Average Error: 43.3 → 0.8
Time: 10.5s
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 r37987 = x;
        double r37988 = exp(r37987);
        double r37989 = -r37987;
        double r37990 = exp(r37989);
        double r37991 = r37988 + r37990;
        double r37992 = 2.0;
        double r37993 = r37991 / r37992;
        double r37994 = y;
        double r37995 = cos(r37994);
        double r37996 = r37993 * r37995;
        double r37997 = r37988 - r37990;
        double r37998 = r37997 / r37992;
        double r37999 = sin(r37994);
        double r38000 = r37998 * r37999;
        double r38001 = /* ERROR: no complex support in C */;
        double r38002 = /* ERROR: no complex support in C */;
        return r38002;
}
↓
double f(double x, double y) {
        double r38003 = 0.3333333333333333;
        double r38004 = x;
        double r38005 = 3.0;
        double r38006 = pow(r38004, r38005);
        double r38007 = r38003 * r38006;
        double r38008 = 0.016666666666666666;
        double r38009 = 5.0;
        double r38010 = pow(r38004, r38009);
        double r38011 = r38008 * r38010;
        double r38012 = 2.0;
        double r38013 = r38012 * r38004;
        double r38014 = r38011 + r38013;
        double r38015 = r38007 + r38014;
        double r38016 = 2.0;
        double r38017 = r38015 / r38016;
        double r38018 = y;
        double r38019 = sin(r38018);
        double r38020 = r38017 * r38019;
        return r38020;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.3

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

  2. Simplified43.3

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

  3. Taylor expanded around 0 0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020036 
(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)))))