Average Error: 43.6 → 0.8
Time: 37.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{{x}^{5} \cdot \frac{1}{60} + \left(2 + \frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x}{2} \cdot \sin y i\right))\]
\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{{x}^{5} \cdot \frac{1}{60} + \left(2 + \frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1589285 = x;
        double r1589286 = exp(r1589285);
        double r1589287 = -r1589285;
        double r1589288 = exp(r1589287);
        double r1589289 = r1589286 + r1589288;
        double r1589290 = 2.0;
        double r1589291 = r1589289 / r1589290;
        double r1589292 = y;
        double r1589293 = cos(r1589292);
        double r1589294 = r1589291 * r1589293;
        double r1589295 = r1589286 - r1589288;
        double r1589296 = r1589295 / r1589290;
        double r1589297 = sin(r1589292);
        double r1589298 = r1589296 * r1589297;
        double r1589299 = /* ERROR: no complex support in C */;
        double r1589300 = /* ERROR: no complex support in C */;
        return r1589300;
}
↓
double f(double x, double y) {
        double r1589301 = x;
        double r1589302 = exp(r1589301);
        double r1589303 = -r1589301;
        double r1589304 = exp(r1589303);
        double r1589305 = r1589302 + r1589304;
        double r1589306 = 2.0;
        double r1589307 = r1589305 / r1589306;
        double r1589308 = y;
        double r1589309 = cos(r1589308);
        double r1589310 = r1589307 * r1589309;
        double r1589311 = 5.0;
        double r1589312 = pow(r1589301, r1589311);
        double r1589313 = 0.016666666666666666;
        double r1589314 = r1589312 * r1589313;
        double r1589315 = 0.3333333333333333;
        double r1589316 = r1589301 * r1589301;
        double r1589317 = r1589315 * r1589316;
        double r1589318 = r1589306 + r1589317;
        double r1589319 = r1589318 * r1589301;
        double r1589320 = r1589314 + r1589319;
        double r1589321 = r1589320 / r1589306;
        double r1589322 = sin(r1589308);
        double r1589323 = r1589321 * r1589322;
        double r1589324 = /* ERROR: no complex support in C */;
        double r1589325 = /* ERROR: no complex support in C */;
        return r1589325;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.6

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019134 
(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)))))