Average Error: 44.1 → 0.7
Time: 29.3s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\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
double f(double x, double y) {
        double r62387 = x;
        double r62388 = exp(r62387);
        double r62389 = -r62387;
        double r62390 = exp(r62389);
        double r62391 = r62388 + r62390;
        double r62392 = 2.0;
        double r62393 = r62391 / r62392;
        double r62394 = y;
        double r62395 = cos(r62394);
        double r62396 = r62393 * r62395;
        double r62397 = r62388 - r62390;
        double r62398 = r62397 / r62392;
        double r62399 = sin(r62394);
        double r62400 = r62398 * r62399;
        double r62401 = /* ERROR: no complex support in C */;
        double r62402 = /* ERROR: no complex support in C */;
        return r62402;
}
↓
double f(double x, double y) {
        double r62403 = 0.3333333333333333;
        double r62404 = x;
        double r62405 = 3.0;
        double r62406 = pow(r62404, r62405);
        double r62407 = 0.016666666666666666;
        double r62408 = 5.0;
        double r62409 = pow(r62404, r62408);
        double r62410 = 2.0;
        double r62411 = r62410 * r62404;
        double r62412 = fma(r62407, r62409, r62411);
        double r62413 = fma(r62403, r62406, r62412);
        double r62414 = 2.0;
        double r62415 = r62413 / r62414;
        double r62416 = y;
        double r62417 = sin(r62416);
        double r62418 = r62415 * r62417;
        return r62418;
}

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. Simplified44.1

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019209 +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)))))