Average Error: 0.0 → 0.0
Time: 22.5s
Precision: 64
\[\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]
↓
\[\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)} \cdot \mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right) - x\]
\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x
↓
\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)} \cdot \mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right) - x
double f(double x) {
        double r3160061 = 2.30753;
        double r3160062 = x;
        double r3160063 = 0.27061;
        double r3160064 = r3160062 * r3160063;
        double r3160065 = r3160061 + r3160064;
        double r3160066 = 1.0;
        double r3160067 = 0.99229;
        double r3160068 = 0.04481;
        double r3160069 = r3160062 * r3160068;
        double r3160070 = r3160067 + r3160069;
        double r3160071 = r3160062 * r3160070;
        double r3160072 = r3160066 + r3160071;
        double r3160073 = r3160065 / r3160072;
        double r3160074 = r3160073 - r3160062;
        return r3160074;
}
↓
double f(double x) {
        double r3160075 = 1.0;
        double r3160076 = x;
        double r3160077 = 0.04481;
        double r3160078 = 0.99229;
        double r3160079 = fma(r3160077, r3160076, r3160078);
        double r3160080 = 1.0;
        double r3160081 = fma(r3160076, r3160079, r3160080);
        double r3160082 = r3160075 / r3160081;
        double r3160083 = 0.27061;
        double r3160084 = 2.30753;
        double r3160085 = fma(r3160076, r3160083, r3160084);
        double r3160086 = r3160082 * r3160085;
        double r3160087 = r3160086 - r3160076;
        return r3160087;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)} - x}\]
  3. Using strategy rm

  4. Applied div-inv0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}} - x\]

  5. Final simplification0.0

    \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)} \cdot \mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right) - x\]

Reproduce

herbie shell --seed 2019200 +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))