Average Error: 9.1 → 0.4
Time: 36.3s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]
↓
\[\left(z \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right) + \left(\left(\log \left(\sqrt[3]{y}\right) + \log \left({y}^{\frac{1}{3}}\right)\right) \cdot x + x \cdot \log \left(\sqrt[3]{y}\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t
↓
\left(z \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right) + \left(\left(\log \left(\sqrt[3]{y}\right) + \log \left({y}^{\frac{1}{3}}\right)\right) \cdot x + x \cdot \log \left(\sqrt[3]{y}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r24784795 = x;
        double r24784796 = y;
        double r24784797 = log(r24784796);
        double r24784798 = r24784795 * r24784797;
        double r24784799 = z;
        double r24784800 = 1.0;
        double r24784801 = r24784800 - r24784796;
        double r24784802 = log(r24784801);
        double r24784803 = r24784799 * r24784802;
        double r24784804 = r24784798 + r24784803;
        double r24784805 = t;
        double r24784806 = r24784804 - r24784805;
        return r24784806;
}
↓
double f(double x, double y, double z, double t) {
        double r24784807 = z;
        double r24784808 = 1.0;
        double r24784809 = log(r24784808);
        double r24784810 = y;
        double r24784811 = r24784810 * r24784808;
        double r24784812 = r24784809 - r24784811;
        double r24784813 = 0.5;
        double r24784814 = r24784808 / r24784810;
        double r24784815 = r24784813 / r24784814;
        double r24784816 = r24784815 / r24784814;
        double r24784817 = r24784812 - r24784816;
        double r24784818 = r24784807 * r24784817;
        double r24784819 = cbrt(r24784810);
        double r24784820 = log(r24784819);
        double r24784821 = 0.3333333333333333;
        double r24784822 = pow(r24784810, r24784821);
        double r24784823 = log(r24784822);
        double r24784824 = r24784820 + r24784823;
        double r24784825 = x;
        double r24784826 = r24784824 * r24784825;
        double r24784827 = r24784825 * r24784820;
        double r24784828 = r24784826 + r24784827;
        double r24784829 = r24784818 + r24784828;
        double r24784830 = t;
        double r24784831 = r24784829 - r24784830;
        return r24784831;
}

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

Original9.1
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 9.1

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

  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

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Simplified0.4

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

  10. Applied pow1/30.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019164 
(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))