Average Error: 0.0 → 0.0
Time: 2.0s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[\frac{-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} + x\]
x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}
↓
\frac{-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} + x
double f(double x) {
        double r82542 = x;
        double r82543 = 2.30753;
        double r82544 = 0.27061;
        double r82545 = r82542 * r82544;
        double r82546 = r82543 + r82545;
        double r82547 = 1.0;
        double r82548 = 0.99229;
        double r82549 = 0.04481;
        double r82550 = r82542 * r82549;
        double r82551 = r82548 + r82550;
        double r82552 = r82551 * r82542;
        double r82553 = r82547 + r82552;
        double r82554 = r82546 / r82553;
        double r82555 = r82542 - r82554;
        return r82555;
}
↓
double f(double x) {
        double r82556 = 0.27061;
        double r82557 = x;
        double r82558 = 2.30753;
        double r82559 = fma(r82556, r82557, r82558);
        double r82560 = -r82559;
        double r82561 = 0.04481;
        double r82562 = 0.99229;
        double r82563 = fma(r82561, r82557, r82562);
        double r82564 = 1.0;
        double r82565 = fma(r82557, r82563, r82564);
        double r82566 = r82560 / r82565;
        double r82567 = r82566 + r82557;
        return r82567;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} + x}\]

  3. Final simplification0.0

    \[\leadsto \frac{-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} + x\]

Reproduce

herbie shell --seed 2020018 +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)))))