Average Error: 0.0 → 0.1
Time: 18.5s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{1}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}} \cdot \frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \frac{1}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}} \cdot \frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}}
double f(double x) {
        double r173818 = x;
        double r173819 = 2.30753;
        double r173820 = 0.27061;
        double r173821 = r173818 * r173820;
        double r173822 = r173819 + r173821;
        double r173823 = 1.0;
        double r173824 = 0.99229;
        double r173825 = 0.04481;
        double r173826 = r173818 * r173825;
        double r173827 = r173824 + r173826;
        double r173828 = r173827 * r173818;
        double r173829 = r173823 + r173828;
        double r173830 = r173822 / r173829;
        double r173831 = r173818 - r173830;
        return r173831;
}
↓
double f(double x) {
        double r173832 = x;
        double r173833 = 1.0;
        double r173834 = 0.04481;
        double r173835 = 0.99229;
        double r173836 = fma(r173834, r173832, r173835);
        double r173837 = 1.0;
        double r173838 = fma(r173836, r173832, r173837);
        double r173839 = sqrt(r173838);
        double r173840 = r173833 / r173839;
        double r173841 = 0.27061;
        double r173842 = 2.30753;
        double r173843 = fma(r173832, r173841, r173842);
        double r173844 = r173843 / r173839;
        double r173845 = r173840 * r173844;
        double r173846 = r173832 - r173845;
        return r173846;
}

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 add-sqr-sqrt0.1

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

  4. Applied *-un-lft-identity0.1

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

  5. Applied times-frac0.1

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

  6. Simplified0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 +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)))))