Average Error: 43.1 → 0.8
Time: 41.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(x, 2, \left(\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{3}\right)\right)\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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(x, 2, \left(\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{3}\right)\right)\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1284138 = x;
        double r1284139 = exp(r1284138);
        double r1284140 = -r1284138;
        double r1284141 = exp(r1284140);
        double r1284142 = r1284139 + r1284141;
        double r1284143 = 2.0;
        double r1284144 = r1284142 / r1284143;
        double r1284145 = y;
        double r1284146 = cos(r1284145);
        double r1284147 = r1284144 * r1284146;
        double r1284148 = r1284139 - r1284141;
        double r1284149 = r1284148 / r1284143;
        double r1284150 = sin(r1284145);
        double r1284151 = r1284149 * r1284150;
        double r1284152 = /* ERROR: no complex support in C */;
        double r1284153 = /* ERROR: no complex support in C */;
        return r1284153;
}
↓
double f(double x, double y) {
        double r1284154 = x;
        double r1284155 = exp(r1284154);
        double r1284156 = -r1284154;
        double r1284157 = exp(r1284156);
        double r1284158 = r1284155 + r1284157;
        double r1284159 = 2.0;
        double r1284160 = r1284158 / r1284159;
        double r1284161 = y;
        double r1284162 = cos(r1284161);
        double r1284163 = r1284160 * r1284162;
        double r1284164 = 5.0;
        double r1284165 = pow(r1284154, r1284164);
        double r1284166 = 0.016666666666666666;
        double r1284167 = r1284154 * r1284154;
        double r1284168 = 0.3333333333333333;
        double r1284169 = r1284154 * r1284168;
        double r1284170 = r1284167 * r1284169;
        double r1284171 = fma(r1284165, r1284166, r1284170);
        double r1284172 = fma(r1284154, r1284159, r1284171);
        double r1284173 = r1284172 / r1284159;
        double r1284174 = sin(r1284161);
        double r1284175 = r1284173 * r1284174;
        double r1284176 = /* ERROR: no complex support in C */;
        double r1284177 = /* ERROR: no complex support in C */;
        return r1284177;
}

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

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))