Average Error: 43.5 → 0.8
Time: 31.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\sin y}{2} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(\frac{1}{3}, {x}^{3}, {x}^{5} \cdot \frac{1}{60}\right)\right)\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\sin y}{2} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(\frac{1}{3}, {x}^{3}, {x}^{5} \cdot \frac{1}{60}\right)\right)
double f(double x, double y) {
        double r64020 = x;
        double r64021 = exp(r64020);
        double r64022 = -r64020;
        double r64023 = exp(r64022);
        double r64024 = r64021 + r64023;
        double r64025 = 2.0;
        double r64026 = r64024 / r64025;
        double r64027 = y;
        double r64028 = cos(r64027);
        double r64029 = r64026 * r64028;
        double r64030 = r64021 - r64023;
        double r64031 = r64030 / r64025;
        double r64032 = sin(r64027);
        double r64033 = r64031 * r64032;
        double r64034 = /* ERROR: no complex support in C */;
        double r64035 = /* ERROR: no complex support in C */;
        return r64035;
}
↓
double f(double x, double y) {
        double r64036 = y;
        double r64037 = sin(r64036);
        double r64038 = 2.0;
        double r64039 = r64037 / r64038;
        double r64040 = 2.0;
        double r64041 = x;
        double r64042 = 0.3333333333333333;
        double r64043 = 3.0;
        double r64044 = pow(r64041, r64043);
        double r64045 = 5.0;
        double r64046 = pow(r64041, r64045);
        double r64047 = 0.016666666666666666;
        double r64048 = r64046 * r64047;
        double r64049 = fma(r64042, r64044, r64048);
        double r64050 = fma(r64040, r64041, r64049);
        double r64051 = r64039 * r64050;
        return r64051;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.5

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

  2. Simplified43.5

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

  3. Taylor expanded around 0 0.8

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

  4. Simplified0.8

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

  5. Final simplification0.8

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))