Average Error: 0.0 → 0.0
Time: 10.0s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right), x, 1\right)} \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)\]
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(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right), x, 1\right)} \cdot \mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)
double f(double x) {
        double r3232423 = x;
        double r3232424 = 2.30753;
        double r3232425 = 0.27061;
        double r3232426 = r3232423 * r3232425;
        double r3232427 = r3232424 + r3232426;
        double r3232428 = 1.0;
        double r3232429 = 0.99229;
        double r3232430 = 0.04481;
        double r3232431 = r3232423 * r3232430;
        double r3232432 = r3232429 + r3232431;
        double r3232433 = r3232432 * r3232423;
        double r3232434 = r3232428 + r3232433;
        double r3232435 = r3232427 / r3232434;
        double r3232436 = r3232423 - r3232435;
        return r3232436;
}
↓
double f(double x) {
        double r3232437 = x;
        double r3232438 = 1.0;
        double r3232439 = 0.04481;
        double r3232440 = 0.99229;
        double r3232441 = fma(r3232437, r3232439, r3232440);
        double r3232442 = 1.0;
        double r3232443 = fma(r3232441, r3232437, r3232442);
        double r3232444 = r3232438 / r3232443;
        double r3232445 = 0.27061;
        double r3232446 = 2.30753;
        double r3232447 = fma(r3232445, r3232437, r3232446);
        double r3232448 = r3232444 * r3232447;
        double r3232449 = r3232437 - r3232448;
        return r3232449;
}

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

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

  4. Applied div-inv0.0

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

  5. Final simplification0.0

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

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)))))