Average Error: 43.4 → 0.8
Time: 11.6s
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 r42439 = x;
        double r42440 = exp(r42439);
        double r42441 = -r42439;
        double r42442 = exp(r42441);
        double r42443 = r42440 + r42442;
        double r42444 = 2.0;
        double r42445 = r42443 / r42444;
        double r42446 = y;
        double r42447 = cos(r42446);
        double r42448 = r42445 * r42447;
        double r42449 = r42440 - r42442;
        double r42450 = r42449 / r42444;
        double r42451 = sin(r42446);
        double r42452 = r42450 * r42451;
        double r42453 = /* ERROR: no complex support in C */;
        double r42454 = /* ERROR: no complex support in C */;
        return r42454;
}
↓
double f(double x, double y) {
        double r42455 = 0.3333333333333333;
        double r42456 = x;
        double r42457 = 3.0;
        double r42458 = pow(r42456, r42457);
        double r42459 = 0.016666666666666666;
        double r42460 = 5.0;
        double r42461 = pow(r42456, r42460);
        double r42462 = 2.0;
        double r42463 = r42462 * r42456;
        double r42464 = fma(r42459, r42461, r42463);
        double r42465 = fma(r42455, r42458, r42464);
        double r42466 = 2.0;
        double r42467 = r42465 / r42466;
        double r42468 = y;
        double r42469 = sin(r42468);
        double r42470 = r42467 * r42469;
        return r42470;
}

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. Simplified43.4

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

  3. Taylor expanded around 0 0.8

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

  4. Simplified0.8

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

  5. Final simplification0.8

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

Reproduce

herbie shell --seed 2020083 +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)))))