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

Average Error: 9.8 → 0.4

Time: 7.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r545940 = x;
        double r545941 = y;
        double r545942 = log(r545941);
        double r545943 = r545940 * r545942;
        double r545944 = z;
        double r545945 = 1.0;
        double r545946 = r545945 - r545941;
        double r545947 = log(r545946);
        double r545948 = r545944 * r545947;
        double r545949 = r545943 + r545948;
        double r545950 = t;
        double r545951 = r545949 - r545950;
        return r545951;
}

↓

double f(double x, double y, double z, double t) {
        double r545952 = x;
        double r545953 = y;
        double r545954 = cbrt(r545953);
        double r545955 = r545954 * r545954;
        double r545956 = log(r545955);
        double r545957 = r545952 * r545956;
        double r545958 = z;
        double r545959 = 1.0;
        double r545960 = log(r545959);
        double r545961 = r545953 * r545959;
        double r545962 = r545960 - r545961;
        double r545963 = r545958 * r545962;
        double r545964 = -0.5;
        double r545965 = 2.0;
        double r545966 = pow(r545959, r545965);
        double r545967 = pow(r545953, r545965);
        double r545968 = r545958 * r545967;
        double r545969 = r545966 / r545968;
        double r545970 = r545964 / r545969;
        double r545971 = 1.0;
        double r545972 = 0.6666666666666666;
        double r545973 = pow(r545953, r545972);
        double r545974 = r545971 / r545973;
        double r545975 = -0.3333333333333333;
        double r545976 = pow(r545974, r545975);
        double r545977 = r545971 / r545954;
        double r545978 = pow(r545977, r545975);
        double r545979 = r545976 * r545978;
        double r545980 = log(r545979);
        double r545981 = r545952 * r545980;
        double r545982 = r545970 + r545981;
        double r545983 = r545963 + r545982;
        double r545984 = r545957 + r545983;
        double r545985 = t;
        double r545986 = r545984 - r545985;
        return r545986;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.8
Target	0.3
Herbie	0.4

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.8

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

2. Taylor expanded around 0 0.4

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

4. Applied add-cube-cbrt 0.4

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

5. Applied log-prod 0.4

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

6. Applied distribute-lft-in 0.4

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

7. Applied associate-+l+ 0.4

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

8. Simplified 0.4

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

9. Taylor expanded around inf 0.4

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

11. Applied add-cube-cbrt 0.4

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

12. Applied add-cube-cbrt 0.4

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

13. Applied times-frac 0.4

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

14. Applied unpow-prod-down 0.4

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

15. Simplified 0.4

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

16. Simplified 0.4

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

17. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020047 
(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))