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

Average Error: 0.4 → 0.4

Time: 2.6s

Precision: 64

Math

\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]

↓

\[\frac{3}{x \cdot -1} \cdot \frac{\sqrt{x} \cdot \left(-1\right)}{9} + \left(3 \cdot \sqrt{x}\right) \cdot \left(y - 1\right)\]

C

double code(double x, double y) {
	return ((3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0));
}

↓

double code(double x, double y) {
	return (((3.0 / (x * -1.0)) * ((sqrt(x) * -1.0) / 9.0)) + ((3.0 * sqrt(x)) * (y - 1.0)));
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.4
Target	0.4
Herbie	0.4

\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

Derivation

1. Initial program 0.4

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
2. Using strategy rm

3. Applied *-commutative 0.4

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{\color{blue}{9 \cdot x}}\right) - 1\right)\]

4. Applied associate-/r* 0.4

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

6. Applied +-commutative 0.4

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(\frac{\frac{1}{9}}{x} + y\right)} - 1\right)\]

7. Applied associate--l+ 0.4

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\frac{\frac{1}{9}}{x} + \left(y - 1\right)\right)}\]

8. Applied distribute-lft-in 0.4

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

10. Applied frac-2neg 0.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \frac{\color{blue}{\frac{-1}{-9}}}{x} + \left(3 \cdot \sqrt{x}\right) \cdot \left(y - 1\right)\]

11. Applied associate-/l/ 0.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{-1}{x \cdot \left(-9\right)}} + \left(3 \cdot \sqrt{x}\right) \cdot \left(y - 1\right)\]

12. Applied associate-*r/ 0.4

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

14. Applied neg-mul-1 0.4

    \[\leadsto \frac{\left(3 \cdot \sqrt{x}\right) \cdot \left(-1\right)}{x \cdot \color{blue}{\left(-1 \cdot 9\right)}} + \left(3 \cdot \sqrt{x}\right) \cdot \left(y - 1\right)\]

15. Applied associate-*r* 0.4

    \[\leadsto \frac{\left(3 \cdot \sqrt{x}\right) \cdot \left(-1\right)}{\color{blue}{\left(x \cdot -1\right) \cdot 9}} + \left(3 \cdot \sqrt{x}\right) \cdot \left(y - 1\right)\]

16. Applied associate-*l* 0.4

    \[\leadsto \frac{\color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(-1\right)\right)}}{\left(x \cdot -1\right) \cdot 9} + \left(3 \cdot \sqrt{x}\right) \cdot \left(y - 1\right)\]

17. Applied times-frac 0.4

    \[\leadsto \color{blue}{\frac{3}{x \cdot -1} \cdot \frac{\sqrt{x} \cdot \left(-1\right)}{9}} + \left(3 \cdot \sqrt{x}\right) \cdot \left(y - 1\right)\]

18. Final simplification 0.4

    \[\leadsto \frac{3}{x \cdot -1} \cdot \frac{\sqrt{x} \cdot \left(-1\right)}{9} + \left(3 \cdot \sqrt{x}\right) \cdot \left(y - 1\right)\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3 (+ (* y (sqrt x)) (* (- (/ 1 (* x 9)) 1) (sqrt x))))

  (* (* 3 (sqrt x)) (- (+ y (/ 1 (* x 9))) 1)))