Average Error: 43.5 → 0.9
Time: 26.1s
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 r44964 = x;
        double r44965 = exp(r44964);
        double r44966 = -r44964;
        double r44967 = exp(r44966);
        double r44968 = r44965 + r44967;
        double r44969 = 2.0;
        double r44970 = r44968 / r44969;
        double r44971 = y;
        double r44972 = cos(r44971);
        double r44973 = r44970 * r44972;
        double r44974 = r44965 - r44967;
        double r44975 = r44974 / r44969;
        double r44976 = sin(r44971);
        double r44977 = r44975 * r44976;
        double r44978 = /* ERROR: no complex support in C */;
        double r44979 = /* ERROR: no complex support in C */;
        return r44979;
}
↓
double f(double x, double y) {
        double r44980 = 0.3333333333333333;
        double r44981 = x;
        double r44982 = 3.0;
        double r44983 = pow(r44981, r44982);
        double r44984 = r44980 * r44983;
        double r44985 = 0.016666666666666666;
        double r44986 = 5.0;
        double r44987 = pow(r44981, r44986);
        double r44988 = r44985 * r44987;
        double r44989 = 2.0;
        double r44990 = r44989 * r44981;
        double r44991 = r44988 + r44990;
        double r44992 = r44984 + r44991;
        double r44993 = 2.0;
        double r44994 = r44992 / r44993;
        double r44995 = y;
        double r44996 = sin(r44995);
        double r44997 = r44994 * r44996;
        return r44997;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.5

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

  2. Simplified43.5

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

  3. Taylor expanded around 0 0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019297 
(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)))))