Average Error: 43.2 → 0.9
Time: 25.4s
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 r61925 = x;
        double r61926 = exp(r61925);
        double r61927 = -r61925;
        double r61928 = exp(r61927);
        double r61929 = r61926 + r61928;
        double r61930 = 2.0;
        double r61931 = r61929 / r61930;
        double r61932 = y;
        double r61933 = cos(r61932);
        double r61934 = r61931 * r61933;
        double r61935 = r61926 - r61928;
        double r61936 = r61935 / r61930;
        double r61937 = sin(r61932);
        double r61938 = r61936 * r61937;
        double r61939 = /* ERROR: no complex support in C */;
        double r61940 = /* ERROR: no complex support in C */;
        return r61940;
}
↓
double f(double x, double y) {
        double r61941 = 0.3333333333333333;
        double r61942 = x;
        double r61943 = 3.0;
        double r61944 = pow(r61942, r61943);
        double r61945 = r61941 * r61944;
        double r61946 = 0.016666666666666666;
        double r61947 = 5.0;
        double r61948 = pow(r61942, r61947);
        double r61949 = r61946 * r61948;
        double r61950 = 2.0;
        double r61951 = r61950 * r61942;
        double r61952 = r61949 + r61951;
        double r61953 = r61945 + r61952;
        double r61954 = 2.0;
        double r61955 = r61953 / r61954;
        double r61956 = y;
        double r61957 = sin(r61956);
        double r61958 = r61955 * r61957;
        return r61958;
}

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 0.9

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

  3. Final simplification0.9

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

Reproduce

herbie shell --seed 197574269 
(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)))))