Average Error: 0.0 → 0.0
Time: 11.2s
Precision: 64
\[\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(x, 0.27061000000000002, 2.30753\right), \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}, -x\right)\]
\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x
↓
\mathsf{fma}\left(\mathsf{fma}\left(x, 0.27061000000000002, 2.30753\right), \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}, -x\right)
double f(double x) {
        double r58799 = 2.30753;
        double r58800 = x;
        double r58801 = 0.27061;
        double r58802 = r58800 * r58801;
        double r58803 = r58799 + r58802;
        double r58804 = 1.0;
        double r58805 = 0.99229;
        double r58806 = 0.04481;
        double r58807 = r58800 * r58806;
        double r58808 = r58805 + r58807;
        double r58809 = r58800 * r58808;
        double r58810 = r58804 + r58809;
        double r58811 = r58803 / r58810;
        double r58812 = r58811 - r58800;
        return r58812;
}
↓
double f(double x) {
        double r58813 = x;
        double r58814 = 0.27061;
        double r58815 = 2.30753;
        double r58816 = fma(r58813, r58814, r58815);
        double r58817 = 1.0;
        double r58818 = 0.04481;
        double r58819 = 0.99229;
        double r58820 = fma(r58818, r58813, r58819);
        double r58821 = 1.0;
        double r58822 = fma(r58813, r58820, r58821);
        double r58823 = r58817 / r58822;
        double r58824 = -r58813;
        double r58825 = fma(r58816, r58823, r58824);
        return r58825;
}

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. Applied fma-neg0.0

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

  6. Final simplification0.0

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

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))