Average Error: 43.4 → 0.7
Time: 47.6s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{x \cdot \left(\left(x \cdot \frac{1}{3}\right) \cdot x + 2\right) + {x}^{5} \cdot \frac{1}{60}}{2} \cdot \sin y i\right))\]
\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{x \cdot \left(\left(x \cdot \frac{1}{3}\right) \cdot x + 2\right) + {x}^{5} \cdot \frac{1}{60}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1714557 = x;
        double r1714558 = exp(r1714557);
        double r1714559 = -r1714557;
        double r1714560 = exp(r1714559);
        double r1714561 = r1714558 + r1714560;
        double r1714562 = 2.0;
        double r1714563 = r1714561 / r1714562;
        double r1714564 = y;
        double r1714565 = cos(r1714564);
        double r1714566 = r1714563 * r1714565;
        double r1714567 = r1714558 - r1714560;
        double r1714568 = r1714567 / r1714562;
        double r1714569 = sin(r1714564);
        double r1714570 = r1714568 * r1714569;
        double r1714571 = /* ERROR: no complex support in C */;
        double r1714572 = /* ERROR: no complex support in C */;
        return r1714572;
}
↓
double f(double x, double y) {
        double r1714573 = x;
        double r1714574 = exp(r1714573);
        double r1714575 = -r1714573;
        double r1714576 = exp(r1714575);
        double r1714577 = r1714574 + r1714576;
        double r1714578 = 2.0;
        double r1714579 = r1714577 / r1714578;
        double r1714580 = y;
        double r1714581 = cos(r1714580);
        double r1714582 = r1714579 * r1714581;
        double r1714583 = 0.3333333333333333;
        double r1714584 = r1714573 * r1714583;
        double r1714585 = r1714584 * r1714573;
        double r1714586 = r1714585 + r1714578;
        double r1714587 = r1714573 * r1714586;
        double r1714588 = 5.0;
        double r1714589 = pow(r1714573, r1714588);
        double r1714590 = 0.016666666666666666;
        double r1714591 = r1714589 * r1714590;
        double r1714592 = r1714587 + r1714591;
        double r1714593 = r1714592 / r1714578;
        double r1714594 = sin(r1714580);
        double r1714595 = r1714593 * r1714594;
        double r1714596 = /* ERROR: no complex support in C */;
        double r1714597 = /* ERROR: no complex support in C */;
        return r1714597;
}

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 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019146 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))