Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C

Average Error: 0.0 → 0.0

Time: 11.9s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r46811 = 2.30753;
        double r46812 = x;
        double r46813 = 0.27061;
        double r46814 = r46812 * r46813;
        double r46815 = r46811 + r46814;
        double r46816 = 1.0;
        double r46817 = 0.99229;
        double r46818 = 0.04481;
        double r46819 = r46812 * r46818;
        double r46820 = r46817 + r46819;
        double r46821 = r46812 * r46820;
        double r46822 = r46816 + r46821;
        double r46823 = r46815 / r46822;
        double r46824 = r46823 - r46812;
        return r46824;
}

↓

double f(double x) {
        double r46825 = x;
        double r46826 = 0.27061;
        double r46827 = 2.30753;
        double r46828 = fma(r46825, r46826, r46827);
        double r46829 = 1.0;
        double r46830 = 0.04481;
        double r46831 = 0.99229;
        double r46832 = fma(r46830, r46825, r46831);
        double r46833 = 1.0;
        double r46834 = fma(r46825, r46832, r46833);
        double r46835 = r46829 / r46834;
        double r46836 = -r46825;
        double r46837 = fma(r46828, r46835, r46836);
        return r46837;
}

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. Simplified 0.0

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

4. Applied div-inv 0.0

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

5. Applied fma-neg 0.0

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

6. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* x (+ 0.99229 (* x 0.04481))))) x))