Average Error: 0.0 → 0.0
Time: 24.3s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{1}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \frac{1}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}}
double f(double x) {
        double r83825 = x;
        double r83826 = 2.30753;
        double r83827 = 0.27061;
        double r83828 = r83825 * r83827;
        double r83829 = r83826 + r83828;
        double r83830 = 1.0;
        double r83831 = 0.99229;
        double r83832 = 0.04481;
        double r83833 = r83825 * r83832;
        double r83834 = r83831 + r83833;
        double r83835 = r83834 * r83825;
        double r83836 = r83830 + r83835;
        double r83837 = r83829 / r83836;
        double r83838 = r83825 - r83837;
        return r83838;
}
↓
double f(double x) {
        double r83839 = x;
        double r83840 = 1.0;
        double r83841 = 0.04481;
        double r83842 = 0.99229;
        double r83843 = fma(r83841, r83839, r83842);
        double r83844 = 1.0;
        double r83845 = fma(r83839, r83843, r83844);
        double r83846 = 0.27061;
        double r83847 = 2.30753;
        double r83848 = fma(r83846, r83839, r83847);
        double r83849 = r83845 / r83848;
        double r83850 = r83840 / r83849;
        double r83851 = r83839 - r83850;
        return r83851;
}

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. Using strategy rm

  3. Applied clear-num0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +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)))))