Average Error: 43.6 → 0.8
Time: 1.1m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot \left(x \cdot \left(\frac{1}{3} \cdot x\right) + 2\right)\right)}{2} \cdot \sin y i\right))\]
\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(x \cdot \left(\frac{1}{3} \cdot x\right) + 2\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1897314 = x;
        double r1897315 = exp(r1897314);
        double r1897316 = -r1897314;
        double r1897317 = exp(r1897316);
        double r1897318 = r1897315 + r1897317;
        double r1897319 = 2.0;
        double r1897320 = r1897318 / r1897319;
        double r1897321 = y;
        double r1897322 = cos(r1897321);
        double r1897323 = r1897320 * r1897322;
        double r1897324 = r1897315 - r1897317;
        double r1897325 = r1897324 / r1897319;
        double r1897326 = sin(r1897321);
        double r1897327 = r1897325 * r1897326;
        double r1897328 = /* ERROR: no complex support in C */;
        double r1897329 = /* ERROR: no complex support in C */;
        return r1897329;
}
↓
double f(double x, double y) {
        double r1897330 = x;
        double r1897331 = exp(r1897330);
        double r1897332 = -r1897330;
        double r1897333 = exp(r1897332);
        double r1897334 = r1897331 + r1897333;
        double r1897335 = 2.0;
        double r1897336 = r1897334 / r1897335;
        double r1897337 = y;
        double r1897338 = cos(r1897337);
        double r1897339 = r1897336 * r1897338;
        double r1897340 = 5.0;
        double r1897341 = pow(r1897330, r1897340);
        double r1897342 = 0.016666666666666666;
        double r1897343 = 0.3333333333333333;
        double r1897344 = r1897343 * r1897330;
        double r1897345 = r1897330 * r1897344;
        double r1897346 = r1897345 + r1897335;
        double r1897347 = r1897330 * r1897346;
        double r1897348 = fma(r1897341, r1897342, r1897347);
        double r1897349 = r1897348 / r1897335;
        double r1897350 = sin(r1897337);
        double r1897351 = r1897349 * r1897350;
        double r1897352 = /* ERROR: no complex support in C */;
        double r1897353 = /* ERROR: no complex support in C */;
        return r1897353;
}

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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019143 +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)))))