Average Error: 0.0 → 0.0
Time: 3.6s
Precision: 64
\[\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\]
↓
\[\frac{\mathsf{fma}\left(x, 0.27061000000000002, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} - x\]
\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x
↓
\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
double f(double x) {
        double r51031 = 2.30753;
        double r51032 = x;
        double r51033 = 0.27061;
        double r51034 = r51032 * r51033;
        double r51035 = r51031 + r51034;
        double r51036 = 1.0;
        double r51037 = 0.99229;
        double r51038 = 0.04481;
        double r51039 = r51032 * r51038;
        double r51040 = r51037 + r51039;
        double r51041 = r51032 * r51040;
        double r51042 = r51036 + r51041;
        double r51043 = r51035 / r51042;
        double r51044 = r51043 - r51032;
        return r51044;
}
↓
double f(double x) {
        double r51045 = x;
        double r51046 = 0.27061;
        double r51047 = 2.30753;
        double r51048 = fma(r51045, r51046, r51047);
        double r51049 = 0.04481;
        double r51050 = 0.99229;
        double r51051 = fma(r51049, r51045, r51050);
        double r51052 = 1.0;
        double r51053 = fma(r51045, r51051, r51052);
        double r51054 = r51048 / r51053;
        double r51055 = r51054 - r51045;
        return r51055;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 +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))