Average Error: 43.3 → 0.8
Time: 16.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]
\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 r144 = x;
        double r145 = exp(r144);
        double r146 = -r144;
        double r147 = exp(r146);
        double r148 = r145 + r147;
        double r149 = 2.0;
        double r150 = r148 / r149;
        double r151 = y;
        double r152 = cos(r151);
        double r153 = r150 * r152;
        double r154 = r145 - r147;
        double r155 = r154 / r149;
        double r156 = sin(r151);
        double r157 = r155 * r156;
        double r158 = /* ERROR: no complex support in C */;
        double r159 = /* ERROR: no complex support in C */;
        return r159;
}
↓
double f(double x, double y) {
        double r160 = 0.3333333333333333;
        double r161 = x;
        double r162 = 3.0;
        double r163 = pow(r161, r162);
        double r164 = 0.016666666666666666;
        double r165 = 5.0;
        double r166 = pow(r161, r165);
        double r167 = 2.0;
        double r168 = r167 * r161;
        double r169 = fma(r164, r166, r168);
        double r170 = fma(r160, r163, r169);
        double r171 = 2.0;
        double r172 = r170 / r171;
        double r173 = y;
        double r174 = sin(r173);
        double r175 = r172 * r174;
        return r175;
}

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 2020025 +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)))))