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

Average Error: 9.8 → 0.3

Time: 7.8s

Precision: binary64

Math

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

↓

\[\left(\left(\left(x \cdot \left(4 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) + x \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2\right)\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - y \cdot \left(z \cdot 1 + y \cdot \left(z \cdot 0.5\right)\right)\right)\right) - t\]

C

double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (x * ((double) log(y)))) + ((double) (z * ((double) log(((double) (1.0 - y)))))))) - t));
}

↓

double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (((double) (((double) (x * ((double) (4.0 * ((double) log(((double) cbrt(((double) cbrt(y)))))))))) + ((double) (x * ((double) (((double) log(((double) cbrt(((double) cbrt(y)))))) * 2.0)))))) + ((double) (x * ((double) log(((double) cbrt(y)))))))) + ((double) (((double) (z * ((double) log(1.0)))) - ((double) (y * ((double) (((double) (z * 1.0)) + ((double) (y * ((double) (z * 0.5)))))))))))) - t));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.8
Target	0.2
Herbie	0.3

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

Derivation

1. Initial program 9.8

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

2. Taylor expanded around 0 0.3

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

3. Simplified 0.3

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

4. Taylor expanded around 0 0.3

   \[\leadsto \left(x \cdot \log y + \color{blue}{\left(\log 1 \cdot z - \left(1 \cdot \left(z \cdot y\right) + 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right)}\right) - t\]

5. Simplified 0.3

   \[\leadsto \left(x \cdot \log y + \color{blue}{\left(z \cdot \log 1 - y \cdot \left(z \cdot 1 + \left(z \cdot 0.5\right) \cdot y\right)\right)}\right) - t\]
6. Using strategy rm

7. Applied add-cube-cbrt 0.3

   \[\leadsto \left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + \left(z \cdot \log 1 - y \cdot \left(z \cdot 1 + \left(z \cdot 0.5\right) \cdot y\right)\right)\right) - t\]

8. Applied log-prod 0.3

   \[\leadsto \left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + \left(z \cdot \log 1 - y \cdot \left(z \cdot 1 + \left(z \cdot 0.5\right) \cdot y\right)\right)\right) - t\]

9. Applied distribute-lft-in 0.3

   \[\leadsto \left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + \left(z \cdot \log 1 - y \cdot \left(z \cdot 1 + \left(z \cdot 0.5\right) \cdot y\right)\right)\right) - t\]

10. Simplified 0.3

    \[\leadsto \left(\left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - y \cdot \left(z \cdot 1 + \left(z \cdot 0.5\right) \cdot y\right)\right)\right) - t\]
11. Using strategy rm

12. Applied add-cube-cbrt 0.3

    \[\leadsto \left(\left(x \cdot \left(2 \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - y \cdot \left(z \cdot 1 + \left(z \cdot 0.5\right) \cdot y\right)\right)\right) - t\]

13. Applied log-prod 0.3

    \[\leadsto \left(\left(x \cdot \left(2 \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - y \cdot \left(z \cdot 1 + \left(z \cdot 0.5\right) \cdot y\right)\right)\right) - t\]

14. Applied distribute-lft-in 0.3

    \[\leadsto \left(\left(x \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + 2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - y \cdot \left(z \cdot 1 + \left(z \cdot 0.5\right) \cdot y\right)\right)\right) - t\]

15. Applied distribute-lft-in 0.3

    \[\leadsto \left(\left(\color{blue}{\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) + x \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - y \cdot \left(z \cdot 1 + \left(z \cdot 0.5\right) \cdot y\right)\right)\right) - t\]

16. Simplified 0.3

    \[\leadsto \left(\left(\left(\color{blue}{x \cdot \left(4 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)} + x \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - y \cdot \left(z \cdot 1 + \left(z \cdot 0.5\right) \cdot y\right)\right)\right) - t\]

17. Final simplification 0.3

    \[\leadsto \left(\left(\left(x \cdot \left(4 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) + x \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2\right)\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - y \cdot \left(z \cdot 1 + y \cdot \left(z \cdot 0.5\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (neg z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))