Average Error: 9.5 → 0.4
Time: 20.6s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]
↓
\[\left(\left(\log 1.0 - \left(\left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2} + y \cdot 1.0\right)\right) \cdot z + \left(\left(\log \left(\sqrt[3]{y}\right) \cdot x + \log \left(\sqrt[3]{y}\right) \cdot x\right) + \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot x + \log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x + x\right)\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t
↓
\left(\left(\log 1.0 - \left(\left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2} + y \cdot 1.0\right)\right) \cdot z + \left(\left(\log \left(\sqrt[3]{y}\right) \cdot x + \log \left(\sqrt[3]{y}\right) \cdot x\right) + \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot x + \log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x + x\right)\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r22183580 = x;
        double r22183581 = y;
        double r22183582 = log(r22183581);
        double r22183583 = r22183580 * r22183582;
        double r22183584 = z;
        double r22183585 = 1.0;
        double r22183586 = r22183585 - r22183581;
        double r22183587 = log(r22183586);
        double r22183588 = r22183584 * r22183587;
        double r22183589 = r22183583 + r22183588;
        double r22183590 = t;
        double r22183591 = r22183589 - r22183590;
        return r22183591;
}
↓
double f(double x, double y, double z, double t) {
        double r22183592 = 1.0;
        double r22183593 = log(r22183592);
        double r22183594 = y;
        double r22183595 = r22183594 / r22183592;
        double r22183596 = r22183595 * r22183595;
        double r22183597 = 0.5;
        double r22183598 = r22183596 * r22183597;
        double r22183599 = r22183594 * r22183592;
        double r22183600 = r22183598 + r22183599;
        double r22183601 = r22183593 - r22183600;
        double r22183602 = z;
        double r22183603 = r22183601 * r22183602;
        double r22183604 = cbrt(r22183594);
        double r22183605 = log(r22183604);
        double r22183606 = x;
        double r22183607 = r22183605 * r22183606;
        double r22183608 = r22183607 + r22183607;
        double r22183609 = cbrt(r22183604);
        double r22183610 = log(r22183609);
        double r22183611 = r22183610 * r22183606;
        double r22183612 = r22183606 + r22183606;
        double r22183613 = r22183610 * r22183612;
        double r22183614 = r22183611 + r22183613;
        double r22183615 = r22183608 + r22183614;
        double r22183616 = r22183603 + r22183615;
        double r22183617 = t;
        double r22183618 = r22183616 - r22183617;
        return r22183618;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.5
Target0.3
Herbie0.4
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{\frac{1}{3}}{1.0 \cdot \left(1.0 \cdot 1.0\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.5

    \[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1.0 - \left(1.0 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1.0}^{2}}\right)\right)}\right) - t\]

  3. Simplified0.4

    \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1.0 - \left(1.0 \cdot y + \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)}\right) - t\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.4

    \[\leadsto \left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + z \cdot \left(\log 1.0 - \left(1.0 \cdot y + \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]

  6. Applied log-prod0.4

    \[\leadsto \left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\log 1.0 - \left(1.0 \cdot y + \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]

  7. Applied distribute-lft-in0.4

    \[\leadsto \left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\log 1.0 - \left(1.0 \cdot y + \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]

  8. Simplified0.4

    \[\leadsto \left(\left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \left(\log 1.0 - \left(1.0 \cdot y + \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.4

    \[\leadsto \left(\left(\left(x \cdot \log \left(\sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right) + z \cdot \left(\log 1.0 - \left(1.0 \cdot y + \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]

  11. Applied log-prod0.4

    \[\leadsto \left(\left(\left(x \cdot \log \left(\sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)}\right) + z \cdot \left(\log 1.0 - \left(1.0 \cdot y + \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]

  12. Applied distribute-lft-in0.4

    \[\leadsto \left(\left(\left(x \cdot \log \left(\sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \color{blue}{\left(x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)}\right) + z \cdot \left(\log 1.0 - \left(1.0 \cdot y + \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]

  13. Simplified0.4

    \[\leadsto \left(\left(\left(x \cdot \log \left(\sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(\color{blue}{\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x + x\right)} + x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + z \cdot \left(\log 1.0 - \left(1.0 \cdot y + \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]

  14. Final simplification0.4

    \[\leadsto \left(\left(\log 1.0 - \left(\left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2} + y \cdot 1.0\right)\right) \cdot z + \left(\left(\log \left(\sqrt[3]{y}\right) \cdot x + \log \left(\sqrt[3]{y}\right) \cdot x\right) + \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot x + \log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x + x\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 1/3 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))