Average Error: 9.0 → 0.6
Time: 30.9s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{1.0} \cdot \frac{y}{1.0}, \frac{-1}{2}, \log 1.0 - y \cdot 1.0\right), z, \left(x \cdot \sqrt[3]{\log y}\right) \cdot \sqrt[3]{\log y \cdot \log y} - t\right)\]
\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t
↓
\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{1.0} \cdot \frac{y}{1.0}, \frac{-1}{2}, \log 1.0 - y \cdot 1.0\right), z, \left(x \cdot \sqrt[3]{\log y}\right) \cdot \sqrt[3]{\log y \cdot \log y} - t\right)
double f(double x, double y, double z, double t) {
        double r19866983 = x;
        double r19866984 = y;
        double r19866985 = log(r19866984);
        double r19866986 = r19866983 * r19866985;
        double r19866987 = z;
        double r19866988 = 1.0;
        double r19866989 = r19866988 - r19866984;
        double r19866990 = log(r19866989);
        double r19866991 = r19866987 * r19866990;
        double r19866992 = r19866986 + r19866991;
        double r19866993 = t;
        double r19866994 = r19866992 - r19866993;
        return r19866994;
}
↓
double f(double x, double y, double z, double t) {
        double r19866995 = y;
        double r19866996 = 1.0;
        double r19866997 = r19866995 / r19866996;
        double r19866998 = r19866997 * r19866997;
        double r19866999 = -0.5;
        double r19867000 = log(r19866996);
        double r19867001 = r19866995 * r19866996;
        double r19867002 = r19867000 - r19867001;
        double r19867003 = fma(r19866998, r19866999, r19867002);
        double r19867004 = z;
        double r19867005 = x;
        double r19867006 = log(r19866995);
        double r19867007 = cbrt(r19867006);
        double r19867008 = r19867005 * r19867007;
        double r19867009 = r19867006 * r19867006;
        double r19867010 = cbrt(r19867009);
        double r19867011 = r19867008 * r19867010;
        double r19867012 = t;
        double r19867013 = r19867011 - r19867012;
        double r19867014 = fma(r19867003, r19867004, r19867013);
        return r19867014;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.0
Target0.3
Herbie0.6
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{\frac{1}{3}}{1.0 \cdot \left(1.0 \cdot 1.0\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.0

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

  2. Simplified9.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(1.0 - y\right), z, \log y \cdot x - t\right)}\]

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  6. Applied add-cube-cbrt0.8

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

  7. Applied associate-*l*0.8

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

  9. Applied cbrt-unprod0.6

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

  10. Final simplification0.6

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

Reproduce

herbie shell --seed 2019163 +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) (* (/ 1/3 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

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