Average Error: 9.6 → 0.4
Time: 10.6s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right)\right) + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + x \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right)\right) + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r367632 = x;
        double r367633 = y;
        double r367634 = log(r367633);
        double r367635 = r367632 * r367634;
        double r367636 = z;
        double r367637 = 1.0;
        double r367638 = r367637 - r367633;
        double r367639 = log(r367638);
        double r367640 = r367636 * r367639;
        double r367641 = r367635 + r367640;
        double r367642 = t;
        double r367643 = r367641 - r367642;
        return r367643;
}
↓
double f(double x, double y, double z, double t) {
        double r367644 = x;
        double r367645 = 2.0;
        double r367646 = y;
        double r367647 = cbrt(r367646);
        double r367648 = log(r367647);
        double r367649 = r367645 * r367648;
        double r367650 = r367644 * r367649;
        double r367651 = 1.0;
        double r367652 = r367651 / r367646;
        double r367653 = -0.3333333333333333;
        double r367654 = pow(r367652, r367653);
        double r367655 = log(r367654);
        double r367656 = r367644 * r367655;
        double r367657 = r367650 + r367656;
        double r367658 = z;
        double r367659 = 1.0;
        double r367660 = log(r367659);
        double r367661 = r367659 * r367646;
        double r367662 = 0.5;
        double r367663 = pow(r367646, r367645);
        double r367664 = pow(r367659, r367645);
        double r367665 = r367663 / r367664;
        double r367666 = r367662 * r367665;
        double r367667 = r367661 + r367666;
        double r367668 = r367660 - r367667;
        double r367669 = r367658 * r367668;
        double r367670 = r367657 + r367669;
        double r367671 = t;
        double r367672 = r367670 - r367671;
        return r367672;
}

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

Original9.6
Target0.3
Herbie0.4
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.6

    \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

  2. Taylor expanded around 0 0.4

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

  4. Applied add-cube-cbrt0.4

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

  5. Applied log-prod0.4

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

  6. Applied distribute-lft-in0.4

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

  7. Simplified0.4

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

  8. Taylor expanded around inf 0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1 y)))) t))