Average Error: 44.1 → 0.7
Time: 34.7s
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 r51545 = x;
        double r51546 = exp(r51545);
        double r51547 = -r51545;
        double r51548 = exp(r51547);
        double r51549 = r51546 + r51548;
        double r51550 = 2.0;
        double r51551 = r51549 / r51550;
        double r51552 = y;
        double r51553 = cos(r51552);
        double r51554 = r51551 * r51553;
        double r51555 = r51546 - r51548;
        double r51556 = r51555 / r51550;
        double r51557 = sin(r51552);
        double r51558 = r51556 * r51557;
        double r51559 = /* ERROR: no complex support in C */;
        double r51560 = /* ERROR: no complex support in C */;
        return r51560;
}
↓
double f(double x, double y) {
        double r51561 = 0.3333333333333333;
        double r51562 = x;
        double r51563 = 3.0;
        double r51564 = pow(r51562, r51563);
        double r51565 = 0.016666666666666666;
        double r51566 = 5.0;
        double r51567 = pow(r51562, r51566);
        double r51568 = 2.0;
        double r51569 = r51568 * r51562;
        double r51570 = fma(r51565, r51567, r51569);
        double r51571 = fma(r51561, r51564, r51570);
        double r51572 = 2.0;
        double r51573 = r51571 / r51572;
        double r51574 = y;
        double r51575 = sin(r51574);
        double r51576 = r51573 * r51575;
        return r51576;
}

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 2019322 +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)))))