Average Error: 0.0 → 0.0
Time: 2.4s
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 r92676 = x;
        double r92677 = 2.30753;
        double r92678 = 0.27061;
        double r92679 = r92676 * r92678;
        double r92680 = r92677 + r92679;
        double r92681 = 1.0;
        double r92682 = 0.99229;
        double r92683 = 0.04481;
        double r92684 = r92676 * r92683;
        double r92685 = r92682 + r92684;
        double r92686 = r92685 * r92676;
        double r92687 = r92681 + r92686;
        double r92688 = r92680 / r92687;
        double r92689 = r92676 - r92688;
        return r92689;
}
↓
double f(double x) {
        double r92690 = 0.27061;
        double r92691 = x;
        double r92692 = 2.30753;
        double r92693 = fma(r92690, r92691, r92692);
        double r92694 = -r92693;
        double r92695 = 0.04481;
        double r92696 = 0.99229;
        double r92697 = fma(r92695, r92691, r92696);
        double r92698 = 1.0;
        double r92699 = fma(r92691, r92697, r92698);
        double r92700 = r92694 / r92699;
        double r92701 = r92700 + r92691;
        return r92701;
}

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 2020056 +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)))))