Average Error: 0.0 → 0.0
Time: 16.6s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\cos re \cdot \mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\cos re \cdot \mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)
double f(double re, double im) {
        double r1217213 = 0.5;
        double r1217214 = re;
        double r1217215 = cos(r1217214);
        double r1217216 = r1217213 * r1217215;
        double r1217217 = im;
        double r1217218 = -r1217217;
        double r1217219 = exp(r1217218);
        double r1217220 = exp(r1217217);
        double r1217221 = r1217219 + r1217220;
        double r1217222 = r1217216 * r1217221;
        return r1217222;
}
↓
double f(double re, double im) {
        double r1217223 = re;
        double r1217224 = cos(r1217223);
        double r1217225 = im;
        double r1217226 = exp(r1217225);
        double r1217227 = 0.5;
        double r1217228 = r1217227 / r1217226;
        double r1217229 = fma(r1217226, r1217227, r1217228);
        double r1217230 = r1217224 * r1217229;
        return r1217230;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))