Average Error: 0.0 → 0.0
Time: 20.5s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}
double f(double x) {
        double r71991 = x;
        double r71992 = 2.30753;
        double r71993 = 0.27061;
        double r71994 = r71991 * r71993;
        double r71995 = r71992 + r71994;
        double r71996 = 1.0;
        double r71997 = 0.99229;
        double r71998 = 0.04481;
        double r71999 = r71991 * r71998;
        double r72000 = r71997 + r71999;
        double r72001 = r72000 * r71991;
        double r72002 = r71996 + r72001;
        double r72003 = r71995 / r72002;
        double r72004 = r71991 - r72003;
        return r72004;
}
↓
double f(double x) {
        double r72005 = x;
        double r72006 = 0.27061;
        double r72007 = 2.30753;
        double r72008 = fma(r72005, r72006, r72007);
        double r72009 = 0.04481;
        double r72010 = 0.99229;
        double r72011 = fma(r72009, r72005, r72010);
        double r72012 = 1.0;
        double r72013 = fma(r72011, r72005, r72012);
        double r72014 = r72008 / r72013;
        double r72015 = r72005 - r72014;
        return r72015;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061000000000002)) (+ 1 (* (+ 0.992290000000000005 (* x 0.044810000000000003)) x)))))