Average Error: 0.0 → 0.0
Time: 12.9s
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 r1625547 = 0.5;
        double r1625548 = re;
        double r1625549 = cos(r1625548);
        double r1625550 = r1625547 * r1625549;
        double r1625551 = im;
        double r1625552 = -r1625551;
        double r1625553 = exp(r1625552);
        double r1625554 = exp(r1625551);
        double r1625555 = r1625553 + r1625554;
        double r1625556 = r1625550 * r1625555;
        return r1625556;
}
↓
double f(double re, double im) {
        double r1625557 = re;
        double r1625558 = cos(r1625557);
        double r1625559 = im;
        double r1625560 = exp(r1625559);
        double r1625561 = r1625558 / r1625560;
        double r1625562 = fma(r1625558, r1625560, r1625561);
        double r1625563 = 0.5;
        double r1625564 = r1625562 * r1625563;
        return r1625564;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))