Average Error: 0.0 → 0.0
Time: 6.0s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \cos re\right)
double f(double re, double im) {
        double r58346 = 0.5;
        double r58347 = re;
        double r58348 = cos(r58347);
        double r58349 = r58346 * r58348;
        double r58350 = im;
        double r58351 = -r58350;
        double r58352 = exp(r58351);
        double r58353 = exp(r58350);
        double r58354 = r58352 + r58353;
        double r58355 = r58349 * r58354;
        return r58355;
}
↓
double f(double re, double im) {
        double r58356 = 0.5;
        double r58357 = im;
        double r58358 = exp(r58357);
        double r58359 = r58356 / r58358;
        double r58360 = fma(r58356, r58358, r58359);
        double r58361 = sqrt(r58360);
        double r58362 = re;
        double r58363 = cos(r58362);
        double r58364 = r58361 * r58363;
        double r58365 = r58361 * r58364;
        return r58365;
}

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. Using strategy rm

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-*l*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))