Average Error: 9.6 → 0.3
Time: 8.2s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(x \cdot \log \left(1 \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \left(\log \left(1 \cdot {y}^{\frac{1}{3}}\right) \cdot x + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(x \cdot \log \left(1 \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \left(\log \left(1 \cdot {y}^{\frac{1}{3}}\right) \cdot x + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r458955 = x;
        double r458956 = y;
        double r458957 = log(r458956);
        double r458958 = r458955 * r458957;
        double r458959 = z;
        double r458960 = 1.0;
        double r458961 = r458960 - r458956;
        double r458962 = log(r458961);
        double r458963 = r458959 * r458962;
        double r458964 = r458958 + r458963;
        double r458965 = t;
        double r458966 = r458964 - r458965;
        return r458966;
}
↓
double f(double x, double y, double z, double t) {
        double r458967 = x;
        double r458968 = 1.0;
        double r458969 = y;
        double r458970 = cbrt(r458969);
        double r458971 = r458970 * r458970;
        double r458972 = r458968 * r458971;
        double r458973 = log(r458972);
        double r458974 = r458967 * r458973;
        double r458975 = 0.3333333333333333;
        double r458976 = pow(r458969, r458975);
        double r458977 = r458968 * r458976;
        double r458978 = log(r458977);
        double r458979 = r458978 * r458967;
        double r458980 = z;
        double r458981 = 1.0;
        double r458982 = log(r458981);
        double r458983 = r458981 * r458969;
        double r458984 = 0.5;
        double r458985 = 2.0;
        double r458986 = pow(r458969, r458985);
        double r458987 = pow(r458981, r458985);
        double r458988 = r458986 / r458987;
        double r458989 = r458984 * r458988;
        double r458990 = r458983 + r458989;
        double r458991 = r458982 - r458990;
        double r458992 = r458980 * r458991;
        double r458993 = r458979 + r458992;
        double r458994 = r458974 + r458993;
        double r458995 = t;
        double r458996 = r458994 - r458995;
        return r458996;
}

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

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 0.3

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

  4. Applied add-cube-cbrt0.3

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

  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-in0.3

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

  7. Applied associate-+l+0.3

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

  8. Simplified0.3

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

  10. Applied *-un-lft-identity0.3

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

  11. Applied cbrt-prod0.3

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

  12. Simplified0.3

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

  13. Simplified0.3

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

  15. Applied *-un-lft-identity0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2020049 
(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))