Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D

Average Error: 0.2 → 0.2

Time: 3.6s

Precision: binary64

Math

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

↓

\[\mathsf{fma}\left(-1, \frac{y}{3 \cdot \sqrt{x}}, 1 - \frac{0.1111111111111111}{x}\right) \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))

↓

(FPCore (x y)
 :precision binary64
 (fma -1.0 (/ y (* 3.0 (sqrt x))) (- 1.0 (/ 0.1111111111111111 x))))

C

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

↓

double code(double x, double y) {
	return fma(-1.0, (y / (3.0 * sqrt(x))), (1.0 - (0.1111111111111111 / x)));
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.2
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.2

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

2. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{y}{3 \cdot \sqrt{x}}, 1 - \frac{0.1111111111111111}{x}\right)} \]

3. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(-1, \frac{y}{3 \cdot \sqrt{x}}, 1 - \frac{0.1111111111111111}{x}\right) \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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