Average Error: 43.8 → 0.9
Time: 13.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r40661 = x;
        double r40662 = exp(r40661);
        double r40663 = -r40661;
        double r40664 = exp(r40663);
        double r40665 = r40662 + r40664;
        double r40666 = 2.0;
        double r40667 = r40665 / r40666;
        double r40668 = y;
        double r40669 = cos(r40668);
        double r40670 = r40667 * r40669;
        double r40671 = r40662 - r40664;
        double r40672 = r40671 / r40666;
        double r40673 = sin(r40668);
        double r40674 = r40672 * r40673;
        double r40675 = /* ERROR: no complex support in C */;
        double r40676 = /* ERROR: no complex support in C */;
        return r40676;
}
↓
double f(double x, double y) {
        double r40677 = 0.3333333333333333;
        double r40678 = x;
        double r40679 = 3.0;
        double r40680 = pow(r40678, r40679);
        double r40681 = r40677 * r40680;
        double r40682 = 0.016666666666666666;
        double r40683 = 5.0;
        double r40684 = pow(r40678, r40683);
        double r40685 = r40682 * r40684;
        double r40686 = 2.0;
        double r40687 = r40686 * r40678;
        double r40688 = r40685 + r40687;
        double r40689 = r40681 + r40688;
        double r40690 = 2.0;
        double r40691 = r40689 / r40690;
        double r40692 = y;
        double r40693 = sin(r40692);
        double r40694 = r40691 * r40693;
        return r40694;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.8

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

  2. Simplified43.8

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

  3. Taylor expanded around 0 0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019353 
(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)))))