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

Average Error: 19.3 → 0.7

Time: 5.3s

Precision: 64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Target

Original	19.3
Target	0.1
Herbie	0.7

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

Derivation

1. Initial program 19.3

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

3. Applied add-sqr-sqrt 19.3

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

4. Applied associate-*l* 19.3

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

5. Applied add-cube-cbrt 19.6

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

6. Applied associate-*l* 19.6

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

7. Applied times-frac 10.4

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

8. Simplified 10.4

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

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. Using strategy rm

14. Applied add-sqr-sqrt 0.6

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

16. Applied add-cube-cbrt 0.7

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

17. Applied cbrt-prod 0.7

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

18. Final simplification 0.7

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

Reproduce

herbie shell --seed 2020066 +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))))