Average Error: 0.0 → 0.0
Time: 1.5s
Precision: 64
\[\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\]
↓
\[\left(2.30753 + x \cdot 0.27061000000000002\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right) \cdot 1} - x\]
\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x
↓
\left(2.30753 + x \cdot 0.27061000000000002\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right) \cdot 1} - x
double f(double x) {
        double r66709 = 2.30753;
        double r66710 = x;
        double r66711 = 0.27061;
        double r66712 = r66710 * r66711;
        double r66713 = r66709 + r66712;
        double r66714 = 1.0;
        double r66715 = 0.99229;
        double r66716 = 0.04481;
        double r66717 = r66710 * r66716;
        double r66718 = r66715 + r66717;
        double r66719 = r66710 * r66718;
        double r66720 = r66714 + r66719;
        double r66721 = r66713 / r66720;
        double r66722 = r66721 - r66710;
        return r66722;
}
↓
double f(double x) {
        double r66723 = 2.30753;
        double r66724 = x;
        double r66725 = 0.27061;
        double r66726 = r66724 * r66725;
        double r66727 = r66723 + r66726;
        double r66728 = 1.0;
        double r66729 = 0.04481;
        double r66730 = 0.99229;
        double r66731 = fma(r66729, r66724, r66730);
        double r66732 = 1.0;
        double r66733 = fma(r66724, r66731, r66732);
        double r66734 = r66733 * r66728;
        double r66735 = r66728 / r66734;
        double r66736 = r66727 * r66735;
        double r66737 = r66736 - r66724;
        return r66737;
}

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 div-inv0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020025 +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))