Average Error: 0.0 → 0.0
Time: 16.6s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(\cos re, e^{im}, \frac{\cos re}{e^{im}}\right) \cdot 0.5\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(\cos re, e^{im}, \frac{\cos re}{e^{im}}\right) \cdot 0.5
double f(double re, double im) {
        double r32100 = 0.5;
        double r32101 = re;
        double r32102 = cos(r32101);
        double r32103 = r32100 * r32102;
        double r32104 = im;
        double r32105 = -r32104;
        double r32106 = exp(r32105);
        double r32107 = exp(r32104);
        double r32108 = r32106 + r32107;
        double r32109 = r32103 * r32108;
        return r32109;
}
↓
double f(double re, double im) {
        double r32110 = re;
        double r32111 = cos(r32110);
        double r32112 = im;
        double r32113 = exp(r32112);
        double r32114 = r32111 / r32113;
        double r32115 = fma(r32111, r32113, r32114);
        double r32116 = 0.5;
        double r32117 = r32115 * r32116;
        return r32117;
}

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 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

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

  5. Simplified0.0

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

  6. Taylor expanded around inf 0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))