Average Error: 0.0 → 0.0
Time: 10.7s
Precision: 64
\[\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\]
↓
\[\mathsf{fma}\left(x, 0.27061000000000002, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} - x\]
\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x
↓
\mathsf{fma}\left(x, 0.27061000000000002, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} - x
double f(double x) {
        double r61808 = 2.30753;
        double r61809 = x;
        double r61810 = 0.27061;
        double r61811 = r61809 * r61810;
        double r61812 = r61808 + r61811;
        double r61813 = 1.0;
        double r61814 = 0.99229;
        double r61815 = 0.04481;
        double r61816 = r61809 * r61815;
        double r61817 = r61814 + r61816;
        double r61818 = r61809 * r61817;
        double r61819 = r61813 + r61818;
        double r61820 = r61812 / r61819;
        double r61821 = r61820 - r61809;
        return r61821;
}
↓
double f(double x) {
        double r61822 = x;
        double r61823 = 0.27061;
        double r61824 = 2.30753;
        double r61825 = fma(r61822, r61823, r61824);
        double r61826 = 1.0;
        double r61827 = 0.04481;
        double r61828 = 0.99229;
        double r61829 = fma(r61827, r61822, r61828);
        double r61830 = 1.0;
        double r61831 = fma(r61822, r61829, r61830);
        double r61832 = r61826 / r61831;
        double r61833 = r61825 * r61832;
        double r61834 = r61833 - r61822;
        return r61834;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 0.27061000000000002, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} - x}\]
  3. Using strategy rm

  4. Applied div-inv0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020042 +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))