Average Error: 0.1 → 0.1
Time: 7.8s
Precision: binary64
\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
↓
\[0.70711 \cdot \left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} - x\right) \]
0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)
↓
0.70711 \cdot \left(\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\right)

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))
↓
(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (fma x 0.27061 2.30753) (fma x (fma x 0.04481 0.99229) 1.0)) x)))
double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}
↓
double code(double x) {
	return 0.70711 * ((fma(x, 0.27061, 2.30753) / fma(x, fma(x, 0.04481, 0.99229), 1.0)) - x);
}

Error

Bits error versus x

Derivation

  1. Initial program 0.1

    \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

  2. Applied *-un-lft-identity_binary640.1

    \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{\color{blue}{1 \cdot \left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)}} - x\right) \]

  3. Applied *-un-lft-identity_binary640.1

    \[\leadsto 0.70711 \cdot \left(\frac{\color{blue}{1 \cdot \left(2.30753 + x \cdot 0.27061\right)}}{1 \cdot \left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)} - x\right) \]

  4. Applied times-frac_binary640.1

    \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{1}{1} \cdot \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} - x\right) \]

  5. Simplified0.1

    \[\leadsto 0.70711 \cdot \left(\color{blue}{1} \cdot \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

  6. Simplified0.1

    \[\leadsto 0.70711 \cdot \left(1 \cdot \color{blue}{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} - x\right) \]

  7. Final simplification0.1

    \[\leadsto 0.70711 \cdot \left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} - x\right) \]

Reproduce

herbie shell --seed 2021280 
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))