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

Average Error: 2.0 → 0.7

Time: 34.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le 2.425819830564266198213947300688724832234 \cdot 10^{164}:\\ \;\;\;\;{\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r287774 = x;
        double r287775 = y;
        double r287776 = z;
        double r287777 = log(r287776);
        double r287778 = r287775 * r287777;
        double r287779 = t;
        double r287780 = 1.0;
        double r287781 = r287779 - r287780;
        double r287782 = a;
        double r287783 = log(r287782);
        double r287784 = r287781 * r287783;
        double r287785 = r287778 + r287784;
        double r287786 = b;
        double r287787 = r287785 - r287786;
        double r287788 = exp(r287787);
        double r287789 = r287774 * r287788;
        double r287790 = r287789 / r287775;
        return r287790;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r287791 = a;
        double r287792 = 2.425819830564266e+164;
        bool r287793 = r287791 <= r287792;
        double r287794 = 1.0;
        double r287795 = 1.0;
        double r287796 = pow(r287791, r287795);
        double r287797 = r287794 / r287796;
        double r287798 = pow(r287797, r287795);
        double r287799 = x;
        double r287800 = z;
        double r287801 = r287794 / r287800;
        double r287802 = log(r287801);
        double r287803 = y;
        double r287804 = r287802 * r287803;
        double r287805 = r287794 / r287791;
        double r287806 = log(r287805);
        double r287807 = t;
        double r287808 = r287806 * r287807;
        double r287809 = b;
        double r287810 = r287808 + r287809;
        double r287811 = r287804 + r287810;
        double r287812 = exp(r287811);
        double r287813 = r287812 * r287803;
        double r287814 = r287799 / r287813;
        double r287815 = r287798 * r287814;
        double r287816 = r287803 * r287802;
        double r287817 = r287816 + r287810;
        double r287818 = exp(r287817);
        double r287819 = sqrt(r287818);
        double r287820 = r287799 / r287819;
        double r287821 = pow(r287805, r287795);
        double r287822 = r287821 / r287819;
        double r287823 = r287822 / r287803;
        double r287824 = r287820 * r287823;
        double r287825 = r287793 ? r287815 : r287824;
        return r287825;
}

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	2.0
Target	11.4
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if a < 2.425819830564266e+164

   1. Initial program 1.2

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

   2. Taylor expanded around inf 1.2

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

   3. Simplified 9.3

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

   5. Applied *-un-lft-identity 9.3

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

   6. Applied add-sqr-sqrt 9.3

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

   7. Applied times-frac 0.5

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

   8. Applied times-frac 2.0

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

   9. Simplified 2.0

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

   10. Taylor expanded around inf 0.9

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

   if 2.425819830564266e+164 < a

   1. Initial program 4.3

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

   2. Taylor expanded around inf 4.2

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

   3. Simplified 3.3

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

   5. Applied *-un-lft-identity 3.3

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

   6. Applied add-sqr-sqrt 3.3

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

   7. Applied times-frac 3.3

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

   8. Applied times-frac 0.1

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

   9. Simplified 0.1

      \[\leadsto \color{blue}{\frac{x}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}} \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 2.425819830564266198213947300688724832234 \cdot 10^{164}:\\ \;\;\;\;{\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(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 -0.88458485041274715) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.22883740731) (/ (* (/ 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))