Average Error: 0.0 → 0.0
Time: 2.0s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[\mathsf{fma}\left(-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right), \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}, x\right)\]
x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}
↓
\mathsf{fma}\left(-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right), \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}, x\right)
double f(double x) {
        double r185698 = x;
        double r185699 = 2.30753;
        double r185700 = 0.27061;
        double r185701 = r185698 * r185700;
        double r185702 = r185699 + r185701;
        double r185703 = 1.0;
        double r185704 = 0.99229;
        double r185705 = 0.04481;
        double r185706 = r185698 * r185705;
        double r185707 = r185704 + r185706;
        double r185708 = r185707 * r185698;
        double r185709 = r185703 + r185708;
        double r185710 = r185702 / r185709;
        double r185711 = r185698 - r185710;
        return r185711;
}
↓
double f(double x) {
        double r185712 = 0.27061;
        double r185713 = x;
        double r185714 = 2.30753;
        double r185715 = fma(r185712, r185713, r185714);
        double r185716 = -r185715;
        double r185717 = 1.0;
        double r185718 = 0.04481;
        double r185719 = 0.99229;
        double r185720 = fma(r185718, r185713, r185719);
        double r185721 = 1.0;
        double r185722 = fma(r185713, r185720, r185721);
        double r185723 = r185717 / r185722;
        double r185724 = fma(r185716, r185723, r185713);
        return r185724;
}

Error

Bits error versus x

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. Simplified0.0

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

  4. Applied div-inv0.0

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

  5. Applied fma-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020100 +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)))))