Average Error: 0.0 → 0.0
Time: 1.8s
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 r82969 = x;
        double r82970 = 2.30753;
        double r82971 = 0.27061;
        double r82972 = r82969 * r82971;
        double r82973 = r82970 + r82972;
        double r82974 = 1.0;
        double r82975 = 0.99229;
        double r82976 = 0.04481;
        double r82977 = r82969 * r82976;
        double r82978 = r82975 + r82977;
        double r82979 = r82978 * r82969;
        double r82980 = r82974 + r82979;
        double r82981 = r82973 / r82980;
        double r82982 = r82969 - r82981;
        return r82982;
}
↓
double f(double x) {
        double r82983 = 0.27061;
        double r82984 = x;
        double r82985 = 2.30753;
        double r82986 = fma(r82983, r82984, r82985);
        double r82987 = -r82986;
        double r82988 = 1.0;
        double r82989 = 0.04481;
        double r82990 = 0.99229;
        double r82991 = fma(r82989, r82984, r82990);
        double r82992 = 1.0;
        double r82993 = fma(r82984, r82991, r82992);
        double r82994 = r82988 / r82993;
        double r82995 = r82987 * r82994;
        double r82996 = r82995 + r82984;
        return r82996;
}

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 2020025 +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)))))