Average Error: 0.0 → 0.1
Time: 5.8s
Precision: binary64
\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)\]
↓
\[0.70711 \cdot \left(\frac{1}{\sqrt{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} \cdot \frac{2.30753 + x \cdot 0.27061}{\sqrt{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} - x\right)\]
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{1}{\sqrt{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} \cdot \frac{2.30753 + x \cdot 0.27061}{\sqrt{1 + x \cdot \left(0.99229 + x \cdot 0.04481\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
  (-
   (*
    (/ 1.0 (sqrt (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))))
    (/
     (+ 2.30753 (* x 0.27061))
     (sqrt (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))))
   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 * (((1.0 / sqrt(1.0 + (x * (0.99229 + (x * 0.04481))))) * ((2.30753 + (x * 0.27061)) / sqrt(1.0 + (x * (0.99229 + (x * 0.04481)))))) - x);
}

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 add-sqr-sqrt_binary64_31690.1

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

  4. Applied *-un-lft-identity_binary64_31470.1

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

  5. Applied times-frac_binary64_31530.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2021093 
(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)))