Average Error: 43.8 → 0.8
Time: 15.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r80214 = x;
        double r80215 = exp(r80214);
        double r80216 = -r80214;
        double r80217 = exp(r80216);
        double r80218 = r80215 + r80217;
        double r80219 = 2.0;
        double r80220 = r80218 / r80219;
        double r80221 = y;
        double r80222 = cos(r80221);
        double r80223 = r80220 * r80222;
        double r80224 = r80215 - r80217;
        double r80225 = r80224 / r80219;
        double r80226 = sin(r80221);
        double r80227 = r80225 * r80226;
        double r80228 = /* ERROR: no complex support in C */;
        double r80229 = /* ERROR: no complex support in C */;
        return r80229;
}
↓
double f(double x, double y) {
        double r80230 = x;
        double r80231 = exp(r80230);
        double r80232 = -r80230;
        double r80233 = exp(r80232);
        double r80234 = r80231 + r80233;
        double r80235 = 2.0;
        double r80236 = r80234 / r80235;
        double r80237 = y;
        double r80238 = cos(r80237);
        double r80239 = r80236 * r80238;
        double r80240 = 0.3333333333333333;
        double r80241 = 3.0;
        double r80242 = pow(r80230, r80241);
        double r80243 = 0.016666666666666666;
        double r80244 = 5.0;
        double r80245 = pow(r80230, r80244);
        double r80246 = 2.0;
        double r80247 = r80246 * r80230;
        double r80248 = fma(r80243, r80245, r80247);
        double r80249 = fma(r80240, r80242, r80248);
        double r80250 = r80249 / r80235;
        double r80251 = sin(r80237);
        double r80252 = r80250 * r80251;
        double r80253 = /* ERROR: no complex support in C */;
        double r80254 = /* ERROR: no complex support in C */;
        return r80254;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.8

    \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  2. Taylor expanded around 0 0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2} \cdot \sin y i\right))\]

  3. Simplified0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}}{2} \cdot \sin y i\right))\]

  4. Final simplification0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))