Average Error: 0.0 → 0.0
Time: 2.6s
Precision: binary64
\[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
↓
\[\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{e^{\mathsf{log1p}\left(x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)\right)}} - x \]
\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x
↓
\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{e^{\mathsf{log1p}\left(x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)\right)}} - x

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))
↓
(FPCore (x)
 :precision binary64
 (- (/ (fma x 0.27061 2.30753) (exp (log1p (* x (fma x 0.04481 0.99229))))) x))
double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}
↓
double code(double x) {
	return (fma(x, 0.27061, 2.30753) / exp(log1p((x * fma(x, 0.04481, 0.99229))))) - x;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} - x} \]

  3. Applied egg-rr0.0

    \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{e^{\mathsf{log1p}\left(x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)\right)}}} - x \]

  4. Final simplification0.0

    \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{e^{\mathsf{log1p}\left(x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)\right)}} - x \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))