Average Error: 43.4 → 0.9
Time: 12.2s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\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 r54797 = x;
        double r54798 = exp(r54797);
        double r54799 = -r54797;
        double r54800 = exp(r54799);
        double r54801 = r54798 + r54800;
        double r54802 = 2.0;
        double r54803 = r54801 / r54802;
        double r54804 = y;
        double r54805 = cos(r54804);
        double r54806 = r54803 * r54805;
        double r54807 = r54798 - r54800;
        double r54808 = r54807 / r54802;
        double r54809 = sin(r54804);
        double r54810 = r54808 * r54809;
        double r54811 = /* ERROR: no complex support in C */;
        double r54812 = /* ERROR: no complex support in C */;
        return r54812;
}
↓
double f(double x, double y) {
        double r54813 = 0.3333333333333333;
        double r54814 = x;
        double r54815 = 3.0;
        double r54816 = pow(r54814, r54815);
        double r54817 = r54813 * r54816;
        double r54818 = 0.016666666666666666;
        double r54819 = 5.0;
        double r54820 = pow(r54814, r54819);
        double r54821 = r54818 * r54820;
        double r54822 = 2.0;
        double r54823 = r54822 * r54814;
        double r54824 = r54821 + r54823;
        double r54825 = r54817 + r54824;
        double r54826 = 2.0;
        double r54827 = r54825 / r54826;
        double r54828 = y;
        double r54829 = sin(r54828);
        double r54830 = r54827 * r54829;
        return r54830;
}

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 0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2020035 
(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)))))