Average Error: 43.6 → 0.8
Time: 54.6s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(x, 2, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1128120 = x;
        double r1128121 = exp(r1128120);
        double r1128122 = -r1128120;
        double r1128123 = exp(r1128122);
        double r1128124 = r1128121 + r1128123;
        double r1128125 = 2.0;
        double r1128126 = r1128124 / r1128125;
        double r1128127 = y;
        double r1128128 = cos(r1128127);
        double r1128129 = r1128126 * r1128128;
        double r1128130 = r1128121 - r1128123;
        double r1128131 = r1128130 / r1128125;
        double r1128132 = sin(r1128127);
        double r1128133 = r1128131 * r1128132;
        double r1128134 = /* ERROR: no complex support in C */;
        double r1128135 = /* ERROR: no complex support in C */;
        return r1128135;
}
↓
double f(double x, double y) {
        double r1128136 = x;
        double r1128137 = exp(r1128136);
        double r1128138 = -r1128136;
        double r1128139 = exp(r1128138);
        double r1128140 = r1128137 + r1128139;
        double r1128141 = 2.0;
        double r1128142 = r1128140 / r1128141;
        double r1128143 = y;
        double r1128144 = cos(r1128143);
        double r1128145 = r1128142 * r1128144;
        double r1128146 = 5.0;
        double r1128147 = pow(r1128136, r1128146);
        double r1128148 = 0.016666666666666666;
        double r1128149 = 0.3333333333333333;
        double r1128150 = r1128136 * r1128136;
        double r1128151 = r1128150 * r1128136;
        double r1128152 = r1128149 * r1128151;
        double r1128153 = fma(r1128147, r1128148, r1128152);
        double r1128154 = fma(r1128136, r1128141, r1128153);
        double r1128155 = r1128154 / r1128141;
        double r1128156 = sin(r1128143);
        double r1128157 = r1128155 * r1128156;
        double r1128158 = /* ERROR: no complex support in C */;
        double r1128159 = /* ERROR: no complex support in C */;
        return r1128159;
}

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. 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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019153 +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)))))