Average Error: 43.2 → 0.8
Time: 12.3s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\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{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r42002 = x;
        double r42003 = exp(r42002);
        double r42004 = -r42002;
        double r42005 = exp(r42004);
        double r42006 = r42003 + r42005;
        double r42007 = 2.0;
        double r42008 = r42006 / r42007;
        double r42009 = y;
        double r42010 = cos(r42009);
        double r42011 = r42008 * r42010;
        double r42012 = r42003 - r42005;
        double r42013 = r42012 / r42007;
        double r42014 = sin(r42009);
        double r42015 = r42013 * r42014;
        double r42016 = /* ERROR: no complex support in C */;
        double r42017 = /* ERROR: no complex support in C */;
        return r42017;
}
↓
double f(double x, double y) {
        double r42018 = x;
        double r42019 = exp(r42018);
        double r42020 = -r42018;
        double r42021 = exp(r42020);
        double r42022 = r42019 + r42021;
        double r42023 = 2.0;
        double r42024 = r42022 / r42023;
        double r42025 = y;
        double r42026 = cos(r42025);
        double r42027 = r42024 * r42026;
        double r42028 = 0.3333333333333333;
        double r42029 = 3.0;
        double r42030 = pow(r42018, r42029);
        double r42031 = r42028 * r42030;
        double r42032 = 0.016666666666666666;
        double r42033 = 5.0;
        double r42034 = pow(r42018, r42033);
        double r42035 = r42032 * r42034;
        double r42036 = 2.0;
        double r42037 = r42036 * r42018;
        double r42038 = r42035 + r42037;
        double r42039 = r42031 + r42038;
        double r42040 = r42039 / r42023;
        double r42041 = sin(r42025);
        double r42042 = r42040 * r42041;
        double r42043 = /* ERROR: no complex support in C */;
        double r42044 = /* ERROR: no complex support in C */;
        return r42044;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.2

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

  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

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

Reproduce

herbie shell --seed 2020046 
(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)))))