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

Average Error: 3.9 → 2.6

Time: 9.3s

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 r359598 = x;
        double r359599 = y;
        double r359600 = 2.0;
        double r359601 = z;
        double r359602 = t;
        double r359603 = a;
        double r359604 = r359602 + r359603;
        double r359605 = sqrt(r359604);
        double r359606 = r359601 * r359605;
        double r359607 = r359606 / r359602;
        double r359608 = b;
        double r359609 = c;
        double r359610 = r359608 - r359609;
        double r359611 = 5.0;
        double r359612 = 6.0;
        double r359613 = r359611 / r359612;
        double r359614 = r359603 + r359613;
        double r359615 = 3.0;
        double r359616 = r359602 * r359615;
        double r359617 = r359600 / r359616;
        double r359618 = r359614 - r359617;
        double r359619 = r359610 * r359618;
        double r359620 = r359607 - r359619;
        double r359621 = r359600 * r359620;
        double r359622 = exp(r359621);
        double r359623 = r359599 * r359622;
        double r359624 = r359598 + r359623;
        double r359625 = r359598 / r359624;
        return r359625;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r359626 = x;
        double r359627 = y;
        double r359628 = 2.0;
        double r359629 = b;
        double r359630 = c;
        double r359631 = r359629 - r359630;
        double r359632 = -r359631;
        double r359633 = a;
        double r359634 = 5.0;
        double r359635 = 6.0;
        double r359636 = r359634 / r359635;
        double r359637 = r359633 + r359636;
        double r359638 = t;
        double r359639 = 3.0;
        double r359640 = r359638 * r359639;
        double r359641 = r359628 / r359640;
        double r359642 = r359637 - r359641;
        double r359643 = z;
        double r359644 = r359638 + r359633;
        double r359645 = sqrt(r359644);
        double r359646 = r359643 * r359645;
        double r359647 = r359646 / r359638;
        double r359648 = fma(r359632, r359642, r359647);
        double r359649 = 3.0;
        double r359650 = pow(r359648, r359649);
        double r359651 = cbrt(r359650);
        double r359652 = r359628 * r359651;
        double r359653 = exp(r359652);
        double r359654 = r359627 * r359653;
        double r359655 = r359626 + r359654;
        double r359656 = r359626 / r359655;
        return r359656;
}

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)))))))))))