Average Error: 0.0 → 0.0
Time: 1.3s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[\frac{-\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)} + x\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
\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)} + x
double f(double x) {
        double r63664 = x;
        double r63665 = 2.30753;
        double r63666 = 0.27061;
        double r63667 = r63664 * r63666;
        double r63668 = r63665 + r63667;
        double r63669 = 1.0;
        double r63670 = 0.99229;
        double r63671 = 0.04481;
        double r63672 = r63664 * r63671;
        double r63673 = r63670 + r63672;
        double r63674 = r63673 * r63664;
        double r63675 = r63669 + r63674;
        double r63676 = r63668 / r63675;
        double r63677 = r63664 - r63676;
        return r63677;
}
↓
double f(double x) {
        double r63678 = 0.27061;
        double r63679 = x;
        double r63680 = 2.30753;
        double r63681 = fma(r63678, r63679, r63680);
        double r63682 = -r63681;
        double r63683 = 0.04481;
        double r63684 = 0.99229;
        double r63685 = fma(r63683, r63679, r63684);
        double r63686 = 1.0;
        double r63687 = fma(r63679, r63685, r63686);
        double r63688 = r63682 / r63687;
        double r63689 = r63688 + r63679;
        return r63689;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019353 +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)))))