Average Error: 0.0 → 0.0
Time: 4.1s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \cdot \cos re\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \cdot \cos re
double f(double re, double im) {
        double r57641 = 0.5;
        double r57642 = re;
        double r57643 = cos(r57642);
        double r57644 = r57641 * r57643;
        double r57645 = im;
        double r57646 = -r57645;
        double r57647 = exp(r57646);
        double r57648 = exp(r57645);
        double r57649 = r57647 + r57648;
        double r57650 = r57644 * r57649;
        return r57650;
}
↓
double f(double re, double im) {
        double r57651 = 0.5;
        double r57652 = im;
        double r57653 = exp(r57652);
        double r57654 = r57651 / r57653;
        double r57655 = fma(r57651, r57653, r57654);
        double r57656 = re;
        double r57657 = cos(r57656);
        double r57658 = r57655 * r57657;
        return r57658;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))