Average Error: 0.0 → 0.0
Time: 12.0s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)} \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)} \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)
double f(double x) {
        double r3274854 = x;
        double r3274855 = 2.30753;
        double r3274856 = 0.27061;
        double r3274857 = r3274854 * r3274856;
        double r3274858 = r3274855 + r3274857;
        double r3274859 = 1.0;
        double r3274860 = 0.99229;
        double r3274861 = 0.04481;
        double r3274862 = r3274854 * r3274861;
        double r3274863 = r3274860 + r3274862;
        double r3274864 = r3274863 * r3274854;
        double r3274865 = r3274859 + r3274864;
        double r3274866 = r3274858 / r3274865;
        double r3274867 = r3274854 - r3274866;
        return r3274867;
}
↓
double f(double x) {
        double r3274868 = x;
        double r3274869 = 1.0;
        double r3274870 = 0.04481;
        double r3274871 = 0.99229;
        double r3274872 = fma(r3274870, r3274868, r3274871);
        double r3274873 = 1.0;
        double r3274874 = fma(r3274872, r3274868, r3274873);
        double r3274875 = r3274869 / r3274874;
        double r3274876 = 0.27061;
        double r3274877 = 2.30753;
        double r3274878 = fma(r3274876, r3274868, r3274877);
        double r3274879 = r3274875 * r3274878;
        double r3274880 = r3274868 - r3274879;
        return r3274880;
}

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. Using strategy rm

  4. Applied div-inv0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +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)))))