Average Error: 0.0 → 0.1
Time: 4.5s
Precision: 64
\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
↓
\[0.707110000000000016 \cdot \left(\frac{\frac{2.30753 + x \cdot 0.27061000000000002}{\sqrt{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}}}{\sqrt{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}} - x\right)\]
0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)
↓
0.707110000000000016 \cdot \left(\frac{\frac{2.30753 + x \cdot 0.27061000000000002}{\sqrt{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}}}{\sqrt{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}} - x\right)
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 * ((((2.30753 + (x * 0.27061)) / sqrt((1.0 + (x * (0.99229 + (x * 0.04481)))))) / 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

    \[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.1

    \[\leadsto 0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{\color{blue}{\sqrt{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} \cdot \sqrt{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}}} - x\right)\]

  4. Applied associate-/r*0.1

    \[\leadsto 0.707110000000000016 \cdot \left(\color{blue}{\frac{\frac{2.30753 + x \cdot 0.27061000000000002}{\sqrt{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}}}{\sqrt{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}}} - x\right)\]

  5. Final simplification0.1

    \[\leadsto 0.707110000000000016 \cdot \left(\frac{\frac{2.30753 + x \cdot 0.27061000000000002}{\sqrt{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}}}{\sqrt{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}} - x\right)\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* x (+ 0.99229 (* x 0.04481))))) x)))