Average Error: 0.1 → 0.2
Time: 2.2s
Precision: binary64
\[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
↓
\[1 - \mathsf{fma}\left(0.253, x, 0.12 \cdot \left(x \cdot x\right)\right) \]
1 - x \cdot \left(0.253 + x \cdot 0.12\right)
↓
1 - \mathsf{fma}\left(0.253, x, 0.12 \cdot \left(x \cdot x\right)\right)

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))
↓
(FPCore (x) :precision binary64 (- 1.0 (fma 0.253 x (* 0.12 (* x x)))))
double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}
↓
double code(double x) {
	return 1.0 - fma(0.253, x, (0.12 * (x * x)));
}

Error

Bits error versus x

Derivation

  1. Initial program 0.1

    \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]

  3. Taylor expanded in x around 0 0.2

    \[\leadsto \color{blue}{1 - \left(0.12 \cdot {x}^{2} + 0.253 \cdot x\right)} \]

  4. Applied egg-rr0.2

    \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(0.253, x, 0.12 \cdot \left(x \cdot x\right)\right)} \]

  5. Final simplification0.2

    \[\leadsto 1 - \mathsf{fma}\left(0.253, x, 0.12 \cdot \left(x \cdot x\right)\right) \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))