Average Error: 0.0 → 0.0
Time: 5.4s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(0.5, e^{im}, \sqrt{0.5} \cdot \frac{\sqrt{0.5}}{e^{im}}\right) \cdot \cos re\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(0.5, e^{im}, \sqrt{0.5} \cdot \frac{\sqrt{0.5}}{e^{im}}\right) \cdot \cos re
double f(double re, double im) {
        double r92257 = 0.5;
        double r92258 = re;
        double r92259 = cos(r92258);
        double r92260 = r92257 * r92259;
        double r92261 = im;
        double r92262 = -r92261;
        double r92263 = exp(r92262);
        double r92264 = exp(r92261);
        double r92265 = r92263 + r92264;
        double r92266 = r92260 * r92265;
        return r92266;
}
↓
double f(double re, double im) {
        double r92267 = 0.5;
        double r92268 = im;
        double r92269 = exp(r92268);
        double r92270 = sqrt(r92267);
        double r92271 = r92270 / r92269;
        double r92272 = r92270 * r92271;
        double r92273 = fma(r92267, r92269, r92272);
        double r92274 = re;
        double r92275 = cos(r92274);
        double r92276 = r92273 * r92275;
        return r92276;
}

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 *-un-lft-identity0.0

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{\color{blue}{1 \cdot e^{im}}}\right) \cdot \cos re\]

  5. Applied add-sqr-sqrt0.0

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{\sqrt{0.5} \cdot \sqrt{0.5}}}{1 \cdot e^{im}}\right) \cdot \cos re\]

  6. Applied times-frac0.0

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{\sqrt{0.5}}{1} \cdot \frac{\sqrt{0.5}}{e^{im}}}\right) \cdot \cos re\]

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))