Average Error: 0.0 → 0.0
Time: 3.0s
Precision: 64
\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)\]
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{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)
double f(double x) {
        double r87407 = 0.70711;
        double r87408 = 2.30753;
        double r87409 = x;
        double r87410 = 0.27061;
        double r87411 = r87409 * r87410;
        double r87412 = r87408 + r87411;
        double r87413 = 1.0;
        double r87414 = 0.99229;
        double r87415 = 0.04481;
        double r87416 = r87409 * r87415;
        double r87417 = r87414 + r87416;
        double r87418 = r87409 * r87417;
        double r87419 = r87413 + r87418;
        double r87420 = r87412 / r87419;
        double r87421 = r87420 - r87409;
        double r87422 = r87407 * r87421;
        return r87422;
}
↓
double f(double x) {
        double r87423 = x;
        double r87424 = -r87423;
        double r87425 = 0.70711;
        double r87426 = 0.27061;
        double r87427 = 2.30753;
        double r87428 = fma(r87426, r87423, r87427);
        double r87429 = r87425 * r87428;
        double r87430 = 0.04481;
        double r87431 = 0.99229;
        double r87432 = fma(r87430, r87423, r87431);
        double r87433 = 1.0;
        double r87434 = fma(r87423, r87432, r87433);
        double r87435 = r87429 / r87434;
        double r87436 = fma(r87424, r87425, r87435);
        return r87436;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 +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)))