Average Error: 0.0 → 0.0
Time: 19.4s
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 r4749942 = 0.70711;
        double r4749943 = 2.30753;
        double r4749944 = x;
        double r4749945 = 0.27061;
        double r4749946 = r4749944 * r4749945;
        double r4749947 = r4749943 + r4749946;
        double r4749948 = 1.0;
        double r4749949 = 0.99229;
        double r4749950 = 0.04481;
        double r4749951 = r4749944 * r4749950;
        double r4749952 = r4749949 + r4749951;
        double r4749953 = r4749944 * r4749952;
        double r4749954 = r4749948 + r4749953;
        double r4749955 = r4749947 / r4749954;
        double r4749956 = r4749955 - r4749944;
        double r4749957 = r4749942 * r4749956;
        return r4749957;
}
↓
double f(double x) {
        double r4749958 = 0.70711;
        double r4749959 = 0.27061;
        double r4749960 = x;
        double r4749961 = 2.30753;
        double r4749962 = fma(r4749959, r4749960, r4749961);
        double r4749963 = 0.04481;
        double r4749964 = 0.99229;
        double r4749965 = fma(r4749960, r4749963, r4749964);
        double r4749966 = 1.0;
        double r4749967 = fma(r4749960, r4749965, r4749966);
        double r4749968 = r4749962 / r4749967;
        double r4749969 = r4749968 * r4749968;
        double r4749970 = r4749968 * r4749969;
        double r4749971 = cbrt(r4749970);
        double r4749972 = r4749971 - r4749960;
        double r4749973 = r4749958 * r4749972;
        return r4749973;
}

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)))