Average Error: 43.8 → 0.9
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 r35644 = x;
        double r35645 = exp(r35644);
        double r35646 = -r35644;
        double r35647 = exp(r35646);
        double r35648 = r35645 + r35647;
        double r35649 = 2.0;
        double r35650 = r35648 / r35649;
        double r35651 = y;
        double r35652 = cos(r35651);
        double r35653 = r35650 * r35652;
        double r35654 = r35645 - r35647;
        double r35655 = r35654 / r35649;
        double r35656 = sin(r35651);
        double r35657 = r35655 * r35656;
        double r35658 = /* ERROR: no complex support in C */;
        double r35659 = /* ERROR: no complex support in C */;
        return r35659;
}
↓
double f(double x, double y) {
        double r35660 = 0.3333333333333333;
        double r35661 = x;
        double r35662 = 3.0;
        double r35663 = pow(r35661, r35662);
        double r35664 = 0.016666666666666666;
        double r35665 = 5.0;
        double r35666 = pow(r35661, r35665);
        double r35667 = 2.0;
        double r35668 = r35667 * r35661;
        double r35669 = fma(r35664, r35666, r35668);
        double r35670 = fma(r35660, r35663, r35669);
        double r35671 = 2.0;
        double r35672 = r35670 / r35671;
        double r35673 = y;
        double r35674 = sin(r35673);
        double r35675 = r35672 * r35674;
        return r35675;
}

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. Simplified0.9

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

  5. Final simplification0.9

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

Reproduce

herbie shell --seed 2019353 +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)))))