Average Error: 0.0 → 0.0
Time: 12.1s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{\mathsf{fma}\left(x \cdot x, 0.04481000000000000260680366181986755691469, \mathsf{fma}\left(0.992290000000000005364597654988756403327, x, 1\right)\right)}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{\mathsf{fma}\left(x \cdot x, 0.04481000000000000260680366181986755691469, \mathsf{fma}\left(0.992290000000000005364597654988756403327, x, 1\right)\right)}
double f(double x) {
        double r100966 = x;
        double r100967 = 2.30753;
        double r100968 = 0.27061;
        double r100969 = r100966 * r100968;
        double r100970 = r100967 + r100969;
        double r100971 = 1.0;
        double r100972 = 0.99229;
        double r100973 = 0.04481;
        double r100974 = r100966 * r100973;
        double r100975 = r100972 + r100974;
        double r100976 = r100975 * r100966;
        double r100977 = r100971 + r100976;
        double r100978 = r100970 / r100977;
        double r100979 = r100966 - r100978;
        return r100979;
}
↓
double f(double x) {
        double r100980 = x;
        double r100981 = 2.30753;
        double r100982 = 0.27061;
        double r100983 = r100980 * r100982;
        double r100984 = r100981 + r100983;
        double r100985 = r100980 * r100980;
        double r100986 = 0.04481;
        double r100987 = 0.99229;
        double r100988 = 1.0;
        double r100989 = fma(r100987, r100980, r100988);
        double r100990 = fma(r100985, r100986, r100989);
        double r100991 = r100984 / r100990;
        double r100992 = r100980 - r100991;
        return r100992;
}

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. Taylor expanded around 0 0.0

    \[\leadsto x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{\color{blue}{0.992290000000000005364597654988756403327 \cdot x + \left(0.04481000000000000260680366181986755691469 \cdot {x}^{2} + 1\right)}}\]

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +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)))))