Average Error: 43.2 → 0.8
Time: 32.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\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 r44347 = x;
        double r44348 = exp(r44347);
        double r44349 = -r44347;
        double r44350 = exp(r44349);
        double r44351 = r44348 + r44350;
        double r44352 = 2.0;
        double r44353 = r44351 / r44352;
        double r44354 = y;
        double r44355 = cos(r44354);
        double r44356 = r44353 * r44355;
        double r44357 = r44348 - r44350;
        double r44358 = r44357 / r44352;
        double r44359 = sin(r44354);
        double r44360 = r44358 * r44359;
        double r44361 = /* ERROR: no complex support in C */;
        double r44362 = /* ERROR: no complex support in C */;
        return r44362;
}
↓
double f(double x, double y) {
        double r44363 = 0.3333333333333333;
        double r44364 = x;
        double r44365 = 3.0;
        double r44366 = pow(r44364, r44365);
        double r44367 = r44363 * r44366;
        double r44368 = 0.016666666666666666;
        double r44369 = 5.0;
        double r44370 = pow(r44364, r44369);
        double r44371 = r44368 * r44370;
        double r44372 = 2.0;
        double r44373 = r44372 * r44364;
        double r44374 = r44371 + r44373;
        double r44375 = r44367 + r44374;
        double r44376 = 2.0;
        double r44377 = r44375 / r44376;
        double r44378 = y;
        double r44379 = sin(r44378);
        double r44380 = r44377 * r44379;
        return r44380;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.2

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

  2. Simplified43.2

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

  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 2019325 
(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)))))