Average Error: 0.0 → 0.0
Time: 22.4s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{1}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \frac{1}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}}
double f(double x) {
        double r71185 = x;
        double r71186 = 2.30753;
        double r71187 = 0.27061;
        double r71188 = r71185 * r71187;
        double r71189 = r71186 + r71188;
        double r71190 = 1.0;
        double r71191 = 0.99229;
        double r71192 = 0.04481;
        double r71193 = r71185 * r71192;
        double r71194 = r71191 + r71193;
        double r71195 = r71194 * r71185;
        double r71196 = r71190 + r71195;
        double r71197 = r71189 / r71196;
        double r71198 = r71185 - r71197;
        return r71198;
}
↓
double f(double x) {
        double r71199 = x;
        double r71200 = 1.0;
        double r71201 = 0.04481;
        double r71202 = 0.99229;
        double r71203 = fma(r71201, r71199, r71202);
        double r71204 = 1.0;
        double r71205 = fma(r71199, r71203, r71204);
        double r71206 = 0.27061;
        double r71207 = 2.30753;
        double r71208 = fma(r71206, r71199, r71207);
        double r71209 = r71205 / r71208;
        double r71210 = r71200 / r71209;
        double r71211 = r71199 - r71210;
        return r71211;
}

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. Using strategy rm

  3. Applied clear-num0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* (+ 0.99229 (* x 0.04481)) x)))))