Average Error: 43.4 → 0.8
Time: 54.2s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right) + {x}^{5} \cdot \frac{1}{60}}{2} \cdot \sin y i\right))\]
\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{x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right) + {x}^{5} \cdot \frac{1}{60}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1689234 = x;
        double r1689235 = exp(r1689234);
        double r1689236 = -r1689234;
        double r1689237 = exp(r1689236);
        double r1689238 = r1689235 + r1689237;
        double r1689239 = 2.0;
        double r1689240 = r1689238 / r1689239;
        double r1689241 = y;
        double r1689242 = cos(r1689241);
        double r1689243 = r1689240 * r1689242;
        double r1689244 = r1689235 - r1689237;
        double r1689245 = r1689244 / r1689239;
        double r1689246 = sin(r1689241);
        double r1689247 = r1689245 * r1689246;
        double r1689248 = /* ERROR: no complex support in C */;
        double r1689249 = /* ERROR: no complex support in C */;
        return r1689249;
}
↓
double f(double x, double y) {
        double r1689250 = x;
        double r1689251 = exp(r1689250);
        double r1689252 = -r1689250;
        double r1689253 = exp(r1689252);
        double r1689254 = r1689251 + r1689253;
        double r1689255 = 2.0;
        double r1689256 = r1689254 / r1689255;
        double r1689257 = y;
        double r1689258 = cos(r1689257);
        double r1689259 = r1689256 * r1689258;
        double r1689260 = r1689250 * r1689250;
        double r1689261 = 0.3333333333333333;
        double r1689262 = r1689260 * r1689261;
        double r1689263 = r1689262 + r1689255;
        double r1689264 = r1689250 * r1689263;
        double r1689265 = 5.0;
        double r1689266 = pow(r1689250, r1689265);
        double r1689267 = 0.016666666666666666;
        double r1689268 = r1689266 * r1689267;
        double r1689269 = r1689264 + r1689268;
        double r1689270 = r1689269 / r1689255;
        double r1689271 = sin(r1689257);
        double r1689272 = r1689270 * r1689271;
        double r1689273 = /* ERROR: no complex support in C */;
        double r1689274 = /* ERROR: no complex support in C */;
        return r1689274;
}

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. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019137 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))