Average Error: 0.0 → 0.0
Time: 2.5s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[x - \left(2.30753 + x \cdot 0.27061000000000002\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right) \cdot 1}\]
x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}
↓
x - \left(2.30753 + x \cdot 0.27061000000000002\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right) \cdot 1}
double f(double x) {
        double r77651 = x;
        double r77652 = 2.30753;
        double r77653 = 0.27061;
        double r77654 = r77651 * r77653;
        double r77655 = r77652 + r77654;
        double r77656 = 1.0;
        double r77657 = 0.99229;
        double r77658 = 0.04481;
        double r77659 = r77651 * r77658;
        double r77660 = r77657 + r77659;
        double r77661 = r77660 * r77651;
        double r77662 = r77656 + r77661;
        double r77663 = r77655 / r77662;
        double r77664 = r77651 - r77663;
        return r77664;
}
↓
double f(double x) {
        double r77665 = x;
        double r77666 = 2.30753;
        double r77667 = 0.27061;
        double r77668 = r77665 * r77667;
        double r77669 = r77666 + r77668;
        double r77670 = 1.0;
        double r77671 = 0.04481;
        double r77672 = 0.99229;
        double r77673 = fma(r77671, r77665, r77672);
        double r77674 = 1.0;
        double r77675 = fma(r77665, r77673, r77674);
        double r77676 = r77675 * r77670;
        double r77677 = r77670 / r77676;
        double r77678 = r77669 * r77677;
        double r77679 = r77665 - r77678;
        return r77679;
}

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. Using strategy rm

  3. Applied div-inv0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020062 +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)))))