Average Error: 43.8 → 0.7
Time: 40.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot \left(2 + \frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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({x}^{5}, \frac{1}{60}, x \cdot \left(2 + \frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1327285 = x;
        double r1327286 = exp(r1327285);
        double r1327287 = -r1327285;
        double r1327288 = exp(r1327287);
        double r1327289 = r1327286 + r1327288;
        double r1327290 = 2.0;
        double r1327291 = r1327289 / r1327290;
        double r1327292 = y;
        double r1327293 = cos(r1327292);
        double r1327294 = r1327291 * r1327293;
        double r1327295 = r1327286 - r1327288;
        double r1327296 = r1327295 / r1327290;
        double r1327297 = sin(r1327292);
        double r1327298 = r1327296 * r1327297;
        double r1327299 = /* ERROR: no complex support in C */;
        double r1327300 = /* ERROR: no complex support in C */;
        return r1327300;
}
↓
double f(double x, double y) {
        double r1327301 = x;
        double r1327302 = exp(r1327301);
        double r1327303 = -r1327301;
        double r1327304 = exp(r1327303);
        double r1327305 = r1327302 + r1327304;
        double r1327306 = 2.0;
        double r1327307 = r1327305 / r1327306;
        double r1327308 = y;
        double r1327309 = cos(r1327308);
        double r1327310 = r1327307 * r1327309;
        double r1327311 = 5.0;
        double r1327312 = pow(r1327301, r1327311);
        double r1327313 = 0.016666666666666666;
        double r1327314 = 2.0;
        double r1327315 = 0.3333333333333333;
        double r1327316 = r1327301 * r1327301;
        double r1327317 = r1327315 * r1327316;
        double r1327318 = r1327314 + r1327317;
        double r1327319 = r1327301 * r1327318;
        double r1327320 = fma(r1327312, r1327313, r1327319);
        double r1327321 = r1327320 / r1327306;
        double r1327322 = sin(r1327308);
        double r1327323 = r1327321 * r1327322;
        double r1327324 = /* ERROR: no complex support in C */;
        double r1327325 = /* ERROR: no complex support in C */;
        return r1327325;
}

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 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))