Average Error: 43.4 → 0.8
Time: 12.7s
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 r43207 = x;
        double r43208 = exp(r43207);
        double r43209 = -r43207;
        double r43210 = exp(r43209);
        double r43211 = r43208 + r43210;
        double r43212 = 2.0;
        double r43213 = r43211 / r43212;
        double r43214 = y;
        double r43215 = cos(r43214);
        double r43216 = r43213 * r43215;
        double r43217 = r43208 - r43210;
        double r43218 = r43217 / r43212;
        double r43219 = sin(r43214);
        double r43220 = r43218 * r43219;
        double r43221 = /* ERROR: no complex support in C */;
        double r43222 = /* ERROR: no complex support in C */;
        return r43222;
}
↓
double f(double x, double y) {
        double r43223 = x;
        double r43224 = exp(r43223);
        double r43225 = -r43223;
        double r43226 = exp(r43225);
        double r43227 = r43224 + r43226;
        double r43228 = 2.0;
        double r43229 = r43227 / r43228;
        double r43230 = y;
        double r43231 = cos(r43230);
        double r43232 = r43229 * r43231;
        double r43233 = 0.3333333333333333;
        double r43234 = 3.0;
        double r43235 = pow(r43223, r43234);
        double r43236 = 0.016666666666666666;
        double r43237 = 5.0;
        double r43238 = pow(r43223, r43237);
        double r43239 = 2.0;
        double r43240 = r43239 * r43223;
        double r43241 = fma(r43236, r43238, r43240);
        double r43242 = fma(r43233, r43235, r43241);
        double r43243 = r43242 / r43228;
        double r43244 = sin(r43230);
        double r43245 = r43243 * r43244;
        double r43246 = /* ERROR: no complex support in C */;
        double r43247 = /* ERROR: no complex support in C */;
        return r43247;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

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

  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 2020100 +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)))))