Average Error: 43.6 → 0.6
Time: 14.1s
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 r60472 = x;
        double r60473 = exp(r60472);
        double r60474 = -r60472;
        double r60475 = exp(r60474);
        double r60476 = r60473 + r60475;
        double r60477 = 2.0;
        double r60478 = r60476 / r60477;
        double r60479 = y;
        double r60480 = cos(r60479);
        double r60481 = r60478 * r60480;
        double r60482 = r60473 - r60475;
        double r60483 = r60482 / r60477;
        double r60484 = sin(r60479);
        double r60485 = r60483 * r60484;
        double r60486 = /* ERROR: no complex support in C */;
        double r60487 = /* ERROR: no complex support in C */;
        return r60487;
}
↓
double f(double x, double y) {
        double r60488 = x;
        double r60489 = exp(r60488);
        double r60490 = -r60488;
        double r60491 = exp(r60490);
        double r60492 = r60489 + r60491;
        double r60493 = 2.0;
        double r60494 = r60492 / r60493;
        double r60495 = y;
        double r60496 = cos(r60495);
        double r60497 = r60494 * r60496;
        double r60498 = 0.3333333333333333;
        double r60499 = 3.0;
        double r60500 = pow(r60488, r60499);
        double r60501 = r60498 * r60500;
        double r60502 = 0.016666666666666666;
        double r60503 = 5.0;
        double r60504 = pow(r60488, r60503);
        double r60505 = r60502 * r60504;
        double r60506 = 2.0;
        double r60507 = r60506 * r60488;
        double r60508 = r60505 + r60507;
        double r60509 = r60501 + r60508;
        double r60510 = r60509 / r60493;
        double r60511 = sin(r60495);
        double r60512 = r60510 * r60511;
        double r60513 = /* ERROR: no complex support in C */;
        double r60514 = /* ERROR: no complex support in C */;
        return r60514;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.6

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

  2. Taylor expanded around 0 0.6

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

  3. Final simplification0.6

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

Reproduce

herbie shell --seed 2020056 
(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)))))