Average Error: 9.9 → 0.3
Time: 20.4s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \log \left({y}^{\frac{1}{3}}\right) \cdot x\right) + z \cdot \left(\log 1 - y \cdot \mathsf{fma}\left(\frac{\frac{1}{2}}{1}, \frac{y}{1}, 1\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \log \left({y}^{\frac{1}{3}}\right) \cdot x\right) + z \cdot \left(\log 1 - y \cdot \mathsf{fma}\left(\frac{\frac{1}{2}}{1}, \frac{y}{1}, 1\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r311026 = x;
        double r311027 = y;
        double r311028 = log(r311027);
        double r311029 = r311026 * r311028;
        double r311030 = z;
        double r311031 = 1.0;
        double r311032 = r311031 - r311027;
        double r311033 = log(r311032);
        double r311034 = r311030 * r311033;
        double r311035 = r311029 + r311034;
        double r311036 = t;
        double r311037 = r311035 - r311036;
        return r311037;
}
↓
double f(double x, double y, double z, double t) {
        double r311038 = x;
        double r311039 = 2.0;
        double r311040 = y;
        double r311041 = cbrt(r311040);
        double r311042 = log(r311041);
        double r311043 = r311039 * r311042;
        double r311044 = r311038 * r311043;
        double r311045 = 0.3333333333333333;
        double r311046 = pow(r311040, r311045);
        double r311047 = log(r311046);
        double r311048 = r311047 * r311038;
        double r311049 = r311044 + r311048;
        double r311050 = z;
        double r311051 = 1.0;
        double r311052 = log(r311051);
        double r311053 = 0.5;
        double r311054 = r311053 / r311051;
        double r311055 = r311040 / r311051;
        double r311056 = fma(r311054, r311055, r311051);
        double r311057 = r311040 * r311056;
        double r311058 = r311052 - r311057;
        double r311059 = r311050 * r311058;
        double r311060 = r311049 + r311059;
        double r311061 = t;
        double r311062 = r311060 - r311061;
        return r311062;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.9
Target0.2
Herbie0.3
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.9

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

  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. Simplified0.3

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  11. Applied pow1/30.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(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))