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 r94013 = x;
        double r94014 = 2.30753;
        double r94015 = 0.27061;
        double r94016 = r94013 * r94015;
        double r94017 = r94014 + r94016;
        double r94018 = 1.0;
        double r94019 = 0.99229;
        double r94020 = 0.04481;
        double r94021 = r94013 * r94020;
        double r94022 = r94019 + r94021;
        double r94023 = r94022 * r94013;
        double r94024 = r94018 + r94023;
        double r94025 = r94017 / r94024;
        double r94026 = r94013 - r94025;
        return r94026;
}
↓
double f(double x) {
        double r94027 = 0.27061;
        double r94028 = x;
        double r94029 = 2.30753;
        double r94030 = fma(r94027, r94028, r94029);
        double r94031 = -r94030;
        double r94032 = 0.04481;
        double r94033 = 0.99229;
        double r94034 = fma(r94032, r94028, r94033);
        double r94035 = 1.0;
        double r94036 = fma(r94028, r94034, r94035);
        double r94037 = r94031 / r94036;
        double r94038 = r94037 + r94028;
        return r94038;
}

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 2020046 +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)))))