Average Error: 0.0 → 0.0
Time: 1.0s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[\frac{-\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)} + x\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
\frac{-\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)} + x
double f(double x) {
        double r46985 = x;
        double r46986 = 2.30753;
        double r46987 = 0.27061;
        double r46988 = r46985 * r46987;
        double r46989 = r46986 + r46988;
        double r46990 = 1.0;
        double r46991 = 0.99229;
        double r46992 = 0.04481;
        double r46993 = r46985 * r46992;
        double r46994 = r46991 + r46993;
        double r46995 = r46994 * r46985;
        double r46996 = r46990 + r46995;
        double r46997 = r46989 / r46996;
        double r46998 = r46985 - r46997;
        return r46998;
}
↓
double f(double x) {
        double r46999 = 0.27061;
        double r47000 = x;
        double r47001 = 2.30753;
        double r47002 = fma(r46999, r47000, r47001);
        double r47003 = -r47002;
        double r47004 = 0.04481;
        double r47005 = 0.99229;
        double r47006 = fma(r47004, r47000, r47005);
        double r47007 = 1.0;
        double r47008 = fma(r47000, r47006, r47007);
        double r47009 = r47003 / r47008;
        double r47010 = r47009 + r47000;
        return r47010;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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)))))