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

Average Error: 20.0 → 0.6

Time: 5.8s

Precision: 64

Math

\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]

↓

\[\left(\frac{\sqrt[3]{x}}{\sqrt[3]{\left|x + y\right|} \cdot \sqrt[3]{\left|x + y\right|}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\left|x + y\right|}}\right) \cdot \left(\frac{\sqrt[3]{x}}{\left|x + y\right|} \cdot \frac{y}{\left(x + y\right) + 1}\right)\]

C

double code(double x, double y) {
	return ((double) (((double) (x * y)) / ((double) (((double) (((double) (x + y)) * ((double) (x + y)))) * ((double) (((double) (x + y)) + 1.0))))));
}

↓

double code(double x, double y) {
	return ((double) (((double) (((double) (((double) cbrt(x)) / ((double) (((double) cbrt(((double) fabs(((double) (x + y)))))) * ((double) cbrt(((double) fabs(((double) (x + y)))))))))) * ((double) (((double) cbrt(x)) / ((double) cbrt(((double) fabs(((double) (x + y)))))))))) * ((double) (((double) (((double) cbrt(x)) / ((double) fabs(((double) (x + y)))))) * ((double) (y / ((double) (((double) (x + y)) + 1.0))))))));
}

Error

Bits error versus x

Bits error versus y

Target

Original	20.0
Target	0.2
Herbie	0.6

\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

1. Initial program 20.0

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
2. Using strategy rm

3. Applied add-sqr-sqrt 20.0

   \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\sqrt{\left(x + y\right) \cdot \left(x + y\right)} \cdot \sqrt{\left(x + y\right) \cdot \left(x + y\right)}\right)} \cdot \left(\left(x + y\right) + 1\right)}\]

4. Applied associate-*l* 20.0

   \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt{\left(x + y\right) \cdot \left(x + y\right)} \cdot \left(\sqrt{\left(x + y\right) \cdot \left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)}}\]

5. Applied add-cube-cbrt 20.3

   \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot y}{\sqrt{\left(x + y\right) \cdot \left(x + y\right)} \cdot \left(\sqrt{\left(x + y\right) \cdot \left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)}\]

6. Applied associate-*l* 20.3

   \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot y\right)}}{\sqrt{\left(x + y\right) \cdot \left(x + y\right)} \cdot \left(\sqrt{\left(x + y\right) \cdot \left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)}\]

7. Applied times-frac 10.7

   \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{\sqrt[3]{x} \cdot y}{\sqrt{\left(x + y\right) \cdot \left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)}}\]

8. Simplified 10.7

   \[\leadsto \color{blue}{\frac{\sqrt[3]{x}}{\frac{\left|x + y\right|}{\sqrt[3]{x}}}} \cdot \frac{\sqrt[3]{x} \cdot y}{\sqrt{\left(x + y\right) \cdot \left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)}\]

9. Simplified 0.7

   \[\leadsto \frac{\sqrt[3]{x}}{\frac{\left|x + y\right|}{\sqrt[3]{x}}} \cdot \color{blue}{\left(\frac{\sqrt[3]{x}}{\left|x + y\right|} \cdot \frac{y}{\left(x + y\right) + 1}\right)}\]
10. Using strategy rm

11. Applied add-cube-cbrt 0.6

    \[\leadsto \frac{\sqrt[3]{x}}{\frac{\color{blue}{\left(\sqrt[3]{\left|x + y\right|} \cdot \sqrt[3]{\left|x + y\right|}\right) \cdot \sqrt[3]{\left|x + y\right|}}}{\sqrt[3]{x}}} \cdot \left(\frac{\sqrt[3]{x}}{\left|x + y\right|} \cdot \frac{y}{\left(x + y\right) + 1}\right)\]

12. Applied associate-/l* 0.6

    \[\leadsto \frac{\sqrt[3]{x}}{\color{blue}{\frac{\sqrt[3]{\left|x + y\right|} \cdot \sqrt[3]{\left|x + y\right|}}{\frac{\sqrt[3]{x}}{\sqrt[3]{\left|x + y\right|}}}}} \cdot \left(\frac{\sqrt[3]{x}}{\left|x + y\right|} \cdot \frac{y}{\left(x + y\right) + 1}\right)\]

13. Applied associate-/r/ 0.6

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{\left|x + y\right|} \cdot \sqrt[3]{\left|x + y\right|}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\left|x + y\right|}}\right)} \cdot \left(\frac{\sqrt[3]{x}}{\left|x + y\right|} \cdot \frac{y}{\left(x + y\right) + 1}\right)\]

14. Final simplification 0.6

    \[\leadsto \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\left|x + y\right|} \cdot \sqrt[3]{\left|x + y\right|}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\left|x + y\right|}}\right) \cdot \left(\frac{\sqrt[3]{x}}{\left|x + y\right|} \cdot \frac{y}{\left(x + y\right) + 1}\right)\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1))))