Average Error: 9.7 → 0.4
Time: 21.1s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot x + \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot x\right) + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot x + \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot x\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 r317940 = x;
        double r317941 = y;
        double r317942 = log(r317941);
        double r317943 = r317940 * r317942;
        double r317944 = z;
        double r317945 = 1.0;
        double r317946 = r317945 - r317941;
        double r317947 = log(r317946);
        double r317948 = r317944 * r317947;
        double r317949 = r317943 + r317948;
        double r317950 = t;
        double r317951 = r317949 - r317950;
        return r317951;
}
↓
double f(double x, double y, double z, double t) {
        double r317952 = 2.0;
        double r317953 = y;
        double r317954 = cbrt(r317953);
        double r317955 = log(r317954);
        double r317956 = r317952 * r317955;
        double r317957 = x;
        double r317958 = r317956 * r317957;
        double r317959 = 1.0;
        double r317960 = r317959 / r317953;
        double r317961 = -0.3333333333333333;
        double r317962 = pow(r317960, r317961);
        double r317963 = log(r317962);
        double r317964 = r317963 * r317957;
        double r317965 = r317958 + r317964;
        double r317966 = z;
        double r317967 = 1.0;
        double r317968 = log(r317967);
        double r317969 = r317967 * r317953;
        double r317970 = 0.5;
        double r317971 = pow(r317953, r317952);
        double r317972 = pow(r317967, r317952);
        double r317973 = r317971 / r317972;
        double r317974 = r317970 * r317973;
        double r317975 = r317969 + r317974;
        double r317976 = r317968 - r317975;
        double r317977 = r317966 * r317976;
        double r317978 = r317965 + r317977;
        double r317979 = t;
        double r317980 = r317978 - r317979;
        return r317980;
}

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 0.3

    \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}\right) - t\]
  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-in0.4

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  9. Taylor expanded around inf 0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 
(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))