Average Error: 9.0 → 0.3
Time: 11.1s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log y, x, z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log y, x, z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)
double f(double x, double y, double z, double t) {
        double r387731 = x;
        double r387732 = y;
        double r387733 = log(r387732);
        double r387734 = r387731 * r387733;
        double r387735 = z;
        double r387736 = 1.0;
        double r387737 = r387736 - r387732;
        double r387738 = log(r387737);
        double r387739 = r387735 * r387738;
        double r387740 = r387734 + r387739;
        double r387741 = t;
        double r387742 = r387740 - r387741;
        return r387742;
}
↓
double f(double x, double y, double z, double t) {
        double r387743 = y;
        double r387744 = log(r387743);
        double r387745 = x;
        double r387746 = z;
        double r387747 = 1.0;
        double r387748 = log(r387747);
        double r387749 = r387747 * r387743;
        double r387750 = 0.5;
        double r387751 = 2.0;
        double r387752 = pow(r387743, r387751);
        double r387753 = pow(r387747, r387751);
        double r387754 = r387752 / r387753;
        double r387755 = r387750 * r387754;
        double r387756 = r387749 + r387755;
        double r387757 = r387748 - r387756;
        double r387758 = r387746 * r387757;
        double r387759 = t;
        double r387760 = r387758 - r387759;
        double r387761 = fma(r387744, r387745, r387760);
        return r387761;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.0
Target0.3
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 9.0

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

  2. Simplified9.0

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

  3. Taylor expanded around 0 0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2020033 +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) (* (/ 0.3333333333333333 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1 y)))) t))