Average Error: 43.2 → 0.7
Time: 25.3s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{{x}^{5}}{60} + \left(\left(\frac{{x}^{3}}{3} + x\right) + x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{{x}^{5}}{60} + \left(\left(\frac{{x}^{3}}{3} + x\right) + x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r113610 = x;
        double r113611 = exp(r113610);
        double r113612 = -r113610;
        double r113613 = exp(r113612);
        double r113614 = r113611 + r113613;
        double r113615 = 2.0;
        double r113616 = r113614 / r113615;
        double r113617 = y;
        double r113618 = cos(r113617);
        double r113619 = r113616 * r113618;
        double r113620 = r113611 - r113613;
        double r113621 = r113620 / r113615;
        double r113622 = sin(r113617);
        double r113623 = r113621 * r113622;
        double r113624 = /* ERROR: no complex support in C */;
        double r113625 = /* ERROR: no complex support in C */;
        return r113625;
}
↓
double f(double x, double y) {
        double r113626 = x;
        double r113627 = 5.0;
        double r113628 = pow(r113626, r113627);
        double r113629 = 60.0;
        double r113630 = r113628 / r113629;
        double r113631 = 3.0;
        double r113632 = pow(r113626, r113631);
        double r113633 = r113632 / r113631;
        double r113634 = r113633 + r113626;
        double r113635 = r113634 + r113626;
        double r113636 = r113630 + r113635;
        double r113637 = 2.0;
        double r113638 = r113636 / r113637;
        double r113639 = y;
        double r113640 = sin(r113639);
        double r113641 = r113638 * r113640;
        return r113641;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.2

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 350497007 
(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)))))