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

Average Error: 0.0 → 0.0

Time: 12.1s

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 r34073 = 2.30753;
        double r34074 = x;
        double r34075 = 0.27061;
        double r34076 = r34074 * r34075;
        double r34077 = r34073 + r34076;
        double r34078 = 1.0;
        double r34079 = 0.99229;
        double r34080 = 0.04481;
        double r34081 = r34074 * r34080;
        double r34082 = r34079 + r34081;
        double r34083 = r34074 * r34082;
        double r34084 = r34078 + r34083;
        double r34085 = r34077 / r34084;
        double r34086 = r34085 - r34074;
        return r34086;
}

↓

double f(double x) {
        double r34087 = x;
        double r34088 = 0.27061;
        double r34089 = 2.30753;
        double r34090 = fma(r34087, r34088, r34089);
        double r34091 = 1.0;
        double r34092 = 0.04481;
        double r34093 = 0.99229;
        double r34094 = fma(r34092, r34087, r34093);
        double r34095 = 1.0;
        double r34096 = fma(r34087, r34094, r34095);
        double r34097 = r34091 / r34096;
        double r34098 = -r34087;
        double r34099 = fma(r34090, r34097, r34098);
        return r34099;
}

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 2019326 +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))