math.cos on complex, real part

Average Error: 0.0 → 0.0

Time: 4.7s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r53875 = 0.5;
        double r53876 = re;
        double r53877 = cos(r53876);
        double r53878 = r53875 * r53877;
        double r53879 = im;
        double r53880 = -r53879;
        double r53881 = exp(r53880);
        double r53882 = exp(r53879);
        double r53883 = r53881 + r53882;
        double r53884 = r53878 * r53883;
        return r53884;
}

↓

double f(double re, double im) {
        double r53885 = 0.5;
        double r53886 = re;
        double r53887 = cos(r53886);
        double r53888 = r53885 * r53887;
        double r53889 = im;
        double r53890 = -r53889;
        double r53891 = exp(r53890);
        double r53892 = sqrt(r53891);
        double r53893 = exp(r53889);
        double r53894 = fma(r53892, r53892, r53893);
        double r53895 = r53888 * r53894;
        return r53895;
}

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-sqrt 0.0

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

4. Applied fma-def 0.0

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

5. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))