Average Error: 43.7 → 0.7
Time: 29.8s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\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
double f(double x, double y) {
        double r49196 = x;
        double r49197 = exp(r49196);
        double r49198 = -r49196;
        double r49199 = exp(r49198);
        double r49200 = r49197 + r49199;
        double r49201 = 2.0;
        double r49202 = r49200 / r49201;
        double r49203 = y;
        double r49204 = cos(r49203);
        double r49205 = r49202 * r49204;
        double r49206 = r49197 - r49199;
        double r49207 = r49206 / r49201;
        double r49208 = sin(r49203);
        double r49209 = r49207 * r49208;
        double r49210 = /* ERROR: no complex support in C */;
        double r49211 = /* ERROR: no complex support in C */;
        return r49211;
}
↓
double f(double x, double y) {
        double r49212 = 0.3333333333333333;
        double r49213 = x;
        double r49214 = 3.0;
        double r49215 = pow(r49213, r49214);
        double r49216 = 0.016666666666666666;
        double r49217 = 5.0;
        double r49218 = pow(r49213, r49217);
        double r49219 = 2.0;
        double r49220 = r49219 * r49213;
        double r49221 = fma(r49216, r49218, r49220);
        double r49222 = fma(r49212, r49215, r49221);
        double r49223 = 2.0;
        double r49224 = r49222 / r49223;
        double r49225 = y;
        double r49226 = sin(r49225);
        double r49227 = r49224 * r49226;
        return r49227;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.7

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

  2. Simplified43.7

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019212 +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)))))