Average Error: 43.8 → 0.8
Time: 14.4s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\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 r79924 = x;
        double r79925 = exp(r79924);
        double r79926 = -r79924;
        double r79927 = exp(r79926);
        double r79928 = r79925 + r79927;
        double r79929 = 2.0;
        double r79930 = r79928 / r79929;
        double r79931 = y;
        double r79932 = cos(r79931);
        double r79933 = r79930 * r79932;
        double r79934 = r79925 - r79927;
        double r79935 = r79934 / r79929;
        double r79936 = sin(r79931);
        double r79937 = r79935 * r79936;
        double r79938 = /* ERROR: no complex support in C */;
        double r79939 = /* ERROR: no complex support in C */;
        return r79939;
}
↓
double f(double x, double y) {
        double r79940 = x;
        double r79941 = exp(r79940);
        double r79942 = -r79940;
        double r79943 = exp(r79942);
        double r79944 = r79941 + r79943;
        double r79945 = 2.0;
        double r79946 = r79944 / r79945;
        double r79947 = y;
        double r79948 = cos(r79947);
        double r79949 = r79946 * r79948;
        double r79950 = 0.3333333333333333;
        double r79951 = 3.0;
        double r79952 = pow(r79940, r79951);
        double r79953 = r79950 * r79952;
        double r79954 = 0.016666666666666666;
        double r79955 = 5.0;
        double r79956 = pow(r79940, r79955);
        double r79957 = r79954 * r79956;
        double r79958 = 2.0;
        double r79959 = r79958 * r79940;
        double r79960 = r79957 + r79959;
        double r79961 = r79953 + r79960;
        double r79962 = r79961 / r79945;
        double r79963 = sin(r79947);
        double r79964 = r79962 * r79963;
        double r79965 = /* ERROR: no complex support in C */;
        double r79966 = /* ERROR: no complex support in C */;
        return r79966;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.8

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

  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 2019352 
(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)))))