Average Error: 43.3 → 0.8
Time: 11.3s
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 r42782 = x;
        double r42783 = exp(r42782);
        double r42784 = -r42782;
        double r42785 = exp(r42784);
        double r42786 = r42783 + r42785;
        double r42787 = 2.0;
        double r42788 = r42786 / r42787;
        double r42789 = y;
        double r42790 = cos(r42789);
        double r42791 = r42788 * r42790;
        double r42792 = r42783 - r42785;
        double r42793 = r42792 / r42787;
        double r42794 = sin(r42789);
        double r42795 = r42793 * r42794;
        double r42796 = /* ERROR: no complex support in C */;
        double r42797 = /* ERROR: no complex support in C */;
        return r42797;
}
↓
double f(double x, double y) {
        double r42798 = 0.3333333333333333;
        double r42799 = x;
        double r42800 = 3.0;
        double r42801 = pow(r42799, r42800);
        double r42802 = 0.016666666666666666;
        double r42803 = 5.0;
        double r42804 = pow(r42799, r42803);
        double r42805 = 2.0;
        double r42806 = r42805 * r42799;
        double r42807 = fma(r42802, r42804, r42806);
        double r42808 = fma(r42798, r42801, r42807);
        double r42809 = 2.0;
        double r42810 = r42808 / r42809;
        double r42811 = y;
        double r42812 = sin(r42811);
        double r42813 = r42810 * r42812;
        return r42813;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.3

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

  2. Simplified43.3

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

  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 2020036 +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)))))