Average Error: 0.0 → 0.0
Time: 3.0s
Precision: 64
\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.707110000000000016, \frac{1}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}\right)\]
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, \frac{1}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}\right)
double f(double x) {
        double r83259 = 0.70711;
        double r83260 = 2.30753;
        double r83261 = x;
        double r83262 = 0.27061;
        double r83263 = r83261 * r83262;
        double r83264 = r83260 + r83263;
        double r83265 = 1.0;
        double r83266 = 0.99229;
        double r83267 = 0.04481;
        double r83268 = r83261 * r83267;
        double r83269 = r83266 + r83268;
        double r83270 = r83261 * r83269;
        double r83271 = r83265 + r83270;
        double r83272 = r83264 / r83271;
        double r83273 = r83272 - r83261;
        double r83274 = r83259 * r83273;
        return r83274;
}
↓
double f(double x) {
        double r83275 = x;
        double r83276 = -r83275;
        double r83277 = 0.70711;
        double r83278 = 1.0;
        double r83279 = 0.04481;
        double r83280 = 0.99229;
        double r83281 = fma(r83279, r83275, r83280);
        double r83282 = 1.0;
        double r83283 = fma(r83275, r83281, r83282);
        double r83284 = 0.27061;
        double r83285 = 2.30753;
        double r83286 = fma(r83284, r83275, r83285);
        double r83287 = r83277 * r83286;
        double r83288 = r83283 / r83287;
        double r83289 = r83278 / r83288;
        double r83290 = fma(r83276, r83277, r83289);
        return r83290;
}

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 clear-num0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020081 +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)))