Average Error: 0.0 → 0.0
Time: 2.9s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[\left(-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} + x\]
x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}
↓
\left(-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} + x
double f(double x) {
        double r101668 = x;
        double r101669 = 2.30753;
        double r101670 = 0.27061;
        double r101671 = r101668 * r101670;
        double r101672 = r101669 + r101671;
        double r101673 = 1.0;
        double r101674 = 0.99229;
        double r101675 = 0.04481;
        double r101676 = r101668 * r101675;
        double r101677 = r101674 + r101676;
        double r101678 = r101677 * r101668;
        double r101679 = r101673 + r101678;
        double r101680 = r101672 / r101679;
        double r101681 = r101668 - r101680;
        return r101681;
}
↓
double f(double x) {
        double r101682 = 0.27061;
        double r101683 = x;
        double r101684 = 2.30753;
        double r101685 = fma(r101682, r101683, r101684);
        double r101686 = -r101685;
        double r101687 = 1.0;
        double r101688 = 0.04481;
        double r101689 = 0.99229;
        double r101690 = fma(r101688, r101683, r101689);
        double r101691 = 1.0;
        double r101692 = fma(r101683, r101690, r101691);
        double r101693 = r101687 / r101692;
        double r101694 = r101686 * r101693;
        double r101695 = r101694 + r101683;
        return r101695;
}

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. Using strategy rm

  4. Applied div-inv0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +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)))))