Average Error: 44.2 → 0.7
Time: 34.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\left(x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2} \cdot \sin y i\right))\]
\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{\left(x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2480361 = x;
        double r2480362 = exp(r2480361);
        double r2480363 = -r2480361;
        double r2480364 = exp(r2480363);
        double r2480365 = r2480362 + r2480364;
        double r2480366 = 2.0;
        double r2480367 = r2480365 / r2480366;
        double r2480368 = y;
        double r2480369 = cos(r2480368);
        double r2480370 = r2480367 * r2480369;
        double r2480371 = r2480362 - r2480364;
        double r2480372 = r2480371 / r2480366;
        double r2480373 = sin(r2480368);
        double r2480374 = r2480372 * r2480373;
        double r2480375 = /* ERROR: no complex support in C */;
        double r2480376 = /* ERROR: no complex support in C */;
        return r2480376;
}
↓
double f(double x, double y) {
        double r2480377 = x;
        double r2480378 = exp(r2480377);
        double r2480379 = -r2480377;
        double r2480380 = exp(r2480379);
        double r2480381 = r2480378 + r2480380;
        double r2480382 = 2.0;
        double r2480383 = r2480381 / r2480382;
        double r2480384 = y;
        double r2480385 = cos(r2480384);
        double r2480386 = r2480383 * r2480385;
        double r2480387 = 0.3333333333333333;
        double r2480388 = r2480377 * r2480377;
        double r2480389 = r2480387 * r2480388;
        double r2480390 = r2480377 * r2480389;
        double r2480391 = 0.016666666666666666;
        double r2480392 = 5.0;
        double r2480393 = pow(r2480377, r2480392);
        double r2480394 = r2480391 * r2480393;
        double r2480395 = r2480390 + r2480394;
        double r2480396 = r2480382 * r2480377;
        double r2480397 = r2480395 + r2480396;
        double r2480398 = r2480397 / r2480382;
        double r2480399 = sin(r2480384);
        double r2480400 = r2480398 * r2480399;
        double r2480401 = /* ERROR: no complex support in C */;
        double r2480402 = /* ERROR: no complex support in C */;
        return r2480402;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 44.2

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019165 
(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)))))