Average Error: 0.0 → 0.2
Time: 3.5s
Precision: 64
\[\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\]
↓
\[\frac{2.30753 + x \cdot 0.27061000000000002}{1 + \log \left({\left(e^{x}\right)}^{\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right)\right)}\right)} - x\]
\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x
↓
\frac{2.30753 + x \cdot 0.27061000000000002}{1 + \log \left({\left(e^{x}\right)}^{\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right)\right)}\right)} - x
double f(double x) {
        double r56826 = 2.30753;
        double r56827 = x;
        double r56828 = 0.27061;
        double r56829 = r56827 * r56828;
        double r56830 = r56826 + r56829;
        double r56831 = 1.0;
        double r56832 = 0.99229;
        double r56833 = 0.04481;
        double r56834 = r56827 * r56833;
        double r56835 = r56832 + r56834;
        double r56836 = r56827 * r56835;
        double r56837 = r56831 + r56836;
        double r56838 = r56830 / r56837;
        double r56839 = r56838 - r56827;
        return r56839;
}
↓
double f(double x) {
        double r56840 = 2.30753;
        double r56841 = x;
        double r56842 = 0.27061;
        double r56843 = r56841 * r56842;
        double r56844 = r56840 + r56843;
        double r56845 = 1.0;
        double r56846 = exp(r56841);
        double r56847 = 0.04481;
        double r56848 = 0.99229;
        double r56849 = fma(r56847, r56841, r56848);
        double r56850 = pow(r56846, r56849);
        double r56851 = log(r56850);
        double r56852 = r56845 + r56851;
        double r56853 = r56844 / r56852;
        double r56854 = r56853 - r56841;
        return r56854;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\]
  2. Using strategy rm

  3. Applied add-log-exp0.2

    \[\leadsto \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \color{blue}{\log \left(e^{x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}\right)}} - x\]

  4. Simplified0.2

    \[\leadsto \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \log \color{blue}{\left({\left(e^{x}\right)}^{\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right)\right)}\right)}} - x\]

  5. Final simplification0.2

    \[\leadsto \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \log \left({\left(e^{x}\right)}^{\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right)\right)}\right)} - x\]

Reproduce

herbie shell --seed 2020039 +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))