Average Error: 43.4 → 0.8
Time: 23.3s
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 r50429 = x;
        double r50430 = exp(r50429);
        double r50431 = -r50429;
        double r50432 = exp(r50431);
        double r50433 = r50430 + r50432;
        double r50434 = 2.0;
        double r50435 = r50433 / r50434;
        double r50436 = y;
        double r50437 = cos(r50436);
        double r50438 = r50435 * r50437;
        double r50439 = r50430 - r50432;
        double r50440 = r50439 / r50434;
        double r50441 = sin(r50436);
        double r50442 = r50440 * r50441;
        double r50443 = /* ERROR: no complex support in C */;
        double r50444 = /* ERROR: no complex support in C */;
        return r50444;
}
↓
double f(double x, double y) {
        double r50445 = 0.3333333333333333;
        double r50446 = x;
        double r50447 = 3.0;
        double r50448 = pow(r50446, r50447);
        double r50449 = r50445 * r50448;
        double r50450 = 0.016666666666666666;
        double r50451 = 5.0;
        double r50452 = pow(r50446, r50451);
        double r50453 = r50450 * r50452;
        double r50454 = 2.0;
        double r50455 = r50454 * r50446;
        double r50456 = r50453 + r50455;
        double r50457 = r50449 + r50456;
        double r50458 = 2.0;
        double r50459 = r50457 / r50458;
        double r50460 = y;
        double r50461 = sin(r50460);
        double r50462 = r50459 * r50461;
        return r50462;
}

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

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 2019304 
(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)))))