Average Error: 43.3 → 0.7
Time: 35.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(2, x, \left(\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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(2, x, \left(\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r869309 = x;
        double r869310 = exp(r869309);
        double r869311 = -r869309;
        double r869312 = exp(r869311);
        double r869313 = r869310 + r869312;
        double r869314 = 2.0;
        double r869315 = r869313 / r869314;
        double r869316 = y;
        double r869317 = cos(r869316);
        double r869318 = r869315 * r869317;
        double r869319 = r869310 - r869312;
        double r869320 = r869319 / r869314;
        double r869321 = sin(r869316);
        double r869322 = r869320 * r869321;
        double r869323 = /* ERROR: no complex support in C */;
        double r869324 = /* ERROR: no complex support in C */;
        return r869324;
}
↓
double f(double x, double y) {
        double r869325 = x;
        double r869326 = exp(r869325);
        double r869327 = -r869325;
        double r869328 = exp(r869327);
        double r869329 = r869326 + r869328;
        double r869330 = 2.0;
        double r869331 = r869329 / r869330;
        double r869332 = y;
        double r869333 = cos(r869332);
        double r869334 = r869331 * r869333;
        double r869335 = 0.016666666666666666;
        double r869336 = 5.0;
        double r869337 = pow(r869325, r869336);
        double r869338 = r869325 * r869325;
        double r869339 = 0.3333333333333333;
        double r869340 = r869338 * r869339;
        double r869341 = r869340 * r869325;
        double r869342 = fma(r869335, r869337, r869341);
        double r869343 = fma(r869330, r869325, r869342);
        double r869344 = r869343 / r869330;
        double r869345 = sin(r869332);
        double r869346 = r869344 * r869345;
        double r869347 = /* ERROR: no complex support in C */;
        double r869348 = /* ERROR: no complex support in C */;
        return r869348;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.3

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))