Average Error: 9.7 → 0.4
Time: 21.7s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(x \cdot \log \left(\sqrt[3]{y} \cdot \left({\left(\sqrt[3]{y}\right)}^{\frac{2}{3}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) + \left(x \cdot \log \left({y}^{\frac{1}{3}}\right) + \left(z \cdot \left(\log 1 - y \cdot 1\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(x \cdot \log \left(\sqrt[3]{y} \cdot \left({\left(\sqrt[3]{y}\right)}^{\frac{2}{3}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) + \left(x \cdot \log \left({y}^{\frac{1}{3}}\right) + \left(z \cdot \left(\log 1 - y \cdot 1\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r555603 = x;
        double r555604 = y;
        double r555605 = log(r555604);
        double r555606 = r555603 * r555605;
        double r555607 = z;
        double r555608 = 1.0;
        double r555609 = r555608 - r555604;
        double r555610 = log(r555609);
        double r555611 = r555607 * r555610;
        double r555612 = r555606 + r555611;
        double r555613 = t;
        double r555614 = r555612 - r555613;
        return r555614;
}
↓
double f(double x, double y, double z, double t) {
        double r555615 = x;
        double r555616 = y;
        double r555617 = cbrt(r555616);
        double r555618 = 0.6666666666666666;
        double r555619 = pow(r555617, r555618);
        double r555620 = cbrt(r555617);
        double r555621 = r555619 * r555620;
        double r555622 = r555617 * r555621;
        double r555623 = log(r555622);
        double r555624 = r555615 * r555623;
        double r555625 = 0.3333333333333333;
        double r555626 = pow(r555616, r555625);
        double r555627 = log(r555626);
        double r555628 = r555615 * r555627;
        double r555629 = z;
        double r555630 = 1.0;
        double r555631 = log(r555630);
        double r555632 = r555616 * r555630;
        double r555633 = r555631 - r555632;
        double r555634 = r555629 * r555633;
        double r555635 = 0.5;
        double r555636 = 2.0;
        double r555637 = pow(r555616, r555636);
        double r555638 = r555629 * r555637;
        double r555639 = pow(r555630, r555636);
        double r555640 = r555638 / r555639;
        double r555641 = r555635 * r555640;
        double r555642 = r555634 - r555641;
        double r555643 = r555628 + r555642;
        double r555644 = r555624 + r555643;
        double r555645 = t;
        double r555646 = r555644 - r555645;
        return r555646;
}

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

Original9.7
Target0.3
Herbie0.4
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.7

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Applied associate-+l+0.4

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

  10. Applied pow1/30.4

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

  12. Applied add-cube-cbrt0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019209 
(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) (* (/ 0.333333333333333315 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

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