Average Error: 43.6 → 0.7
Time: 28.6s
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 r50024 = x;
        double r50025 = exp(r50024);
        double r50026 = -r50024;
        double r50027 = exp(r50026);
        double r50028 = r50025 + r50027;
        double r50029 = 2.0;
        double r50030 = r50028 / r50029;
        double r50031 = y;
        double r50032 = cos(r50031);
        double r50033 = r50030 * r50032;
        double r50034 = r50025 - r50027;
        double r50035 = r50034 / r50029;
        double r50036 = sin(r50031);
        double r50037 = r50035 * r50036;
        double r50038 = /* ERROR: no complex support in C */;
        double r50039 = /* ERROR: no complex support in C */;
        return r50039;
}
↓
double f(double x, double y) {
        double r50040 = y;
        double r50041 = sin(r50040);
        double r50042 = 2.0;
        double r50043 = r50041 / r50042;
        double r50044 = 2.0;
        double r50045 = x;
        double r50046 = 0.3333333333333333;
        double r50047 = 3.0;
        double r50048 = pow(r50045, r50047);
        double r50049 = 5.0;
        double r50050 = pow(r50045, r50049);
        double r50051 = 0.016666666666666666;
        double r50052 = r50050 * r50051;
        double r50053 = fma(r50046, r50048, r50052);
        double r50054 = fma(r50044, r50045, r50053);
        double r50055 = r50043 * r50054;
        return r50055;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.6

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

  2. Simplified43.6

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019194 +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)))))