Average Error: 0.0 → 0.0
Time: 24.9s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}
double f(double x) {
        double r4193313 = x;
        double r4193314 = 2.30753;
        double r4193315 = 0.27061;
        double r4193316 = r4193313 * r4193315;
        double r4193317 = r4193314 + r4193316;
        double r4193318 = 1.0;
        double r4193319 = 0.99229;
        double r4193320 = 0.04481;
        double r4193321 = r4193313 * r4193320;
        double r4193322 = r4193319 + r4193321;
        double r4193323 = r4193322 * r4193313;
        double r4193324 = r4193318 + r4193323;
        double r4193325 = r4193317 / r4193324;
        double r4193326 = r4193313 - r4193325;
        return r4193326;
}
↓
double f(double x) {
        double r4193327 = x;
        double r4193328 = 0.27061;
        double r4193329 = 2.30753;
        double r4193330 = fma(r4193328, r4193327, r4193329);
        double r4193331 = 0.04481;
        double r4193332 = 0.99229;
        double r4193333 = fma(r4193331, r4193327, r4193332);
        double r4193334 = 1.0;
        double r4193335 = fma(r4193333, r4193327, r4193334);
        double r4193336 = r4193330 / r4193335;
        double r4193337 = r4193327 - r4193336;
        return r4193337;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019168 +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)))))