Average Error: 0.0 → 0.0
Time: 14.9s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \sqrt[3]{\frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)} \cdot \left(\frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)} \cdot \frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \sqrt[3]{\frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)} \cdot \left(\frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)} \cdot \frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)}
double f(double x) {
        double r3611466 = x;
        double r3611467 = 2.30753;
        double r3611468 = 0.27061;
        double r3611469 = r3611466 * r3611468;
        double r3611470 = r3611467 + r3611469;
        double r3611471 = 1.0;
        double r3611472 = 0.99229;
        double r3611473 = 0.04481;
        double r3611474 = r3611466 * r3611473;
        double r3611475 = r3611472 + r3611474;
        double r3611476 = r3611475 * r3611466;
        double r3611477 = r3611471 + r3611476;
        double r3611478 = r3611470 / r3611477;
        double r3611479 = r3611466 - r3611478;
        return r3611479;
}
↓
double f(double x) {
        double r3611480 = x;
        double r3611481 = 0.27061;
        double r3611482 = 2.30753;
        double r3611483 = fma(r3611481, r3611480, r3611482);
        double r3611484 = 0.04481;
        double r3611485 = 0.99229;
        double r3611486 = fma(r3611484, r3611480, r3611485);
        double r3611487 = 1.0;
        double r3611488 = fma(r3611480, r3611486, r3611487);
        double r3611489 = r3611483 / r3611488;
        double r3611490 = r3611489 * r3611489;
        double r3611491 = r3611489 * r3611490;
        double r3611492 = cbrt(r3611491);
        double r3611493 = r3611480 - r3611492;
        return r3611493;
}

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 add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube21.1

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

  6. Applied cbrt-undiv21.1

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

  7. Simplified0.0

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

  8. Final simplification0.0

    \[\leadsto x - \sqrt[3]{\frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)} \cdot \left(\frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)} \cdot \frac{\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)}\right)}\]

Reproduce

herbie shell --seed 2019169 +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)))))