Average Error: 43.1 → 0.8
Time: 12.4s
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 r49432 = x;
        double r49433 = exp(r49432);
        double r49434 = -r49432;
        double r49435 = exp(r49434);
        double r49436 = r49433 + r49435;
        double r49437 = 2.0;
        double r49438 = r49436 / r49437;
        double r49439 = y;
        double r49440 = cos(r49439);
        double r49441 = r49438 * r49440;
        double r49442 = r49433 - r49435;
        double r49443 = r49442 / r49437;
        double r49444 = sin(r49439);
        double r49445 = r49443 * r49444;
        double r49446 = /* ERROR: no complex support in C */;
        double r49447 = /* ERROR: no complex support in C */;
        return r49447;
}
↓
double f(double x, double y) {
        double r49448 = 0.3333333333333333;
        double r49449 = x;
        double r49450 = 3.0;
        double r49451 = pow(r49449, r49450);
        double r49452 = r49448 * r49451;
        double r49453 = 0.016666666666666666;
        double r49454 = 5.0;
        double r49455 = pow(r49449, r49454);
        double r49456 = r49453 * r49455;
        double r49457 = 2.0;
        double r49458 = r49457 * r49449;
        double r49459 = r49456 + r49458;
        double r49460 = r49452 + r49459;
        double r49461 = 2.0;
        double r49462 = r49460 / r49461;
        double r49463 = y;
        double r49464 = sin(r49463);
        double r49465 = r49462 * r49464;
        return r49465;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.1

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

  2. Simplified43.1

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

  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 2020020 
(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)))))