Average Error: 43.4 → 0.7
Time: 12.7s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]
\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 r36420 = x;
        double r36421 = exp(r36420);
        double r36422 = -r36420;
        double r36423 = exp(r36422);
        double r36424 = r36421 + r36423;
        double r36425 = 2.0;
        double r36426 = r36424 / r36425;
        double r36427 = y;
        double r36428 = cos(r36427);
        double r36429 = r36426 * r36428;
        double r36430 = r36421 - r36423;
        double r36431 = r36430 / r36425;
        double r36432 = sin(r36427);
        double r36433 = r36431 * r36432;
        double r36434 = /* ERROR: no complex support in C */;
        double r36435 = /* ERROR: no complex support in C */;
        return r36435;
}
↓
double f(double x, double y) {
        double r36436 = 0.3333333333333333;
        double r36437 = x;
        double r36438 = 3.0;
        double r36439 = pow(r36437, r36438);
        double r36440 = 0.016666666666666666;
        double r36441 = 5.0;
        double r36442 = pow(r36437, r36441);
        double r36443 = 2.0;
        double r36444 = r36443 * r36437;
        double r36445 = fma(r36440, r36442, r36444);
        double r36446 = fma(r36436, r36439, r36445);
        double r36447 = 2.0;
        double r36448 = r36446 / r36447;
        double r36449 = y;
        double r36450 = sin(r36449);
        double r36451 = r36448 * r36450;
        return r36451;
}

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 0.7

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

  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 2019344 +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)))))