Average Error: 0.0 → 0.0
Time: 41.5s
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 r2777521 = x;
        double r2777522 = 2.30753;
        double r2777523 = 0.27061;
        double r2777524 = r2777521 * r2777523;
        double r2777525 = r2777522 + r2777524;
        double r2777526 = 1.0;
        double r2777527 = 0.99229;
        double r2777528 = 0.04481;
        double r2777529 = r2777521 * r2777528;
        double r2777530 = r2777527 + r2777529;
        double r2777531 = r2777530 * r2777521;
        double r2777532 = r2777526 + r2777531;
        double r2777533 = r2777525 / r2777532;
        double r2777534 = r2777521 - r2777533;
        return r2777534;
}
↓
double f(double x) {
        double r2777535 = x;
        double r2777536 = 1.0;
        double r2777537 = 0.04481;
        double r2777538 = 0.99229;
        double r2777539 = fma(r2777537, r2777535, r2777538);
        double r2777540 = 1.0;
        double r2777541 = fma(r2777539, r2777535, r2777540);
        double r2777542 = r2777536 / r2777541;
        double r2777543 = 0.27061;
        double r2777544 = 2.30753;
        double r2777545 = fma(r2777543, r2777535, r2777544);
        double r2777546 = r2777542 * r2777545;
        double r2777547 = r2777535 - r2777546;
        return r2777547;
}

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 2019200 +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)))))