Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2

Average Error: 2.1 → 26.9

Time: 38.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r70786 = x;
        double r70787 = y;
        double r70788 = z;
        double r70789 = log(r70788);
        double r70790 = r70787 * r70789;
        double r70791 = t;
        double r70792 = 1.0;
        double r70793 = r70791 - r70792;
        double r70794 = a;
        double r70795 = log(r70794);
        double r70796 = r70793 * r70795;
        double r70797 = r70790 + r70796;
        double r70798 = b;
        double r70799 = r70797 - r70798;
        double r70800 = exp(r70799);
        double r70801 = r70786 * r70800;
        double r70802 = r70801 / r70787;
        return r70802;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r70803 = 1.0;
        double r70804 = a;
        double r70805 = r70803 / r70804;
        double r70806 = 1.0;
        double r70807 = pow(r70805, r70806);
        double r70808 = y;
        double r70809 = r70807 / r70808;
        double r70810 = x;
        double r70811 = r70809 * r70810;
        double r70812 = z;
        double r70813 = r70803 / r70812;
        double r70814 = pow(r70813, r70808);
        double r70815 = b;
        double r70816 = exp(r70815);
        double r70817 = r70814 * r70816;
        double r70818 = t;
        double r70819 = pow(r70804, r70818);
        double r70820 = r70817 / r70819;
        double r70821 = r70811 / r70820;
        return r70821;
}

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

Derivation

1. Split input into 2 regimes

2. if a < 2.018699136318795e+64

   1. Initial program 0.8

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

   2. Taylor expanded around inf 0.8

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

   3. Simplified 0.2

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

   5. Applied div-inv 0.2

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

   7. Applied add-sqr-sqrt 0.2

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

   8. Applied associate-*r* 0.2

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

   if 2.018699136318795e+64 < a

   1. Initial program 3.8

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

   2. Taylor expanded around inf 3.8

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

   3. Simplified 3.0

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

   5. Applied div-inv 3.0

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

   7. Applied add-sqr-sqrt 3.1

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

   8. Applied associate-*r* 3.1

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

   10. Applied sqrt-div 3.1

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

   11. Applied sqrt-div 3.1

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

   12. Applied associate-*r/ 3.1

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

   13. Applied frac-times 3.1

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

   14. Applied associate-*l/ 4.3

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

   15. Simplified 1.3

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

4. Final simplification 26.9

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

Reproduce

herbie shell --seed 2019294 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))