Average Error: 44.1 → 0.8
Time: 39.3s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(x, 2, \left(\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(\frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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(x, 2, \left(\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(\frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1216231 = x;
        double r1216232 = exp(r1216231);
        double r1216233 = -r1216231;
        double r1216234 = exp(r1216233);
        double r1216235 = r1216232 + r1216234;
        double r1216236 = 2.0;
        double r1216237 = r1216235 / r1216236;
        double r1216238 = y;
        double r1216239 = cos(r1216238);
        double r1216240 = r1216237 * r1216239;
        double r1216241 = r1216232 - r1216234;
        double r1216242 = r1216241 / r1216236;
        double r1216243 = sin(r1216238);
        double r1216244 = r1216242 * r1216243;
        double r1216245 = /* ERROR: no complex support in C */;
        double r1216246 = /* ERROR: no complex support in C */;
        return r1216246;
}
↓
double f(double x, double y) {
        double r1216247 = x;
        double r1216248 = exp(r1216247);
        double r1216249 = -r1216247;
        double r1216250 = exp(r1216249);
        double r1216251 = r1216248 + r1216250;
        double r1216252 = 2.0;
        double r1216253 = r1216251 / r1216252;
        double r1216254 = y;
        double r1216255 = cos(r1216254);
        double r1216256 = r1216253 * r1216255;
        double r1216257 = 5.0;
        double r1216258 = pow(r1216247, r1216257);
        double r1216259 = 0.016666666666666666;
        double r1216260 = 0.3333333333333333;
        double r1216261 = r1216247 * r1216247;
        double r1216262 = r1216261 * r1216247;
        double r1216263 = r1216260 * r1216262;
        double r1216264 = fma(r1216258, r1216259, r1216263);
        double r1216265 = fma(r1216247, r1216252, r1216264);
        double r1216266 = r1216265 / r1216252;
        double r1216267 = sin(r1216254);
        double r1216268 = r1216266 * r1216267;
        double r1216269 = /* ERROR: no complex support in C */;
        double r1216270 = /* ERROR: no complex support in C */;
        return r1216270;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 44.1

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019129 +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)))))