Average Error: 0.0 → 0.0
Time: 3.9s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[x - \frac{1}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}\right)}^{3}}}\]
x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}
↓
x - \frac{1}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}\right)}^{3}}}
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 - (1.0 / cbrt(pow((fma(x, fma(0.04481, x, 0.99229), 1.0) / fma(0.27061, x, 2.30753)), 3.0))));
}

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 clear-num0.0

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

  4. Simplified0.0

    \[\leadsto x - \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube21.8

    \[\leadsto x - \frac{1}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right) \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)\right) \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}}\]

  7. Applied add-cbrt-cube21.8

    \[\leadsto x - \frac{1}{\frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}}{\sqrt[3]{\left(\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right) \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)\right) \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}\]

  8. Applied cbrt-undiv21.8

    \[\leadsto x - \frac{1}{\color{blue}{\sqrt[3]{\frac{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\left(\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right) \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)\right) \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}}\]

  9. Simplified0.0

    \[\leadsto x - \frac{1}{\sqrt[3]{\color{blue}{{\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}\right)}^{3}}}}\]

  10. Final simplification0.0

    \[\leadsto x - \frac{1}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}\right)}^{3}}}\]

Reproduce

herbie shell --seed 2020060 +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)))))