Average Error: 0.0 → 0.0
Time: 7.9s
Precision: 64
\[\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]
↓
\[\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right), x, 1\right)} \cdot \mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right) - x\]
\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x
↓
\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right), x, 1\right)} \cdot \mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right) - x
double f(double x) {
        double r2372835 = 2.30753;
        double r2372836 = x;
        double r2372837 = 0.27061;
        double r2372838 = r2372836 * r2372837;
        double r2372839 = r2372835 + r2372838;
        double r2372840 = 1.0;
        double r2372841 = 0.99229;
        double r2372842 = 0.04481;
        double r2372843 = r2372836 * r2372842;
        double r2372844 = r2372841 + r2372843;
        double r2372845 = r2372836 * r2372844;
        double r2372846 = r2372840 + r2372845;
        double r2372847 = r2372839 / r2372846;
        double r2372848 = r2372847 - r2372836;
        return r2372848;
}
↓
double f(double x) {
        double r2372849 = 1.0;
        double r2372850 = x;
        double r2372851 = 0.04481;
        double r2372852 = 0.99229;
        double r2372853 = fma(r2372850, r2372851, r2372852);
        double r2372854 = 1.0;
        double r2372855 = fma(r2372853, r2372850, r2372854);
        double r2372856 = r2372849 / r2372855;
        double r2372857 = 0.27061;
        double r2372858 = 2.30753;
        double r2372859 = fma(r2372850, r2372857, r2372858);
        double r2372860 = r2372856 * r2372859;
        double r2372861 = r2372860 - r2372850;
        return r2372861;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  4. Applied div-inv0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))