Average Error: 0.0 → 0.0
Time: 12.5s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\sqrt{e^{-im}}, \sqrt{e^{-im}}, e^{im}\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\sqrt{e^{-im}}, \sqrt{e^{-im}}, e^{im}\right)
double f(double re, double im) {
        double r51186 = 0.5;
        double r51187 = re;
        double r51188 = cos(r51187);
        double r51189 = r51186 * r51188;
        double r51190 = im;
        double r51191 = -r51190;
        double r51192 = exp(r51191);
        double r51193 = exp(r51190);
        double r51194 = r51192 + r51193;
        double r51195 = r51189 * r51194;
        return r51195;
}
↓
double f(double re, double im) {
        double r51196 = 0.5;
        double r51197 = re;
        double r51198 = cos(r51197);
        double r51199 = r51196 * r51198;
        double r51200 = im;
        double r51201 = -r51200;
        double r51202 = exp(r51201);
        double r51203 = sqrt(r51202);
        double r51204 = exp(r51200);
        double r51205 = fma(r51203, r51203, r51204);
        double r51206 = r51199 * r51205;
        return r51206;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.0

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

  4. Applied fma-def0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))