Average Error: 0.0 → 0.0
Time: 2.5s
Precision: 64
\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.707110000000000016, \frac{\sqrt{0.707110000000000016} \cdot \left(\sqrt{0.707110000000000016} \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)\]
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, \frac{\sqrt{0.707110000000000016} \cdot \left(\sqrt{0.707110000000000016} \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)
double f(double x) {
        double r104180 = 0.70711;
        double r104181 = 2.30753;
        double r104182 = x;
        double r104183 = 0.27061;
        double r104184 = r104182 * r104183;
        double r104185 = r104181 + r104184;
        double r104186 = 1.0;
        double r104187 = 0.99229;
        double r104188 = 0.04481;
        double r104189 = r104182 * r104188;
        double r104190 = r104187 + r104189;
        double r104191 = r104182 * r104190;
        double r104192 = r104186 + r104191;
        double r104193 = r104185 / r104192;
        double r104194 = r104193 - r104182;
        double r104195 = r104180 * r104194;
        return r104195;
}
↓
double f(double x) {
        double r104196 = x;
        double r104197 = -r104196;
        double r104198 = 0.70711;
        double r104199 = sqrt(r104198);
        double r104200 = 0.27061;
        double r104201 = 2.30753;
        double r104202 = fma(r104200, r104196, r104201);
        double r104203 = r104199 * r104202;
        double r104204 = r104199 * r104203;
        double r104205 = 0.04481;
        double r104206 = 0.99229;
        double r104207 = fma(r104205, r104196, r104206);
        double r104208 = 1.0;
        double r104209 = fma(r104196, r104207, r104208);
        double r104210 = r104204 / r104209;
        double r104211 = fma(r104197, r104198, r104210);
        return r104211;
}

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 add-sqr-sqrt0.0

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

  5. Applied associate-*l*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020056 +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)))