Average Error: 0.0 → 0.0
Time: 13.3s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(\left(e^{im}\right), 0.5, \left(\frac{0.5}{e^{im}}\right)\right) \cdot \cos re\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(\left(e^{im}\right), 0.5, \left(\frac{0.5}{e^{im}}\right)\right) \cdot \cos re
double f(double re, double im) {
        double r879194 = 0.5;
        double r879195 = re;
        double r879196 = cos(r879195);
        double r879197 = r879194 * r879196;
        double r879198 = im;
        double r879199 = -r879198;
        double r879200 = exp(r879199);
        double r879201 = exp(r879198);
        double r879202 = r879200 + r879201;
        double r879203 = r879197 * r879202;
        return r879203;
}
↓
double f(double re, double im) {
        double r879204 = im;
        double r879205 = exp(r879204);
        double r879206 = 0.5;
        double r879207 = r879206 / r879205;
        double r879208 = fma(r879205, r879206, r879207);
        double r879209 = re;
        double r879210 = cos(r879209);
        double r879211 = r879208 * r879210;
        return r879211;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\left(e^{im}\right), 0.5, \left(\frac{0.5}{e^{im}}\right)\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\left(e^{im}\right), 0.5, \left(\frac{0.5}{e^{im}}\right)\right) \cdot \cos re\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))