Average Error: 43.4 → 0.8
Time: 28.6s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\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 r53018 = x;
        double r53019 = exp(r53018);
        double r53020 = -r53018;
        double r53021 = exp(r53020);
        double r53022 = r53019 + r53021;
        double r53023 = 2.0;
        double r53024 = r53022 / r53023;
        double r53025 = y;
        double r53026 = cos(r53025);
        double r53027 = r53024 * r53026;
        double r53028 = r53019 - r53021;
        double r53029 = r53028 / r53023;
        double r53030 = sin(r53025);
        double r53031 = r53029 * r53030;
        double r53032 = /* ERROR: no complex support in C */;
        double r53033 = /* ERROR: no complex support in C */;
        return r53033;
}
↓
double f(double x, double y) {
        double r53034 = 0.3333333333333333;
        double r53035 = x;
        double r53036 = 3.0;
        double r53037 = pow(r53035, r53036);
        double r53038 = r53034 * r53037;
        double r53039 = 0.016666666666666666;
        double r53040 = 5.0;
        double r53041 = pow(r53035, r53040);
        double r53042 = r53039 * r53041;
        double r53043 = 2.0;
        double r53044 = r53043 * r53035;
        double r53045 = r53042 + r53044;
        double r53046 = r53038 + r53045;
        double r53047 = 2.0;
        double r53048 = r53046 / r53047;
        double r53049 = y;
        double r53050 = sin(r53049);
        double r53051 = r53048 * r53050;
        return r53051;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

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

  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. Final simplification0.8

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

Reproduce

herbie shell --seed 2019298 
(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)))))