Average Error: 9.4 → 0.8
Time: 20.0s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(z, \log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), \left(\sqrt[3]{\log y \cdot x} \cdot \sqrt[3]{\log y \cdot x}\right) \cdot \sqrt[3]{\log y \cdot x}\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(z, \log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), \left(\sqrt[3]{\log y \cdot x} \cdot \sqrt[3]{\log y \cdot x}\right) \cdot \sqrt[3]{\log y \cdot x}\right) - t
double f(double x, double y, double z, double t) {
        double r15085943 = x;
        double r15085944 = y;
        double r15085945 = log(r15085944);
        double r15085946 = r15085943 * r15085945;
        double r15085947 = z;
        double r15085948 = 1.0;
        double r15085949 = r15085948 - r15085944;
        double r15085950 = log(r15085949);
        double r15085951 = r15085947 * r15085950;
        double r15085952 = r15085946 + r15085951;
        double r15085953 = t;
        double r15085954 = r15085952 - r15085953;
        return r15085954;
}
↓
double f(double x, double y, double z, double t) {
        double r15085955 = z;
        double r15085956 = 1.0;
        double r15085957 = log(r15085956);
        double r15085958 = y;
        double r15085959 = 0.5;
        double r15085960 = r15085956 / r15085958;
        double r15085961 = r15085960 * r15085960;
        double r15085962 = r15085959 / r15085961;
        double r15085963 = fma(r15085956, r15085958, r15085962);
        double r15085964 = r15085957 - r15085963;
        double r15085965 = log(r15085958);
        double r15085966 = x;
        double r15085967 = r15085965 * r15085966;
        double r15085968 = cbrt(r15085967);
        double r15085969 = r15085968 * r15085968;
        double r15085970 = r15085969 * r15085968;
        double r15085971 = fma(r15085955, r15085964, r15085970);
        double r15085972 = t;
        double r15085973 = r15085971 - r15085972;
        return r15085973;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.4
Target0.3
Herbie0.8
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.4

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

  2. Simplified9.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(z, \log \left(1 - y\right), \log y \cdot x\right) - t}\]

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

    \[\leadsto \mathsf{fma}\left(z, \color{blue}{\log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right)}, \log y \cdot x\right) - t\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.8

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

  7. Final simplification0.8

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))