Average Error: 9.0 → 0.6
Time: 28.4s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(z, \mathsf{fma}\left(\frac{-1}{2}, \frac{y}{1.0} \cdot \frac{y}{1.0}, \log 1.0 - y \cdot 1.0\right), \left(x \cdot \sqrt[3]{\log y}\right) \cdot \sqrt[3]{\log y \cdot \log y} - t\right)\]
\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t
↓
\mathsf{fma}\left(z, \mathsf{fma}\left(\frac{-1}{2}, \frac{y}{1.0} \cdot \frac{y}{1.0}, \log 1.0 - y \cdot 1.0\right), \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 r18942992 = x;
        double r18942993 = y;
        double r18942994 = log(r18942993);
        double r18942995 = r18942992 * r18942994;
        double r18942996 = z;
        double r18942997 = 1.0;
        double r18942998 = r18942997 - r18942993;
        double r18942999 = log(r18942998);
        double r18943000 = r18942996 * r18942999;
        double r18943001 = r18942995 + r18943000;
        double r18943002 = t;
        double r18943003 = r18943001 - r18943002;
        return r18943003;
}
↓
double f(double x, double y, double z, double t) {
        double r18943004 = z;
        double r18943005 = -0.5;
        double r18943006 = y;
        double r18943007 = 1.0;
        double r18943008 = r18943006 / r18943007;
        double r18943009 = r18943008 * r18943008;
        double r18943010 = log(r18943007);
        double r18943011 = r18943006 * r18943007;
        double r18943012 = r18943010 - r18943011;
        double r18943013 = fma(r18943005, r18943009, r18943012);
        double r18943014 = x;
        double r18943015 = log(r18943006);
        double r18943016 = cbrt(r18943015);
        double r18943017 = r18943014 * r18943016;
        double r18943018 = r18943015 * r18943015;
        double r18943019 = cbrt(r18943018);
        double r18943020 = r18943017 * r18943019;
        double r18943021 = t;
        double r18943022 = r18943020 - r18943021;
        double r18943023 = fma(r18943004, r18943013, r18943022);
        return r18943023;
}

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

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  6. Applied add-cube-cbrt0.8

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

  7. Applied associate-*l*0.8

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

  9. Applied cbrt-unprod0.6

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

  10. Final simplification0.6

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

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))