Average Error: 0.2 → 0.2
Time: 6.0s
Precision: binary64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
↓
\[1 - \mathsf{fma}\left(y, \sqrt{0.1111111111111111 \cdot \frac{1}{x}}, \frac{0.1111111111111111}{x}\right) \]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
1 - \mathsf{fma}\left(y, \sqrt{0.1111111111111111 \cdot \frac{1}{x}}, \frac{0.1111111111111111}{x}\right)

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))
↓
(FPCore (x y)
 :precision binary64
 (-
  1.0
  (fma y (sqrt (* 0.1111111111111111 (/ 1.0 x))) (/ 0.1111111111111111 x))))
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 1.0 - fma(y, sqrt(0.1111111111111111 * (1.0 / x)), (0.1111111111111111 / x));
}

Error

Bits error versus x

Bits error versus y

Target

Original0.2
Target0.2
Herbie0.2
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

Derivation

  1. Initial program 0.2

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

  2. Taylor expanded in y around 0 0.3

    \[\leadsto \color{blue}{1 - \left(0.1111111111111111 \cdot \frac{1}{x} + 0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\right)} \]

  3. Simplified0.3

    \[\leadsto \color{blue}{1 - \mathsf{fma}\left(0.3333333333333333, y \cdot \sqrt{\frac{1}{x}}, \frac{0.1111111111111111}{x}\right)} \]

  4. Taylor expanded in y around 0 0.3

    \[\leadsto 1 - \color{blue}{\left(0.1111111111111111 \cdot \frac{1}{x} + 0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\right)} \]

  5. Simplified0.2

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

  6. Applied div-inv_binary640.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2022077 
(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)))))