Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B

Average Error: 9.3 → 0.4

Time: 8.1s

Precision: 64

Math

\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

↓

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

C

double f(double x, double y, double z, double t) {
        double r437039 = x;
        double r437040 = y;
        double r437041 = log(r437040);
        double r437042 = r437039 * r437041;
        double r437043 = z;
        double r437044 = 1.0;
        double r437045 = r437044 - r437040;
        double r437046 = log(r437045);
        double r437047 = r437043 * r437046;
        double r437048 = r437042 + r437047;
        double r437049 = t;
        double r437050 = r437048 - r437049;
        return r437050;
}

↓

double f(double x, double y, double z, double t) {
        double r437051 = x;
        double r437052 = y;
        double r437053 = 0.6666666666666666;
        double r437054 = pow(r437052, r437053);
        double r437055 = 1.0;
        double r437056 = pow(r437054, r437055);
        double r437057 = log(r437056);
        double r437058 = r437051 * r437057;
        double r437059 = cbrt(r437052);
        double r437060 = log(r437059);
        double r437061 = r437060 * r437051;
        double r437062 = z;
        double r437063 = 1.0;
        double r437064 = log(r437063);
        double r437065 = r437063 * r437052;
        double r437066 = 0.5;
        double r437067 = 2.0;
        double r437068 = pow(r437052, r437067);
        double r437069 = pow(r437063, r437067);
        double r437070 = r437068 / r437069;
        double r437071 = r437066 * r437070;
        double r437072 = r437065 + r437071;
        double r437073 = r437064 - r437072;
        double r437074 = r437062 * r437073;
        double r437075 = r437061 + r437074;
        double r437076 = r437058 + r437075;
        double r437077 = t;
        double r437078 = r437076 - r437077;
        return r437078;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.3
Target	0.3
Herbie	0.4

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.3

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

2. Taylor expanded around 0 0.4

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}\right) - t\]
3. Using strategy rm

4. Applied add-cube-cbrt 0.4

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

5. Applied log-prod 0.4

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

6. Applied distribute-lft-in 0.4

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

7. Applied associate-+l+ 0.4

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

8. Simplified 0.4

   \[\leadsto \left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \color{blue}{\left(\log \left(\sqrt[3]{y}\right) \cdot x + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)}\right) - t\]
9. Using strategy rm

10. Applied pow1 0.4

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

11. Applied pow1 0.4

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

12. Applied pow-prod-down 0.4

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

13. Simplified 0.4

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

14. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020021 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1 y)))) t))