Average Error: 0.0 → 0.0
Time: 8.9s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right), x, 1\right)}\]
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 r3104780 = x;
        double r3104781 = 2.30753;
        double r3104782 = 0.27061;
        double r3104783 = r3104780 * r3104782;
        double r3104784 = r3104781 + r3104783;
        double r3104785 = 1.0;
        double r3104786 = 0.99229;
        double r3104787 = 0.04481;
        double r3104788 = r3104780 * r3104787;
        double r3104789 = r3104786 + r3104788;
        double r3104790 = r3104789 * r3104780;
        double r3104791 = r3104785 + r3104790;
        double r3104792 = r3104784 / r3104791;
        double r3104793 = r3104780 - r3104792;
        return r3104793;
}
↓
double f(double x) {
        double r3104794 = x;
        double r3104795 = 0.27061;
        double r3104796 = 2.30753;
        double r3104797 = fma(r3104795, r3104794, r3104796);
        double r3104798 = 0.04481;
        double r3104799 = 0.99229;
        double r3104800 = fma(r3104794, r3104798, r3104799);
        double r3104801 = 1.0;
        double r3104802 = fma(r3104800, r3104794, r3104801);
        double r3104803 = r3104797 / r3104802;
        double r3104804 = r3104794 - r3104803;
        return r3104804;
}

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 2019192 +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)))))