Average Error: 0.0 → 0.0
Time: 3.8s
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 r347 = 0.70711;
        double r348 = 2.30753;
        double r349 = x;
        double r350 = 0.27061;
        double r351 = r349 * r350;
        double r352 = r348 + r351;
        double r353 = 1.0;
        double r354 = 0.99229;
        double r355 = 0.04481;
        double r356 = r349 * r355;
        double r357 = r354 + r356;
        double r358 = r349 * r357;
        double r359 = r353 + r358;
        double r360 = r352 / r359;
        double r361 = r360 - r349;
        double r362 = r347 * r361;
        return r362;
}
↓
double f(double x) {
        double r363 = x;
        double r364 = -r363;
        double r365 = 0.70711;
        double r366 = 0.27061;
        double r367 = 2.30753;
        double r368 = fma(r366, r363, r367);
        double r369 = 0.04481;
        double r370 = 0.99229;
        double r371 = fma(r369, r363, r370);
        double r372 = 1.0;
        double r373 = fma(r363, r371, r372);
        double r374 = r368 / r373;
        double r375 = r365 * r374;
        double r376 = fma(r364, r365, r375);
        return r376;
}

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)))