Average Error: 43.4 → 0.8
Time: 37.8s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)\right)}{2} \cdot \sin y i\right))\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2078139 = x;
        double r2078140 = exp(r2078139);
        double r2078141 = -r2078139;
        double r2078142 = exp(r2078141);
        double r2078143 = r2078140 + r2078142;
        double r2078144 = 2.0;
        double r2078145 = r2078143 / r2078144;
        double r2078146 = y;
        double r2078147 = cos(r2078146);
        double r2078148 = r2078145 * r2078147;
        double r2078149 = r2078140 - r2078142;
        double r2078150 = r2078149 / r2078144;
        double r2078151 = sin(r2078146);
        double r2078152 = r2078150 * r2078151;
        double r2078153 = /* ERROR: no complex support in C */;
        double r2078154 = /* ERROR: no complex support in C */;
        return r2078154;
}
↓
double f(double x, double y) {
        double r2078155 = x;
        double r2078156 = exp(r2078155);
        double r2078157 = -r2078155;
        double r2078158 = exp(r2078157);
        double r2078159 = r2078156 + r2078158;
        double r2078160 = 2.0;
        double r2078161 = r2078159 / r2078160;
        double r2078162 = y;
        double r2078163 = cos(r2078162);
        double r2078164 = r2078161 * r2078163;
        double r2078165 = 2.0;
        double r2078166 = 5.0;
        double r2078167 = pow(r2078155, r2078166);
        double r2078168 = 0.016666666666666666;
        double r2078169 = 0.3333333333333333;
        double r2078170 = r2078155 * r2078155;
        double r2078171 = r2078169 * r2078170;
        double r2078172 = r2078155 * r2078171;
        double r2078173 = fma(r2078167, r2078168, r2078172);
        double r2078174 = fma(r2078165, r2078155, r2078173);
        double r2078175 = r2078174 / r2078160;
        double r2078176 = sin(r2078162);
        double r2078177 = r2078175 * r2078176;
        double r2078178 = /* ERROR: no complex support in C */;
        double r2078179 = /* ERROR: no complex support in C */;
        return r2078179;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019172 +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)))))