Average Error: 9.2 → 0.3
Time: 8.6s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \left(\log 1 - \left(1 \cdot 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 r343730 = x;
        double r343731 = y;
        double r343732 = log(r343731);
        double r343733 = r343730 * r343732;
        double r343734 = z;
        double r343735 = 1.0;
        double r343736 = r343735 - r343731;
        double r343737 = log(r343736);
        double r343738 = r343734 * r343737;
        double r343739 = r343733 + r343738;
        double r343740 = t;
        double r343741 = r343739 - r343740;
        return r343741;
}
↓
double f(double x, double y, double z, double t) {
        double r343742 = x;
        double r343743 = 2.0;
        double r343744 = y;
        double r343745 = cbrt(r343744);
        double r343746 = log(r343745);
        double r343747 = r343743 * r343746;
        double r343748 = r343742 * r343746;
        double r343749 = fma(r343742, r343747, r343748);
        double r343750 = z;
        double r343751 = 1.0;
        double r343752 = log(r343751);
        double r343753 = r343751 * r343744;
        double r343754 = 0.5;
        double r343755 = pow(r343744, r343743);
        double r343756 = pow(r343751, r343743);
        double r343757 = r343755 / r343756;
        double r343758 = r343754 * r343757;
        double r343759 = r343753 + r343758;
        double r343760 = r343752 - r343759;
        double r343761 = r343750 * r343760;
        double r343762 = r343749 + r343761;
        double r343763 = t;
        double r343764 = r343762 - r343763;
        return r343764;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.2
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 9.2

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

  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. Using strategy rm

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.4

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

  6. Applied distribute-lft-in0.4

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

  7. Simplified0.4

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

  9. Applied fma-def0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020027 +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))