Average Error: 0.0 → 0.0
Time: 22.2s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)\right)}^{3}}}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)\right)}^{3}}}
double f(double x) {
        double r77404 = x;
        double r77405 = 2.30753;
        double r77406 = 0.27061;
        double r77407 = r77404 * r77406;
        double r77408 = r77405 + r77407;
        double r77409 = 1.0;
        double r77410 = 0.99229;
        double r77411 = 0.04481;
        double r77412 = r77404 * r77411;
        double r77413 = r77410 + r77412;
        double r77414 = r77413 * r77404;
        double r77415 = r77409 + r77414;
        double r77416 = r77408 / r77415;
        double r77417 = r77404 - r77416;
        return r77417;
}
↓
double f(double x) {
        double r77418 = x;
        double r77419 = 0.27061;
        double r77420 = 2.30753;
        double r77421 = fma(r77418, r77419, r77420);
        double r77422 = 0.04481;
        double r77423 = 0.99229;
        double r77424 = fma(r77422, r77418, r77423);
        double r77425 = 1.0;
        double r77426 = fma(r77424, r77418, r77425);
        double r77427 = 3.0;
        double r77428 = pow(r77426, r77427);
        double r77429 = cbrt(r77428);
        double r77430 = r77421 / r77429;
        double r77431 = r77418 - r77430;
        return r77431;
}

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

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

  4. Applied add-cbrt-cube0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061000000000002)) (+ 1 (* (+ 0.992290000000000005 (* x 0.044810000000000003)) x)))))