Average Error: 43.3 → 0.8
Time: 13.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\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{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r53350 = x;
        double r53351 = exp(r53350);
        double r53352 = -r53350;
        double r53353 = exp(r53352);
        double r53354 = r53351 + r53353;
        double r53355 = 2.0;
        double r53356 = r53354 / r53355;
        double r53357 = y;
        double r53358 = cos(r53357);
        double r53359 = r53356 * r53358;
        double r53360 = r53351 - r53353;
        double r53361 = r53360 / r53355;
        double r53362 = sin(r53357);
        double r53363 = r53361 * r53362;
        double r53364 = /* ERROR: no complex support in C */;
        double r53365 = /* ERROR: no complex support in C */;
        return r53365;
}
↓
double f(double x, double y) {
        double r53366 = x;
        double r53367 = exp(r53366);
        double r53368 = -r53366;
        double r53369 = exp(r53368);
        double r53370 = r53367 + r53369;
        double r53371 = 2.0;
        double r53372 = r53370 / r53371;
        double r53373 = y;
        double r53374 = cos(r53373);
        double r53375 = r53372 * r53374;
        double r53376 = 0.3333333333333333;
        double r53377 = 3.0;
        double r53378 = pow(r53366, r53377);
        double r53379 = r53376 * r53378;
        double r53380 = 0.016666666666666666;
        double r53381 = 5.0;
        double r53382 = pow(r53366, r53381);
        double r53383 = r53380 * r53382;
        double r53384 = 2.0;
        double r53385 = r53384 * r53366;
        double r53386 = r53383 + r53385;
        double r53387 = r53379 + r53386;
        double r53388 = r53387 / r53371;
        double r53389 = sin(r53373);
        double r53390 = r53388 * r53389;
        double r53391 = /* ERROR: no complex support in C */;
        double r53392 = /* ERROR: no complex support in C */;
        return r53392;
}

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. 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. Final simplification0.8

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

Reproduce

herbie shell --seed 2020036 
(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)))))