Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Average Error: 1.9 → 1.2

Time: 14.7s

Precision: 64

Math

\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

↓

\[\left(\frac{{\left(\frac{1}{\left(\sqrt[3]{\sqrt{a}} \cdot \sqrt[3]{\sqrt{a}}\right) \cdot \sqrt[3]{a}}\right)}^{1}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot x\right) \cdot \frac{\frac{{\left(\frac{1}{\sqrt[3]{a}}\right)}^{1}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r423926 = x;
        double r423927 = y;
        double r423928 = z;
        double r423929 = log(r423928);
        double r423930 = r423927 * r423929;
        double r423931 = t;
        double r423932 = 1.0;
        double r423933 = r423931 - r423932;
        double r423934 = a;
        double r423935 = log(r423934);
        double r423936 = r423933 * r423935;
        double r423937 = r423930 + r423936;
        double r423938 = b;
        double r423939 = r423937 - r423938;
        double r423940 = exp(r423939);
        double r423941 = r423926 * r423940;
        double r423942 = r423941 / r423927;
        return r423942;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r423943 = 1.0;
        double r423944 = a;
        double r423945 = sqrt(r423944);
        double r423946 = cbrt(r423945);
        double r423947 = r423946 * r423946;
        double r423948 = cbrt(r423944);
        double r423949 = r423947 * r423948;
        double r423950 = r423943 / r423949;
        double r423951 = 1.0;
        double r423952 = pow(r423950, r423951);
        double r423953 = y;
        double r423954 = z;
        double r423955 = r423943 / r423954;
        double r423956 = log(r423955);
        double r423957 = r423943 / r423944;
        double r423958 = log(r423957);
        double r423959 = t;
        double r423960 = b;
        double r423961 = fma(r423958, r423959, r423960);
        double r423962 = fma(r423953, r423956, r423961);
        double r423963 = exp(r423962);
        double r423964 = cbrt(r423963);
        double r423965 = r423964 * r423964;
        double r423966 = r423952 / r423965;
        double r423967 = x;
        double r423968 = r423966 * r423967;
        double r423969 = r423943 / r423948;
        double r423970 = pow(r423969, r423951);
        double r423971 = r423970 / r423964;
        double r423972 = r423971 / r423953;
        double r423973 = r423968 * r423972;
        return r423973;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	1.9
Target	11.4
Herbie	1.2

\[\begin{array}{l} \mathbf{if}\;t \lt -0.8845848504127471478852839936735108494759:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.228837407310493290424346923828125:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Derivation

1. Initial program 1.9

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

2. Taylor expanded around inf 2.0

   \[\leadsto \frac{x \cdot \color{blue}{e^{1 \cdot \log \left(\frac{1}{a}\right) - \left(y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)\right)}}}{y}\]

3. Simplified 1.2

   \[\leadsto \frac{x \cdot \color{blue}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]
4. Using strategy rm

5. Applied *-un-lft-identity 1.2

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

6. Applied times-frac 1.3

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

7. Simplified 1.3

   \[\leadsto \color{blue}{x} \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{y}\]
8. Using strategy rm

9. Applied *-un-lft-identity 1.3

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

10. Applied add-cube-cbrt 1.3

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

11. Applied add-cube-cbrt 1.4

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

12. Applied *-un-lft-identity 1.4

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

13. Applied times-frac 1.4

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

14. Applied unpow-prod-down 1.4

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

15. Applied times-frac 1.4

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

16. Applied times-frac 1.4

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

17. Applied associate-*r* 1.2

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

18. Simplified 1.2

    \[\leadsto \color{blue}{\left(\frac{{\left(\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)}^{1}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot x\right)} \cdot \frac{\frac{{\left(\frac{1}{\sqrt[3]{a}}\right)}^{1}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]
19. Using strategy rm

20. Applied add-sqr-sqrt 1.2

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

21. Applied cbrt-prod 1.2

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

22. Final simplification 1.2

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

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))