Average Error: 9.3 → 0.3
Time: 42.9s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(x, \log y, z \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(x, \log y, z \cdot \left(\log 1 - \mathsf{fma}\left(1, 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 r331992 = x;
        double r331993 = y;
        double r331994 = log(r331993);
        double r331995 = r331992 * r331994;
        double r331996 = z;
        double r331997 = 1.0;
        double r331998 = r331997 - r331993;
        double r331999 = log(r331998);
        double r332000 = r331996 * r331999;
        double r332001 = r331995 + r332000;
        double r332002 = t;
        double r332003 = r332001 - r332002;
        return r332003;
}
↓
double f(double x, double y, double z, double t) {
        double r332004 = x;
        double r332005 = y;
        double r332006 = log(r332005);
        double r332007 = z;
        double r332008 = 1.0;
        double r332009 = log(r332008);
        double r332010 = 0.5;
        double r332011 = 2.0;
        double r332012 = pow(r332005, r332011);
        double r332013 = pow(r332008, r332011);
        double r332014 = r332012 / r332013;
        double r332015 = r332010 * r332014;
        double r332016 = fma(r332008, r332005, r332015);
        double r332017 = r332009 - r332016;
        double r332018 = r332007 * r332017;
        double r332019 = fma(r332004, r332006, r332018);
        double r332020 = t;
        double r332021 = r332019 - r332020;
        return r332021;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.3
Target0.2
Herbie0.3
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.3

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

  2. Simplified9.3

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019199 +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))