Average Error: 43.7 → 0.8
Time: 15.2s
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 r54744 = x;
        double r54745 = exp(r54744);
        double r54746 = -r54744;
        double r54747 = exp(r54746);
        double r54748 = r54745 + r54747;
        double r54749 = 2.0;
        double r54750 = r54748 / r54749;
        double r54751 = y;
        double r54752 = cos(r54751);
        double r54753 = r54750 * r54752;
        double r54754 = r54745 - r54747;
        double r54755 = r54754 / r54749;
        double r54756 = sin(r54751);
        double r54757 = r54755 * r54756;
        double r54758 = /* ERROR: no complex support in C */;
        double r54759 = /* ERROR: no complex support in C */;
        return r54759;
}
↓
double f(double x, double y) {
        double r54760 = x;
        double r54761 = exp(r54760);
        double r54762 = -r54760;
        double r54763 = exp(r54762);
        double r54764 = r54761 + r54763;
        double r54765 = 2.0;
        double r54766 = r54764 / r54765;
        double r54767 = y;
        double r54768 = cos(r54767);
        double r54769 = r54766 * r54768;
        double r54770 = 0.3333333333333333;
        double r54771 = 3.0;
        double r54772 = pow(r54760, r54771);
        double r54773 = r54770 * r54772;
        double r54774 = 0.016666666666666666;
        double r54775 = 5.0;
        double r54776 = pow(r54760, r54775);
        double r54777 = r54774 * r54776;
        double r54778 = 2.0;
        double r54779 = r54778 * r54760;
        double r54780 = r54777 + r54779;
        double r54781 = r54773 + r54780;
        double r54782 = r54781 / r54765;
        double r54783 = sin(r54767);
        double r54784 = r54782 * r54783;
        double r54785 = /* ERROR: no complex support in C */;
        double r54786 = /* ERROR: no complex support in C */;
        return r54786;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.7

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

  2. Taylor expanded around 0 0.8

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 2020002 
(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)))))