Average Error: 43.2 → 0.8
Time: 36.2s
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 r49180 = x;
        double r49181 = exp(r49180);
        double r49182 = -r49180;
        double r49183 = exp(r49182);
        double r49184 = r49181 + r49183;
        double r49185 = 2.0;
        double r49186 = r49184 / r49185;
        double r49187 = y;
        double r49188 = cos(r49187);
        double r49189 = r49186 * r49188;
        double r49190 = r49181 - r49183;
        double r49191 = r49190 / r49185;
        double r49192 = sin(r49187);
        double r49193 = r49191 * r49192;
        double r49194 = /* ERROR: no complex support in C */;
        double r49195 = /* ERROR: no complex support in C */;
        return r49195;
}
↓
double f(double x, double y) {
        double r49196 = 0.3333333333333333;
        double r49197 = x;
        double r49198 = 3.0;
        double r49199 = pow(r49197, r49198);
        double r49200 = r49196 * r49199;
        double r49201 = 0.016666666666666666;
        double r49202 = 5.0;
        double r49203 = pow(r49197, r49202);
        double r49204 = r49201 * r49203;
        double r49205 = 2.0;
        double r49206 = r49205 * r49197;
        double r49207 = r49204 + r49206;
        double r49208 = r49200 + r49207;
        double r49209 = 2.0;
        double r49210 = r49208 / r49209;
        double r49211 = y;
        double r49212 = sin(r49211);
        double r49213 = r49210 * r49212;
        return r49213;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.2

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

  2. Simplified43.2

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  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 2019325 
(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)))))