Average Error: 43.3 → 0.8
Time: 11.8s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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 r46255 = x;
        double r46256 = exp(r46255);
        double r46257 = -r46255;
        double r46258 = exp(r46257);
        double r46259 = r46256 + r46258;
        double r46260 = 2.0;
        double r46261 = r46259 / r46260;
        double r46262 = y;
        double r46263 = cos(r46262);
        double r46264 = r46261 * r46263;
        double r46265 = r46256 - r46258;
        double r46266 = r46265 / r46260;
        double r46267 = sin(r46262);
        double r46268 = r46266 * r46267;
        double r46269 = /* ERROR: no complex support in C */;
        double r46270 = /* ERROR: no complex support in C */;
        return r46270;
}
↓
double f(double x, double y) {
        double r46271 = x;
        double r46272 = exp(r46271);
        double r46273 = -r46271;
        double r46274 = exp(r46273);
        double r46275 = r46272 + r46274;
        double r46276 = 2.0;
        double r46277 = r46275 / r46276;
        double r46278 = y;
        double r46279 = cos(r46278);
        double r46280 = r46277 * r46279;
        double r46281 = 0.3333333333333333;
        double r46282 = 3.0;
        double r46283 = pow(r46271, r46282);
        double r46284 = 0.016666666666666666;
        double r46285 = 5.0;
        double r46286 = pow(r46271, r46285);
        double r46287 = 2.0;
        double r46288 = r46287 * r46271;
        double r46289 = fma(r46284, r46286, r46288);
        double r46290 = fma(r46281, r46283, r46289);
        double r46291 = r46290 / r46276;
        double r46292 = sin(r46278);
        double r46293 = r46291 * r46292;
        double r46294 = /* ERROR: no complex support in C */;
        double r46295 = /* ERROR: no complex support in C */;
        return r46295;
}

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 0.8

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

  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 2020025 +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)))))