Average Error: 0.0 → 0.0
Time: 2.6s
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 r85044 = x;
        double r85045 = 2.30753;
        double r85046 = 0.27061;
        double r85047 = r85044 * r85046;
        double r85048 = r85045 + r85047;
        double r85049 = 1.0;
        double r85050 = 0.99229;
        double r85051 = 0.04481;
        double r85052 = r85044 * r85051;
        double r85053 = r85050 + r85052;
        double r85054 = r85053 * r85044;
        double r85055 = r85049 + r85054;
        double r85056 = r85048 / r85055;
        double r85057 = r85044 - r85056;
        return r85057;
}
↓
double f(double x) {
        double r85058 = 0.27061;
        double r85059 = x;
        double r85060 = 2.30753;
        double r85061 = fma(r85058, r85059, r85060);
        double r85062 = -r85061;
        double r85063 = 0.04481;
        double r85064 = 0.99229;
        double r85065 = fma(r85063, r85059, r85064);
        double r85066 = 1.0;
        double r85067 = fma(r85059, r85065, r85066);
        double r85068 = r85062 / r85067;
        double r85069 = r85068 + r85059;
        return r85069;
}

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 2020089 +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)))))