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

Average Error: 3.9 → 2.6

Time: 9.2s

Precision: 64

Math

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

↓

\[\frac{x}{x + y \cdot e^{2 \cdot \sqrt[3]{{\left(\mathsf{fma}\left(-\left(b - c\right), \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}^{3}}}}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r372820 = x;
        double r372821 = y;
        double r372822 = 2.0;
        double r372823 = z;
        double r372824 = t;
        double r372825 = a;
        double r372826 = r372824 + r372825;
        double r372827 = sqrt(r372826);
        double r372828 = r372823 * r372827;
        double r372829 = r372828 / r372824;
        double r372830 = b;
        double r372831 = c;
        double r372832 = r372830 - r372831;
        double r372833 = 5.0;
        double r372834 = 6.0;
        double r372835 = r372833 / r372834;
        double r372836 = r372825 + r372835;
        double r372837 = 3.0;
        double r372838 = r372824 * r372837;
        double r372839 = r372822 / r372838;
        double r372840 = r372836 - r372839;
        double r372841 = r372832 * r372840;
        double r372842 = r372829 - r372841;
        double r372843 = r372822 * r372842;
        double r372844 = exp(r372843);
        double r372845 = r372821 * r372844;
        double r372846 = r372820 + r372845;
        double r372847 = r372820 / r372846;
        return r372847;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r372848 = x;
        double r372849 = y;
        double r372850 = 2.0;
        double r372851 = b;
        double r372852 = c;
        double r372853 = r372851 - r372852;
        double r372854 = -r372853;
        double r372855 = a;
        double r372856 = 5.0;
        double r372857 = 6.0;
        double r372858 = r372856 / r372857;
        double r372859 = r372855 + r372858;
        double r372860 = t;
        double r372861 = 3.0;
        double r372862 = r372860 * r372861;
        double r372863 = r372850 / r372862;
        double r372864 = r372859 - r372863;
        double r372865 = z;
        double r372866 = r372860 + r372855;
        double r372867 = sqrt(r372866);
        double r372868 = r372865 * r372867;
        double r372869 = r372868 / r372860;
        double r372870 = fma(r372854, r372864, r372869);
        double r372871 = 3.0;
        double r372872 = pow(r372870, r372871);
        double r372873 = cbrt(r372872);
        double r372874 = r372850 * r372873;
        double r372875 = exp(r372874);
        double r372876 = r372849 * r372875;
        double r372877 = r372848 + r372876;
        double r372878 = r372848 / r372877;
        return r372878;
}

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

Bits error versus c

Target

Original	3.9
Target	3.1
Herbie	2.6

\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581057561884576920117070548 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333333703407674875052180141211 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547088010424937268931048836 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

1. Initial program 3.9

   \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
2. Using strategy rm

3. Applied add-cbrt-cube 3.9

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot \color{blue}{\sqrt[3]{\left(3 \cdot 3\right) \cdot 3}}}\right)\right)}}\]

4. Applied add-cbrt-cube 7.0

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{\color{blue}{\sqrt[3]{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\left(3 \cdot 3\right) \cdot 3}}\right)\right)}}\]

5. Applied cbrt-unprod 7.0

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{\color{blue}{\sqrt[3]{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\left(3 \cdot 3\right) \cdot 3\right)}}}\right)\right)}}\]

6. Applied add-cbrt-cube 7.0

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{\color{blue}{\sqrt[3]{\left(2 \cdot 2\right) \cdot 2}}}{\sqrt[3]{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\left(3 \cdot 3\right) \cdot 3\right)}}\right)\right)}}\]

7. Applied cbrt-undiv 7.1

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \color{blue}{\sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\left(3 \cdot 3\right) \cdot 3\right)}}}\right)\right)}}\]

8. Simplified 7.1

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \sqrt[3]{\color{blue}{{\left(\frac{2}{t \cdot 3}\right)}^{3}}}\right)\right)}}\]
9. Using strategy rm

10. Applied add-cube-cbrt 7.1

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \sqrt[3]{{\left(\frac{2}{t \cdot 3}\right)}^{3}}\right)\right)}}\]

11. Applied times-frac 5.9

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \sqrt[3]{{\left(\frac{2}{t \cdot 3}\right)}^{3}}\right)\right)}}\]

12. Applied fma-neg 5.2

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \sqrt[3]{{\left(\frac{2}{t \cdot 3}\right)}^{3}}\right)\right)}}}\]

13. Simplified 1.9

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, \color{blue}{\left(-\mathsf{fma}\left(1, a + \frac{5}{6}, -\frac{2}{t \cdot 3}\right)\right) \cdot \left(b - c\right)}\right)}}\]
14. Using strategy rm

15. Applied add-cbrt-cube 1.9

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, \left(-\mathsf{fma}\left(1, a + \frac{5}{6}, -\frac{2}{t \cdot 3}\right)\right) \cdot \left(b - c\right)\right) \cdot \mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, \left(-\mathsf{fma}\left(1, a + \frac{5}{6}, -\frac{2}{t \cdot 3}\right)\right) \cdot \left(b - c\right)\right)\right) \cdot \mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, \left(-\mathsf{fma}\left(1, a + \frac{5}{6}, -\frac{2}{t \cdot 3}\right)\right) \cdot \left(b - c\right)\right)}}}}\]

16. Simplified 2.6

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(-\left(b - c\right), \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}^{3}}}}}\]

17. Final simplification 2.6

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \sqrt[3]{{\left(\mathsf{fma}\left(-\left(b - c\right), \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}^{3}}}}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2 (/ (- (* (* z (sqrt (+ t a))) (* (* 3 t) (- a (/ 5 6)))) (* (- (* (+ (/ 5 6) a) (* 3 t)) 2) (* (- a (/ 5 6)) (* (- b c) t)))) (* (* (* t t) 3) (- a (/ 5 6))))))))) (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3))))))))))))

  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))