Average Error: 0.0 → 0.0
Time: 7.5s
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 r2324961 = 2.30753;
        double r2324962 = x;
        double r2324963 = 0.27061;
        double r2324964 = r2324962 * r2324963;
        double r2324965 = r2324961 + r2324964;
        double r2324966 = 1.0;
        double r2324967 = 0.99229;
        double r2324968 = 0.04481;
        double r2324969 = r2324962 * r2324968;
        double r2324970 = r2324967 + r2324969;
        double r2324971 = r2324962 * r2324970;
        double r2324972 = r2324966 + r2324971;
        double r2324973 = r2324965 / r2324972;
        double r2324974 = r2324973 - r2324962;
        return r2324974;
}
↓
double f(double x) {
        double r2324975 = 1.0;
        double r2324976 = x;
        double r2324977 = 0.04481;
        double r2324978 = 0.99229;
        double r2324979 = fma(r2324976, r2324977, r2324978);
        double r2324980 = 1.0;
        double r2324981 = fma(r2324979, r2324976, r2324980);
        double r2324982 = r2324975 / r2324981;
        double r2324983 = 0.27061;
        double r2324984 = 2.30753;
        double r2324985 = fma(r2324976, r2324983, r2324984);
        double r2324986 = r2324982 * r2324985;
        double r2324987 = r2324986 - r2324976;
        return r2324987;
}

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))