2727378 / r2727379;
        double r2727391 = r2727389 * r2727390;
        return r2727391;
}

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 2.1

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

  2. Simplified1.9

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

  4. Applied add-cube-cbrt1.9

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

  5. Applied add-sqr-sqrt1.9

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

  6. Applied times-frac1.9

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

  7. Applied associate-*r*1.0

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

  9. Applied add-cube-cbrt1.0

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

  10. Applied cbrt-prod1.0

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

  12. Applied add-cube-cbrt1.0

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

  13. Final simplification1.0

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

Reproduce

herbie shell --seed 2019143 +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))