Average Error: 43.9 → 0.7
Time: 11.8s
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 r40646 = x;
        double r40647 = exp(r40646);
        double r40648 = -r40646;
        double r40649 = exp(r40648);
        double r40650 = r40647 + r40649;
        double r40651 = 2.0;
        double r40652 = r40650 / r40651;
        double r40653 = y;
        double r40654 = cos(r40653);
        double r40655 = r40652 * r40654;
        double r40656 = r40647 - r40649;
        double r40657 = r40656 / r40651;
        double r40658 = sin(r40653);
        double r40659 = r40657 * r40658;
        double r40660 = /* ERROR: no complex support in C */;
        double r40661 = /* ERROR: no complex support in C */;
        return r40661;
}
↓
double f(double x, double y) {
        double r40662 = 0.3333333333333333;
        double r40663 = x;
        double r40664 = 3.0;
        double r40665 = pow(r40663, r40664);
        double r40666 = 0.016666666666666666;
        double r40667 = 5.0;
        double r40668 = pow(r40663, r40667);
        double r40669 = 2.0;
        double r40670 = r40669 * r40663;
        double r40671 = fma(r40666, r40668, r40670);
        double r40672 = fma(r40662, r40665, r40671);
        double r40673 = 2.0;
        double r40674 = r40672 / r40673;
        double r40675 = y;
        double r40676 = sin(r40675);
        double r40677 = r40674 * r40676;
        return r40677;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.9

    \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  2. Simplified43.9

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

  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 2020001 +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)))))