Average Error: 43.4 → 0.8
Time: 32.9s
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 r48511 = x;
        double r48512 = exp(r48511);
        double r48513 = -r48511;
        double r48514 = exp(r48513);
        double r48515 = r48512 + r48514;
        double r48516 = 2.0;
        double r48517 = r48515 / r48516;
        double r48518 = y;
        double r48519 = cos(r48518);
        double r48520 = r48517 * r48519;
        double r48521 = r48512 - r48514;
        double r48522 = r48521 / r48516;
        double r48523 = sin(r48518);
        double r48524 = r48522 * r48523;
        double r48525 = /* ERROR: no complex support in C */;
        double r48526 = /* ERROR: no complex support in C */;
        return r48526;
}
↓
double f(double x, double y) {
        double r48527 = 0.3333333333333333;
        double r48528 = x;
        double r48529 = 3.0;
        double r48530 = pow(r48528, r48529);
        double r48531 = r48527 * r48530;
        double r48532 = 0.016666666666666666;
        double r48533 = 5.0;
        double r48534 = pow(r48528, r48533);
        double r48535 = r48532 * r48534;
        double r48536 = 2.0;
        double r48537 = r48536 * r48528;
        double r48538 = r48535 + r48537;
        double r48539 = r48531 + r48538;
        double r48540 = 2.0;
        double r48541 = r48539 / r48540;
        double r48542 = y;
        double r48543 = sin(r48542);
        double r48544 = r48541 * r48543;
        return r48544;
}

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. Simplified43.4

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

  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 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)))))