Average Error: 0.0 → 0.0
Time: 13.4s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)} \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)} \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)
double f(double x) {
        double r3061153 = x;
        double r3061154 = 2.30753;
        double r3061155 = 0.27061;
        double r3061156 = r3061153 * r3061155;
        double r3061157 = r3061154 + r3061156;
        double r3061158 = 1.0;
        double r3061159 = 0.99229;
        double r3061160 = 0.04481;
        double r3061161 = r3061153 * r3061160;
        double r3061162 = r3061159 + r3061161;
        double r3061163 = r3061162 * r3061153;
        double r3061164 = r3061158 + r3061163;
        double r3061165 = r3061157 / r3061164;
        double r3061166 = r3061153 - r3061165;
        return r3061166;
}
↓
double f(double x) {
        double r3061167 = x;
        double r3061168 = 1.0;
        double r3061169 = 0.04481;
        double r3061170 = 0.99229;
        double r3061171 = fma(r3061169, r3061167, r3061170);
        double r3061172 = 1.0;
        double r3061173 = fma(r3061171, r3061167, r3061172);
        double r3061174 = r3061168 / r3061173;
        double r3061175 = 0.27061;
        double r3061176 = 2.30753;
        double r3061177 = fma(r3061175, r3061167, r3061176);
        double r3061178 = r3061174 * r3061177;
        double r3061179 = r3061167 - r3061178;
        return r3061179;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  4. Applied div-inv0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))