Average Error: 0.0 → 0.0
Time: 9.4s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}
double f(double x) {
        double r116127 = x;
        double r116128 = 2.30753;
        double r116129 = 0.27061;
        double r116130 = r116127 * r116129;
        double r116131 = r116128 + r116130;
        double r116132 = 1.0;
        double r116133 = 0.99229;
        double r116134 = 0.04481;
        double r116135 = r116127 * r116134;
        double r116136 = r116133 + r116135;
        double r116137 = r116136 * r116127;
        double r116138 = r116132 + r116137;
        double r116139 = r116131 / r116138;
        double r116140 = r116127 - r116139;
        return r116140;
}
↓
double f(double x) {
        double r116141 = x;
        double r116142 = 0.27061;
        double r116143 = 2.30753;
        double r116144 = fma(r116141, r116142, r116143);
        double r116145 = 0.04481;
        double r116146 = 0.99229;
        double r116147 = fma(r116145, r116141, r116146);
        double r116148 = 1.0;
        double r116149 = fma(r116147, r116141, r116148);
        double r116150 = r116144 / r116149;
        double r116151 = r116141 - r116150;
        return r116151;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{x - \frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}}\]

  3. Final simplification0.0

    \[\leadsto x - \frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}\]

Reproduce

herbie shell --seed 2019351 +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)))))