Average Error: 44.1 → 0.7
Time: 32.9s
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 r50151 = x;
        double r50152 = exp(r50151);
        double r50153 = -r50151;
        double r50154 = exp(r50153);
        double r50155 = r50152 + r50154;
        double r50156 = 2.0;
        double r50157 = r50155 / r50156;
        double r50158 = y;
        double r50159 = cos(r50158);
        double r50160 = r50157 * r50159;
        double r50161 = r50152 - r50154;
        double r50162 = r50161 / r50156;
        double r50163 = sin(r50158);
        double r50164 = r50162 * r50163;
        double r50165 = /* ERROR: no complex support in C */;
        double r50166 = /* ERROR: no complex support in C */;
        return r50166;
}
↓
double f(double x, double y) {
        double r50167 = x;
        double r50168 = exp(r50167);
        double r50169 = -r50167;
        double r50170 = exp(r50169);
        double r50171 = r50168 + r50170;
        double r50172 = 2.0;
        double r50173 = r50171 / r50172;
        double r50174 = y;
        double r50175 = cos(r50174);
        double r50176 = r50173 * r50175;
        double r50177 = 0.3333333333333333;
        double r50178 = 3.0;
        double r50179 = pow(r50167, r50178);
        double r50180 = r50177 * r50179;
        double r50181 = 0.016666666666666666;
        double r50182 = 5.0;
        double r50183 = pow(r50167, r50182);
        double r50184 = r50181 * r50183;
        double r50185 = 2.0;
        double r50186 = r50185 * r50167;
        double r50187 = r50184 + r50186;
        double r50188 = r50180 + r50187;
        double r50189 = r50188 / r50172;
        double r50190 = sin(r50174);
        double r50191 = r50189 * r50190;
        double r50192 = /* ERROR: no complex support in C */;
        double r50193 = /* ERROR: no complex support in C */;
        return r50193;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 44.1

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

  2. Taylor expanded around 0 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2019209 
(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)))))