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

Average Error: 0.4 → 0.4

Time: 3.4s

Precision: binary64

Math

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

↓

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

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 sub-neg 0.4

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

4. Applied distribute-lft-in 0.4

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

5. Simplified 0.4

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

6. Simplified 0.4

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

8. Applied pow1 0.4

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

9. Applied pow1 0.4

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

10. Applied pow-prod-down 0.4

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

11. Applied pow1 0.4

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

12. Applied pow-prod-down 0.4

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

13. Simplified 0.4

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

14. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020179 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))