Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B

Average Error: 9.4 → 0.4

Time: 8.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r437789 = x;
        double r437790 = y;
        double r437791 = log(r437790);
        double r437792 = r437789 * r437791;
        double r437793 = z;
        double r437794 = 1.0;
        double r437795 = r437794 - r437790;
        double r437796 = log(r437795);
        double r437797 = r437793 * r437796;
        double r437798 = r437792 + r437797;
        double r437799 = t;
        double r437800 = r437798 - r437799;
        return r437800;
}

↓

double f(double x, double y, double z, double t) {
        double r437801 = 1.0;
        double r437802 = log(r437801);
        double r437803 = y;
        double r437804 = 0.5;
        double r437805 = 2.0;
        double r437806 = pow(r437803, r437805);
        double r437807 = pow(r437801, r437805);
        double r437808 = r437806 / r437807;
        double r437809 = r437804 * r437808;
        double r437810 = fma(r437801, r437803, r437809);
        double r437811 = r437802 - r437810;
        double r437812 = z;
        double r437813 = x;
        double r437814 = cbrt(r437803);
        double r437815 = log(r437814);
        double r437816 = r437805 * r437815;
        double r437817 = r437813 * r437816;
        double r437818 = 0.3333333333333333;
        double r437819 = r437813 * r437818;
        double r437820 = log(r437803);
        double r437821 = r437819 * r437820;
        double r437822 = r437817 + r437821;
        double r437823 = t;
        double r437824 = r437822 - r437823;
        double r437825 = fma(r437811, r437812, r437824);
        double r437826 = -r437823;
        double r437827 = 1.0;
        double r437828 = fma(r437826, r437827, r437823);
        double r437829 = r437825 + r437828;
        return r437829;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.4
Target	0.3
Herbie	0.4

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.4

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

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-cbrt 0.9

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

5. Applied add-sqr-sqrt 32.5

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

6. Applied prod-diff 32.5

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

7. Simplified 0.3

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

8. Simplified 0.3

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

10. Applied add-cube-cbrt 0.3

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

11. Applied log-prod 0.4

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

12. Applied distribute-lft-in 0.4

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

13. Simplified 0.4

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

15. Applied pow1/3 0.4

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

16. Applied log-pow 0.4

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

17. Applied associate-*r* 0.4

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

18. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020001 +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))