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

Average Error: 1.9 → 1.0

Time: 43.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r18513725 = x;
        double r18513726 = y;
        double r18513727 = z;
        double r18513728 = log(r18513727);
        double r18513729 = r18513726 * r18513728;
        double r18513730 = t;
        double r18513731 = 1.0;
        double r18513732 = r18513730 - r18513731;
        double r18513733 = a;
        double r18513734 = log(r18513733);
        double r18513735 = r18513732 * r18513734;
        double r18513736 = r18513729 + r18513735;
        double r18513737 = b;
        double r18513738 = r18513736 - r18513737;
        double r18513739 = exp(r18513738);
        double r18513740 = r18513725 * r18513739;
        double r18513741 = r18513740 / r18513726;
        return r18513741;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r18513742 = x;
        double r18513743 = y;
        double r18513744 = cbrt(r18513743);
        double r18513745 = 1.0;
        double r18513746 = a;
        double r18513747 = log(r18513746);
        double r18513748 = z;
        double r18513749 = log(r18513748);
        double r18513750 = -r18513749;
        double r18513751 = t;
        double r18513752 = -r18513747;
        double r18513753 = b;
        double r18513754 = fma(r18513751, r18513752, r18513753);
        double r18513755 = fma(r18513743, r18513750, r18513754);
        double r18513756 = fma(r18513745, r18513747, r18513755);
        double r18513757 = -r18513756;
        double r18513758 = exp(r18513757);
        double r18513759 = sqrt(r18513758);
        double r18513760 = r18513744 / r18513759;
        double r18513761 = r18513742 / r18513760;
        double r18513762 = 1.0;
        double r18513763 = cbrt(r18513744);
        double r18513764 = r18513763 * r18513763;
        double r18513765 = r18513763 * r18513764;
        double r18513766 = cbrt(r18513765);
        double r18513767 = r18513766 * r18513764;
        double r18513768 = r18513767 * r18513744;
        double r18513769 = r18513768 / r18513759;
        double r18513770 = r18513762 / r18513769;
        double r18513771 = r18513761 * r18513770;
        return r18513771;
}

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	11.2
Herbie	1.0

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

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 1.9

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

3. Simplified 2.0

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

5. Applied add-sqr-sqrt 2.0

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

6. Applied add-cube-cbrt 2.0

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

7. Applied times-frac 2.0

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

8. Applied *-un-lft-identity 2.0

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

9. Applied times-frac 1.0

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

11. Applied add-cube-cbrt 1.0

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

13. Applied add-cube-cbrt 1.0

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

14. Final simplification 1.0

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))