Average Error: 43.4 → 0.9
Time: 12.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\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
double f(double x, double y) {
        double r49300 = x;
        double r49301 = exp(r49300);
        double r49302 = -r49300;
        double r49303 = exp(r49302);
        double r49304 = r49301 + r49303;
        double r49305 = 2.0;
        double r49306 = r49304 / r49305;
        double r49307 = y;
        double r49308 = cos(r49307);
        double r49309 = r49306 * r49308;
        double r49310 = r49301 - r49303;
        double r49311 = r49310 / r49305;
        double r49312 = sin(r49307);
        double r49313 = r49311 * r49312;
        double r49314 = /* ERROR: no complex support in C */;
        double r49315 = /* ERROR: no complex support in C */;
        return r49315;
}
↓
double f(double x, double y) {
        double r49316 = 0.3333333333333333;
        double r49317 = x;
        double r49318 = 3.0;
        double r49319 = pow(r49317, r49318);
        double r49320 = 0.016666666666666666;
        double r49321 = 5.0;
        double r49322 = pow(r49317, r49321);
        double r49323 = 2.0;
        double r49324 = r49323 * r49317;
        double r49325 = fma(r49320, r49322, r49324);
        double r49326 = fma(r49316, r49319, r49325);
        double r49327 = 2.0;
        double r49328 = r49326 / r49327;
        double r49329 = y;
        double r49330 = sin(r49329);
        double r49331 = r49328 * r49330;
        return r49331;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

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

  2. Simplified43.4

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.9

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

  4. Simplified0.9

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

  5. Final simplification0.9

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

Reproduce

herbie shell --seed 2020059 +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)))))