Average Error: 43.8 → 0.8
Time: 14.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r49170 = x;
        double r49171 = exp(r49170);
        double r49172 = -r49170;
        double r49173 = exp(r49172);
        double r49174 = r49171 + r49173;
        double r49175 = 2.0;
        double r49176 = r49174 / r49175;
        double r49177 = y;
        double r49178 = cos(r49177);
        double r49179 = r49176 * r49178;
        double r49180 = r49171 - r49173;
        double r49181 = r49180 / r49175;
        double r49182 = sin(r49177);
        double r49183 = r49181 * r49182;
        double r49184 = /* ERROR: no complex support in C */;
        double r49185 = /* ERROR: no complex support in C */;
        return r49185;
}
↓
double f(double x, double y) {
        double r49186 = x;
        double r49187 = exp(r49186);
        double r49188 = -r49186;
        double r49189 = exp(r49188);
        double r49190 = r49187 + r49189;
        double r49191 = 2.0;
        double r49192 = r49190 / r49191;
        double r49193 = y;
        double r49194 = cos(r49193);
        double r49195 = r49192 * r49194;
        double r49196 = 0.3333333333333333;
        double r49197 = 3.0;
        double r49198 = pow(r49186, r49197);
        double r49199 = 0.016666666666666666;
        double r49200 = 5.0;
        double r49201 = pow(r49186, r49200);
        double r49202 = 2.0;
        double r49203 = r49202 * r49186;
        double r49204 = fma(r49199, r49201, r49203);
        double r49205 = fma(r49196, r49198, r49204);
        double r49206 = r49205 / r49191;
        double r49207 = sin(r49193);
        double r49208 = r49206 * r49207;
        double r49209 = /* ERROR: no complex support in C */;
        double r49210 = /* ERROR: no complex support in C */;
        return r49210;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.8

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(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)))))