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

Average Error: 1.9 → 1.0

Time: 17.5s

Precision: 64

Math

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

↓

\[\frac{x}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\frac{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{\left(2 \cdot 1\right)}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}} \cdot \frac{\sqrt{1}}{\sqrt[3]{y}}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r695100 = x;
        double r695101 = y;
        double r695102 = z;
        double r695103 = log(r695102);
        double r695104 = r695101 * r695103;
        double r695105 = t;
        double r695106 = 1.0;
        double r695107 = r695105 - r695106;
        double r695108 = a;
        double r695109 = log(r695108);
        double r695110 = r695107 * r695109;
        double r695111 = r695104 + r695110;
        double r695112 = b;
        double r695113 = r695111 - r695112;
        double r695114 = exp(r695113);
        double r695115 = r695100 * r695114;
        double r695116 = r695115 / r695101;
        return r695116;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r695117 = x;
        double r695118 = y;
        double r695119 = cbrt(r695118);
        double r695120 = r695119 * r695119;
        double r695121 = 1.0;
        double r695122 = a;
        double r695123 = r695121 / r695122;
        double r695124 = sqrt(r695123);
        double r695125 = 2.0;
        double r695126 = 1.0;
        double r695127 = r695125 * r695126;
        double r695128 = pow(r695124, r695127);
        double r695129 = z;
        double r695130 = r695121 / r695129;
        double r695131 = log(r695130);
        double r695132 = r695118 * r695131;
        double r695133 = log(r695123);
        double r695134 = t;
        double r695135 = r695133 * r695134;
        double r695136 = b;
        double r695137 = r695135 + r695136;
        double r695138 = r695132 + r695137;
        double r695139 = exp(r695138);
        double r695140 = cbrt(r695139);
        double r695141 = r695140 * r695140;
        double r695142 = r695128 / r695141;
        double r695143 = r695142 / r695140;
        double r695144 = r695120 / r695143;
        double r695145 = r695117 / r695144;
        double r695146 = sqrt(r695121);
        double r695147 = r695146 / r695119;
        double r695148 = r695145 * r695147;
        return r695148;
}

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	11.0
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

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

3. Simplified 1.3

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

5. Applied div-inv 1.3

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

7. Applied add-cube-cbrt 1.3

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

8. Applied add-sqr-sqrt 1.3

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

9. Applied unpow-prod-down 1.3

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

10. Applied times-frac 1.3

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

11. Applied associate-*r* 1.3

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

13. Applied add-cube-cbrt 1.4

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

14. Applied add-sqr-sqrt 1.4

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

15. Applied times-frac 1.4

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

16. Applied associate-*r* 1.4

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

17. Simplified 1.0

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

18. Final simplification 1.0

    \[\leadsto \frac{x}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\frac{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{\left(2 \cdot 1\right)}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}} \cdot \frac{\sqrt{1}}{\sqrt[3]{y}}\]

Reproduce

herbie shell --seed 2019362 
(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 -0.8845848504127471) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ 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))