Average Error: 0.0 → 0.0
Time: 16.5s
Precision: 64
\[\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]
↓
\[\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}} - x\]
\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x
↓
\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}} - x
double f(double x) {
        double r61328 = 2.30753;
        double r61329 = x;
        double r61330 = 0.27061;
        double r61331 = r61329 * r61330;
        double r61332 = r61328 + r61331;
        double r61333 = 1.0;
        double r61334 = 0.99229;
        double r61335 = 0.04481;
        double r61336 = r61329 * r61335;
        double r61337 = r61334 + r61336;
        double r61338 = r61329 * r61337;
        double r61339 = r61333 + r61338;
        double r61340 = r61332 / r61339;
        double r61341 = r61340 - r61329;
        return r61341;
}
↓
double f(double x) {
        double r61342 = 1.0;
        double r61343 = 0.04481;
        double r61344 = x;
        double r61345 = 0.99229;
        double r61346 = fma(r61343, r61344, r61345);
        double r61347 = 1.0;
        double r61348 = fma(r61346, r61344, r61347);
        double r61349 = 0.27061;
        double r61350 = 2.30753;
        double r61351 = fma(r61349, r61344, r61350);
        double r61352 = r61348 / r61351;
        double r61353 = r61342 / r61352;
        double r61354 = r61353 - r61344;
        return r61354;
}

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. Using strategy rm

  3. Applied clear-num0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

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))