Average Error: 43.5 → 0.6
Time: 20.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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 r55109 = x;
        double r55110 = exp(r55109);
        double r55111 = -r55109;
        double r55112 = exp(r55111);
        double r55113 = r55110 + r55112;
        double r55114 = 2.0;
        double r55115 = r55113 / r55114;
        double r55116 = y;
        double r55117 = cos(r55116);
        double r55118 = r55115 * r55117;
        double r55119 = r55110 - r55112;
        double r55120 = r55119 / r55114;
        double r55121 = sin(r55116);
        double r55122 = r55120 * r55121;
        double r55123 = /* ERROR: no complex support in C */;
        double r55124 = /* ERROR: no complex support in C */;
        return r55124;
}
↓
double f(double x, double y) {
        double r55125 = x;
        double r55126 = exp(r55125);
        double r55127 = -r55125;
        double r55128 = exp(r55127);
        double r55129 = r55126 + r55128;
        double r55130 = 2.0;
        double r55131 = r55129 / r55130;
        double r55132 = y;
        double r55133 = cos(r55132);
        double r55134 = r55131 * r55133;
        double r55135 = 0.3333333333333333;
        double r55136 = 3.0;
        double r55137 = pow(r55125, r55136);
        double r55138 = 0.016666666666666666;
        double r55139 = 5.0;
        double r55140 = pow(r55125, r55139);
        double r55141 = 2.0;
        double r55142 = r55141 * r55125;
        double r55143 = fma(r55138, r55140, r55142);
        double r55144 = fma(r55135, r55137, r55143);
        double r55145 = r55144 / r55130;
        double r55146 = sin(r55132);
        double r55147 = r55145 * r55146;
        double r55148 = /* ERROR: no complex support in C */;
        double r55149 = /* ERROR: no complex support in C */;
        return r55149;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.5

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020042 +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)))))