Average Error: 0.0 → 0.0
Time: 8.0s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right), x, 1\right)}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right), x, 1\right)}
double f(double x) {
        double r3201142 = x;
        double r3201143 = 2.30753;
        double r3201144 = 0.27061;
        double r3201145 = r3201142 * r3201144;
        double r3201146 = r3201143 + r3201145;
        double r3201147 = 1.0;
        double r3201148 = 0.99229;
        double r3201149 = 0.04481;
        double r3201150 = r3201142 * r3201149;
        double r3201151 = r3201148 + r3201150;
        double r3201152 = r3201151 * r3201142;
        double r3201153 = r3201147 + r3201152;
        double r3201154 = r3201146 / r3201153;
        double r3201155 = r3201142 - r3201154;
        return r3201155;
}
↓
double f(double x) {
        double r3201156 = x;
        double r3201157 = 0.27061;
        double r3201158 = 2.30753;
        double r3201159 = fma(r3201157, r3201156, r3201158);
        double r3201160 = 0.04481;
        double r3201161 = 0.99229;
        double r3201162 = fma(r3201156, r3201160, r3201161);
        double r3201163 = 1.0;
        double r3201164 = fma(r3201162, r3201156, r3201163);
        double r3201165 = r3201159 / r3201164;
        double r3201166 = r3201156 - r3201165;
        return r3201166;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))