Average Error: 43.7 → 0.7
Time: 10.7s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r36101 = x;
        double r36102 = exp(r36101);
        double r36103 = -r36101;
        double r36104 = exp(r36103);
        double r36105 = r36102 + r36104;
        double r36106 = 2.0;
        double r36107 = r36105 / r36106;
        double r36108 = y;
        double r36109 = cos(r36108);
        double r36110 = r36107 * r36109;
        double r36111 = r36102 - r36104;
        double r36112 = r36111 / r36106;
        double r36113 = sin(r36108);
        double r36114 = r36112 * r36113;
        double r36115 = /* ERROR: no complex support in C */;
        double r36116 = /* ERROR: no complex support in C */;
        return r36116;
}
↓
double f(double x, double y) {
        double r36117 = 0.3333333333333333;
        double r36118 = x;
        double r36119 = 3.0;
        double r36120 = pow(r36118, r36119);
        double r36121 = r36117 * r36120;
        double r36122 = 0.016666666666666666;
        double r36123 = 5.0;
        double r36124 = pow(r36118, r36123);
        double r36125 = r36122 * r36124;
        double r36126 = 2.0;
        double r36127 = r36126 * r36118;
        double r36128 = r36125 + r36127;
        double r36129 = r36121 + r36128;
        double r36130 = 2.0;
        double r36131 = r36129 / r36130;
        double r36132 = y;
        double r36133 = sin(r36132);
        double r36134 = r36131 * r36133;
        return r36134;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.7

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

  2. Simplified43.7

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.7

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

  4. Final simplification0.7

    \[\leadsto \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]

Reproduce

herbie shell --seed 2020089 
(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)))))