Average Error: 1.9 → 1.8
Time: 46.6s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\frac{x}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{e^{\mathsf{fma}\left(\log a, t - 1.0, \log z \cdot y\right) - b}}}}{\sqrt[3]{y}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}
↓
\frac{\frac{x}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{e^{\mathsf{fma}\left(\log a, t - 1.0, \log z \cdot y\right) - b}}}}{\sqrt[3]{y}}
double f(double x, double y, double z, double t, double a, double b) {
        double r4020420 = x;
        double r4020421 = y;
        double r4020422 = z;
        double r4020423 = log(r4020422);
        double r4020424 = r4020421 * r4020423;
        double r4020425 = t;
        double r4020426 = 1.0;
        double r4020427 = r4020425 - r4020426;
        double r4020428 = a;
        double r4020429 = log(r4020428);
        double r4020430 = r4020427 * r4020429;
        double r4020431 = r4020424 + r4020430;
        double r4020432 = b;
        double r4020433 = r4020431 - r4020432;
        double r4020434 = exp(r4020433);
        double r4020435 = r4020420 * r4020434;
        double r4020436 = r4020435 / r4020421;
        return r4020436;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r4020437 = x;
        double r4020438 = y;
        double r4020439 = cbrt(r4020438);
        double r4020440 = r4020439 * r4020439;
        double r4020441 = a;
        double r4020442 = log(r4020441);
        double r4020443 = t;
        double r4020444 = 1.0;
        double r4020445 = r4020443 - r4020444;
        double r4020446 = z;
        double r4020447 = log(r4020446);
        double r4020448 = r4020447 * r4020438;
        double r4020449 = fma(r4020442, r4020445, r4020448);
        double r4020450 = b;
        double r4020451 = r4020449 - r4020450;
        double r4020452 = exp(r4020451);
        double r4020453 = r4020440 / r4020452;
        double r4020454 = r4020437 / r4020453;
        double r4020455 = r4020454 / r4020439;
        return r4020455;
}

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 1.9

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

  3. Applied add-cube-cbrt2.0

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

  5. Applied add-cube-cbrt2.0

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

  6. Applied associate-/r*2.0

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

  7. Simplified1.8

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

  8. Final simplification1.8

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

Reproduce

herbie shell --seed 2019162 +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))