Average Error: 0.0 → 0.0
Time: 11.2s
Precision: 64
\[\frac{2.30753 + x \cdot 0.27061}{1.0 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\]
↓
\[\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}} - x\]
\frac{2.30753 + x \cdot 0.27061}{1.0 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x
↓
\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}} - x
double f(double x) {
        double r3456798 = 2.30753;
        double r3456799 = x;
        double r3456800 = 0.27061;
        double r3456801 = r3456799 * r3456800;
        double r3456802 = r3456798 + r3456801;
        double r3456803 = 1.0;
        double r3456804 = 0.99229;
        double r3456805 = 0.04481;
        double r3456806 = r3456799 * r3456805;
        double r3456807 = r3456804 + r3456806;
        double r3456808 = r3456799 * r3456807;
        double r3456809 = r3456803 + r3456808;
        double r3456810 = r3456802 / r3456809;
        double r3456811 = r3456810 - r3456799;
        return r3456811;
}
↓
double f(double x) {
        double r3456812 = 1.0;
        double r3456813 = x;
        double r3456814 = 0.04481;
        double r3456815 = 0.99229;
        double r3456816 = fma(r3456813, r3456814, r3456815);
        double r3456817 = 1.0;
        double r3456818 = fma(r3456816, r3456813, r3456817);
        double r3456819 = 0.27061;
        double r3456820 = 2.30753;
        double r3456821 = fma(r3456813, r3456819, r3456820);
        double r3456822 = r3456818 / r3456821;
        double r3456823 = r3456812 / r3456822;
        double r3456824 = r3456823 - r3456813;
        return r3456824;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{2.30753 + x \cdot 0.27061}{1.0 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)} - x}\]
  3. Using strategy rm

  4. Applied clear-num0.0

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}} - x\]

  5. Final simplification0.0

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}} - x\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))