Average Error: 0.0 → 0.0
Time: 2.4s
Precision: 64
\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.707110000000000016, 0.707110000000000016 \cdot \frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)\]
0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)
↓
\mathsf{fma}\left(-x, 0.707110000000000016, 0.707110000000000016 \cdot \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)}\right)
double f(double x) {
        double r94191 = 0.70711;
        double r94192 = 2.30753;
        double r94193 = x;
        double r94194 = 0.27061;
        double r94195 = r94193 * r94194;
        double r94196 = r94192 + r94195;
        double r94197 = 1.0;
        double r94198 = 0.99229;
        double r94199 = 0.04481;
        double r94200 = r94193 * r94199;
        double r94201 = r94198 + r94200;
        double r94202 = r94193 * r94201;
        double r94203 = r94197 + r94202;
        double r94204 = r94196 / r94203;
        double r94205 = r94204 - r94193;
        double r94206 = r94191 * r94205;
        return r94206;
}
↓
double f(double x) {
        double r94207 = x;
        double r94208 = -r94207;
        double r94209 = 0.70711;
        double r94210 = 0.27061;
        double r94211 = 2.30753;
        double r94212 = fma(r94210, r94207, r94211);
        double r94213 = 0.04481;
        double r94214 = 0.99229;
        double r94215 = fma(r94213, r94207, r94214);
        double r94216 = 1.0;
        double r94217 = fma(r94207, r94215, r94216);
        double r94218 = r94212 / r94217;
        double r94219 = r94209 * r94218;
        double r94220 = fma(r94208, r94209, r94219);
        return r94220;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}\right)\]

  5. Applied times-frac0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \color{blue}{\frac{0.707110000000000016}{1} \cdot \frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}}\right)\]

  6. Simplified0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \color{blue}{0.707110000000000016} \cdot \frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)\]

  7. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, 0.707110000000000016 \cdot \frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* x (+ 0.99229 (* x 0.04481))))) x)))