Average Error: 0.0 → 0.0
Time: 8.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\frac{\cos re}{e^{im}} \cdot 0.5 + e^{im} \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\frac{\cos re}{e^{im}} \cdot 0.5 + e^{im} \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r67852 = 0.5;
        double r67853 = re;
        double r67854 = cos(r67853);
        double r67855 = r67852 * r67854;
        double r67856 = im;
        double r67857 = -r67856;
        double r67858 = exp(r67857);
        double r67859 = exp(r67856);
        double r67860 = r67858 + r67859;
        double r67861 = r67855 * r67860;
        return r67861;
}
↓
double f(double re, double im) {
        double r67862 = re;
        double r67863 = cos(r67862);
        double r67864 = im;
        double r67865 = exp(r67864);
        double r67866 = r67863 / r67865;
        double r67867 = 0.5;
        double r67868 = r67866 * r67867;
        double r67869 = r67867 * r67863;
        double r67870 = r67865 * r67869;
        double r67871 = r67868 + r67870;
        return r67871;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 distribute-lft-in0.0

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

  4. Simplified0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))