Average Error: 43.1 → 0.8
Time: 13.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r44140 = x;
        double r44141 = exp(r44140);
        double r44142 = -r44140;
        double r44143 = exp(r44142);
        double r44144 = r44141 + r44143;
        double r44145 = 2.0;
        double r44146 = r44144 / r44145;
        double r44147 = y;
        double r44148 = cos(r44147);
        double r44149 = r44146 * r44148;
        double r44150 = r44141 - r44143;
        double r44151 = r44150 / r44145;
        double r44152 = sin(r44147);
        double r44153 = r44151 * r44152;
        double r44154 = /* ERROR: no complex support in C */;
        double r44155 = /* ERROR: no complex support in C */;
        return r44155;
}
↓
double f(double x, double y) {
        double r44156 = x;
        double r44157 = exp(r44156);
        double r44158 = -r44156;
        double r44159 = exp(r44158);
        double r44160 = r44157 + r44159;
        double r44161 = 2.0;
        double r44162 = r44160 / r44161;
        double r44163 = y;
        double r44164 = cos(r44163);
        double r44165 = r44162 * r44164;
        double r44166 = 0.3333333333333333;
        double r44167 = 3.0;
        double r44168 = pow(r44156, r44167);
        double r44169 = 0.016666666666666666;
        double r44170 = 5.0;
        double r44171 = pow(r44156, r44170);
        double r44172 = 2.0;
        double r44173 = r44172 * r44156;
        double r44174 = fma(r44169, r44171, r44173);
        double r44175 = fma(r44166, r44168, r44174);
        double r44176 = r44175 / r44161;
        double r44177 = sin(r44163);
        double r44178 = r44176 * r44177;
        double r44179 = /* ERROR: no complex support in C */;
        double r44180 = /* ERROR: no complex support in C */;
        return r44180;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.1

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020020 +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)))))