Average Error: 0.0 → 0.0
Time: 3.5s
Precision: binary64
\[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
↓
\[\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
\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)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\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) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) 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) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - 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. Applied *-un-lft-identity_binary640.0

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

  3. Applied associate-/r*_binary640.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022077 
(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))