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

Average Error: 2.0 → 1.2

Time: 33.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r650187 = x;
        double r650188 = y;
        double r650189 = z;
        double r650190 = log(r650189);
        double r650191 = r650188 * r650190;
        double r650192 = t;
        double r650193 = 1.0;
        double r650194 = r650192 - r650193;
        double r650195 = a;
        double r650196 = log(r650195);
        double r650197 = r650194 * r650196;
        double r650198 = r650191 + r650197;
        double r650199 = b;
        double r650200 = r650198 - r650199;
        double r650201 = exp(r650200);
        double r650202 = r650187 * r650201;
        double r650203 = r650202 / r650188;
        return r650203;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r650204 = 1.0;
        double r650205 = a;
        double r650206 = 1.0;
        double r650207 = pow(r650205, r650206);
        double r650208 = pow(r650207, r650206);
        double r650209 = z;
        double r650210 = r650204 / r650209;
        double r650211 = log(r650210);
        double r650212 = y;
        double r650213 = r650211 * r650212;
        double r650214 = r650204 / r650205;
        double r650215 = log(r650214);
        double r650216 = t;
        double r650217 = r650215 * r650216;
        double r650218 = b;
        double r650219 = r650217 + r650218;
        double r650220 = r650213 + r650219;
        double r650221 = exp(r650220);
        double r650222 = r650221 * r650212;
        double r650223 = x;
        double r650224 = r650222 / r650223;
        double r650225 = r650208 * r650224;
        double r650226 = r650204 / r650225;
        return r650226;
}

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	11.2
Herbie	1.2

\[\begin{array}{l} \mathbf{if}\;t \lt -0.88458485041274715:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.22883740731:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Derivation

1. Initial program 2.0

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

2. Taylor expanded around inf 2.0

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

3. Simplified 1.2

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

5. Applied add-cube-cbrt 1.3

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

6. Applied add-sqr-sqrt 1.3

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

7. Applied *-un-lft-identity 1.3

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

8. Applied times-frac 1.3

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

9. Applied unpow-prod-down 1.3

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

10. Applied times-frac 1.3

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

11. Applied associate-*r* 1.3

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

13. Applied clear-num 1.3

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

14. Simplified 6.2

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

15. Taylor expanded around inf 1.2

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

16. Final simplification 1.2

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

Reproduce

herbie shell --seed 2020056 
(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))