Average Error: 1.8 → 1.9
Time: 39.5s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x}{\frac{\frac{\frac{y}{\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, \log z \cdot y\right) - b}}}}{\sqrt[3]{{e}^{\left(\mathsf{fma}\left(\log a, t - 1.0, \log z \cdot y\right) - b\right)}}}}{\sqrt[3]{{e}^{\left(\mathsf{fma}\left(\log a, t - 1.0, \log z \cdot y\right) - b\right)}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}
↓
\frac{x}{\frac{\frac{\frac{y}{\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1.0, \log z \cdot y\right) - b}}}}{\sqrt[3]{{e}^{\left(\mathsf{fma}\left(\log a, t - 1.0, \log z \cdot y\right) - b\right)}}}}{\sqrt[3]{{e}^{\left(\mathsf{fma}\left(\log a, t - 1.0, \log z \cdot y\right) - b\right)}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r1980351 = x;
        double r1980352 = y;
        double r1980353 = z;
        double r1980354 = log(r1980353);
        double r1980355 = r1980352 * r1980354;
        double r1980356 = t;
        double r1980357 = 1.0;
        double r1980358 = r1980356 - r1980357;
        double r1980359 = a;
        double r1980360 = log(r1980359);
        double r1980361 = r1980358 * r1980360;
        double r1980362 = r1980355 + r1980361;
        double r1980363 = b;
        double r1980364 = r1980362 - r1980363;
        double r1980365 = exp(r1980364);
        double r1980366 = r1980351 * r1980365;
        double r1980367 = r1980366 / r1980352;
        return r1980367;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r1980368 = x;
        double r1980369 = y;
        double r1980370 = a;
        double r1980371 = log(r1980370);
        double r1980372 = t;
        double r1980373 = 1.0;
        double r1980374 = r1980372 - r1980373;
        double r1980375 = z;
        double r1980376 = log(r1980375);
        double r1980377 = r1980376 * r1980369;
        double r1980378 = fma(r1980371, r1980374, r1980377);
        double r1980379 = b;
        double r1980380 = r1980378 - r1980379;
        double r1980381 = exp(r1980380);
        double r1980382 = cbrt(r1980381);
        double r1980383 = r1980369 / r1980382;
        double r1980384 = exp(1.0);
        double r1980385 = pow(r1980384, r1980380);
        double r1980386 = cbrt(r1980385);
        double r1980387 = r1980383 / r1980386;
        double r1980388 = r1980387 / r1980386;
        double r1980389 = r1980368 / r1980388;
        return r1980389;
}

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 *-un-lft-identity1.8

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

  6. Applied exp-prod1.8

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

  7. Simplified1.8

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

  9. Applied *-un-lft-identity1.8

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

  10. Applied exp-prod1.9

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

  11. Simplified1.9

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

  13. Applied associate-/l*1.9

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

  14. Simplified1.9

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

  15. Final simplification1.9

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

Reproduce

herbie shell --seed 2019156 +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))