Average Error: 0.0 → 0.1
Time: 3.3s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[x - \frac{\frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\sqrt{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}}}{\sqrt{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}}\]
x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}
↓
x - \frac{\frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\sqrt{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}}}{\sqrt{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}}
double code(double x) {
	return (x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x))));
}
↓
double code(double x) {
	return (x - ((fma(0.27061, x, 2.30753) / sqrt((1.0 + ((0.99229 + (x * 0.04481)) * x)))) / sqrt((1.0 + ((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

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

  3. Applied add-sqr-sqrt0.0

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

  4. Applied associate-/r*0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* (+ 0.99229 (* x 0.04481)) x)))))