Average Error: 43.4 → 0.8
Time: 12.0s
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 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 = 0.3333333333333333;
        double r54761 = x;
        double r54762 = 3.0;
        double r54763 = pow(r54761, r54762);
        double r54764 = r54760 * r54763;
        double r54765 = 0.016666666666666666;
        double r54766 = 5.0;
        double r54767 = pow(r54761, r54766);
        double r54768 = r54765 * r54767;
        double r54769 = 2.0;
        double r54770 = r54769 * r54761;
        double r54771 = r54768 + r54770;
        double r54772 = r54764 + r54771;
        double r54773 = 2.0;
        double r54774 = r54772 / r54773;
        double r54775 = y;
        double r54776 = sin(r54775);
        double r54777 = r54774 * r54776;
        return r54777;
}

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 2020060 
(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)))))