Average Error: 43.6 → 0.7
Time: 51.8s
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 r54968 = x;
        double r54969 = exp(r54968);
        double r54970 = -r54968;
        double r54971 = exp(r54970);
        double r54972 = r54969 + r54971;
        double r54973 = 2.0;
        double r54974 = r54972 / r54973;
        double r54975 = y;
        double r54976 = cos(r54975);
        double r54977 = r54974 * r54976;
        double r54978 = r54969 - r54971;
        double r54979 = r54978 / r54973;
        double r54980 = sin(r54975);
        double r54981 = r54979 * r54980;
        double r54982 = /* ERROR: no complex support in C */;
        double r54983 = /* ERROR: no complex support in C */;
        return r54983;
}
↓
double f(double x, double y) {
        double r54984 = x;
        double r54985 = exp(r54984);
        double r54986 = -r54984;
        double r54987 = exp(r54986);
        double r54988 = r54985 + r54987;
        double r54989 = 2.0;
        double r54990 = r54988 / r54989;
        double r54991 = y;
        double r54992 = cos(r54991);
        double r54993 = r54990 * r54992;
        double r54994 = 0.3333333333333333;
        double r54995 = 3.0;
        double r54996 = pow(r54984, r54995);
        double r54997 = r54994 * r54996;
        double r54998 = 0.016666666666666666;
        double r54999 = 5.0;
        double r55000 = pow(r54984, r54999);
        double r55001 = r54998 * r55000;
        double r55002 = 2.0;
        double r55003 = r55002 * r54984;
        double r55004 = r55001 + r55003;
        double r55005 = r54997 + r55004;
        double r55006 = r55005 / r54989;
        double r55007 = sin(r54991);
        double r55008 = r55006 * r55007;
        double r55009 = /* ERROR: no complex support in C */;
        double r55010 = /* ERROR: no complex support in C */;
        return r55010;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.6

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

  2. Taylor expanded around 0 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2020045 
(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)))))