Average Error: 0.0 → 0.0
Time: 5.2s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\frac{e^{im}}{\sqrt[3]{0.5}}}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\frac{e^{im}}{\sqrt[3]{0.5}}}\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 r94215 = 0.5;
        double r94216 = re;
        double r94217 = cos(r94216);
        double r94218 = r94215 * r94217;
        double r94219 = im;
        double r94220 = -r94219;
        double r94221 = exp(r94220);
        double r94222 = exp(r94219);
        double r94223 = r94221 + r94222;
        double r94224 = r94218 * r94223;
        return r94224;
}
↓
double f(double re, double im) {
        double r94225 = 0.5;
        double r94226 = im;
        double r94227 = exp(r94226);
        double r94228 = cbrt(r94225);
        double r94229 = r94228 * r94228;
        double r94230 = r94227 / r94228;
        double r94231 = r94229 / r94230;
        double r94232 = fma(r94225, r94227, r94231);
        double r94233 = sqrt(r94232);
        double r94234 = r94225 / r94227;
        double r94235 = fma(r94225, r94227, r94234);
        double r94236 = sqrt(r94235);
        double r94237 = re;
        double r94238 = cos(r94237);
        double r94239 = r94236 * r94238;
        double r94240 = r94233 * r94239;
        return r94240;
}

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. Using strategy rm

  7. Applied add-cube-cbrt0.0

    \[\leadsto \sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{\left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{0.5}}}{e^{im}}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \cos re\right)\]

  8. Applied associate-/l*0.0

    \[\leadsto \sqrt{\mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\frac{e^{im}}{\sqrt[3]{0.5}}}}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \cos re\right)\]

  9. Final simplification0.0

    \[\leadsto \sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\frac{e^{im}}{\sqrt[3]{0.5}}}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \cos re\right)\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))