Average Error: 43.3 → 0.8
Time: 17.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r216 = x;
        double r217 = exp(r216);
        double r218 = -r216;
        double r219 = exp(r218);
        double r220 = r217 + r219;
        double r221 = 2.0;
        double r222 = r220 / r221;
        double r223 = y;
        double r224 = cos(r223);
        double r225 = r222 * r224;
        double r226 = r217 - r219;
        double r227 = r226 / r221;
        double r228 = sin(r223);
        double r229 = r227 * r228;
        double r230 = /* ERROR: no complex support in C */;
        double r231 = /* ERROR: no complex support in C */;
        return r231;
}
↓
double f(double x, double y) {
        double r232 = 0.3333333333333333;
        double r233 = x;
        double r234 = 3.0;
        double r235 = pow(r233, r234);
        double r236 = r232 * r235;
        double r237 = 0.016666666666666666;
        double r238 = 5.0;
        double r239 = pow(r233, r238);
        double r240 = r237 * r239;
        double r241 = 2.0;
        double r242 = r241 * r233;
        double r243 = r240 + r242;
        double r244 = r236 + r243;
        double r245 = 2.0;
        double r246 = r244 / r245;
        double r247 = y;
        double r248 = sin(r247);
        double r249 = r246 * r248;
        return r249;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.3

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

  2. Simplified43.3

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

  3. Taylor expanded around 0 0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020025 
(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)))))