Average Error: 43.1 → 0.8
Time: 43.5s
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 r63124 = x;
        double r63125 = exp(r63124);
        double r63126 = -r63124;
        double r63127 = exp(r63126);
        double r63128 = r63125 + r63127;
        double r63129 = 2.0;
        double r63130 = r63128 / r63129;
        double r63131 = y;
        double r63132 = cos(r63131);
        double r63133 = r63130 * r63132;
        double r63134 = r63125 - r63127;
        double r63135 = r63134 / r63129;
        double r63136 = sin(r63131);
        double r63137 = r63135 * r63136;
        double r63138 = /* ERROR: no complex support in C */;
        double r63139 = /* ERROR: no complex support in C */;
        return r63139;
}
↓
double f(double x, double y) {
        double r63140 = x;
        double r63141 = 5.0;
        double r63142 = pow(r63140, r63141);
        double r63143 = 60.0;
        double r63144 = r63142 / r63143;
        double r63145 = 3.0;
        double r63146 = pow(r63140, r63145);
        double r63147 = r63146 / r63145;
        double r63148 = r63147 + r63140;
        double r63149 = r63148 + r63140;
        double r63150 = r63144 + r63149;
        double r63151 = 2.0;
        double r63152 = r63150 / r63151;
        double r63153 = y;
        double r63154 = sin(r63153);
        double r63155 = r63152 * r63154;
        return r63155;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.1

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

  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. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019303 
(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)))))