Average Error: 0.0 → 0.0
Time: 20.6s
Precision: 64
\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1.0 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)\]
↓
\[0.70711 \cdot \left(\sqrt[3]{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1.0\right)} \cdot \left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1.0\right)} \cdot \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1.0\right)}\right)} - x\right)\]
0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1.0 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)
↓
0.70711 \cdot \left(\sqrt[3]{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1.0\right)} \cdot \left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1.0\right)} \cdot \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1.0\right)}\right)} - x\right)
double f(double x) {
        double r4230933 = 0.70711;
        double r4230934 = 2.30753;
        double r4230935 = x;
        double r4230936 = 0.27061;
        double r4230937 = r4230935 * r4230936;
        double r4230938 = r4230934 + r4230937;
        double r4230939 = 1.0;
        double r4230940 = 0.99229;
        double r4230941 = 0.04481;
        double r4230942 = r4230935 * r4230941;
        double r4230943 = r4230940 + r4230942;
        double r4230944 = r4230935 * r4230943;
        double r4230945 = r4230939 + r4230944;
        double r4230946 = r4230938 / r4230945;
        double r4230947 = r4230946 - r4230935;
        double r4230948 = r4230933 * r4230947;
        return r4230948;
}
↓
double f(double x) {
        double r4230949 = 0.70711;
        double r4230950 = 0.27061;
        double r4230951 = x;
        double r4230952 = 2.30753;
        double r4230953 = fma(r4230950, r4230951, r4230952);
        double r4230954 = 0.04481;
        double r4230955 = 0.99229;
        double r4230956 = fma(r4230951, r4230954, r4230955);
        double r4230957 = 1.0;
        double r4230958 = fma(r4230951, r4230956, r4230957);
        double r4230959 = r4230953 / r4230958;
        double r4230960 = r4230959 * r4230959;
        double r4230961 = r4230959 * r4230960;
        double r4230962 = cbrt(r4230961);
        double r4230963 = r4230962 - r4230951;
        double r4230964 = r4230949 * r4230963;
        return r4230964;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1.0 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1.0\right)} - x\right) \cdot 0.70711}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube0.0

    \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1.0\right)} \cdot \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1.0\right)}\right) \cdot \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1.0\right)}}} - x\right) \cdot 0.70711\]

  5. Final simplification0.0

    \[\leadsto 0.70711 \cdot \left(\sqrt[3]{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1.0\right)} \cdot \left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1.0\right)} \cdot \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1.0\right)}\right)} - x\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))