Average Error: 0.0 → 0.0
Time: 4.6s
Precision: binary64
\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)\]
↓
\[\frac{1.6316775383 + x \cdot 0.1913510371}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} + x \cdot -0.70711\]
0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)
↓
\frac{1.6316775383 + x \cdot 0.1913510371}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} + x \cdot -0.70711
(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))
↓
(FPCore (x)
 :precision binary64
 (+
  (/
   (+ 1.6316775383 (* x 0.1913510371))
   (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
  (* x -0.70711)))
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 ((1.6316775383 + (x * 0.1913510371)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) + (x * -0.70711);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  3. Applied sub-neg_binary64_31460.0

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

  4. Applied distribute-rgt-in_binary64_31850.0

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

  5. Simplified0.0

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

  7. Applied associate-*l/_binary64_32040.0

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

  8. Taylor expanded around 0 0.0

    \[\leadsto \frac{\color{blue}{0.1913510371 \cdot x + 1.6316775383}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} + x \cdot -0.70711\]

  9. Simplified0.0

    \[\leadsto \frac{\color{blue}{1.6316775383 + x \cdot 0.1913510371}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} + x \cdot -0.70711\]

  10. Final simplification0.0

    \[\leadsto \frac{1.6316775383 + x \cdot 0.1913510371}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} + x \cdot -0.70711\]

Reproduce

herbie shell --seed 2020231 
(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)))