Average Error: 43.9 → 0.9
Time: 20.8s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\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 r39936 = x;
        double r39937 = exp(r39936);
        double r39938 = -r39936;
        double r39939 = exp(r39938);
        double r39940 = r39937 + r39939;
        double r39941 = 2.0;
        double r39942 = r39940 / r39941;
        double r39943 = y;
        double r39944 = cos(r39943);
        double r39945 = r39942 * r39944;
        double r39946 = r39937 - r39939;
        double r39947 = r39946 / r39941;
        double r39948 = sin(r39943);
        double r39949 = r39947 * r39948;
        double r39950 = /* ERROR: no complex support in C */;
        double r39951 = /* ERROR: no complex support in C */;
        return r39951;
}
↓
double f(double x, double y) {
        double r39952 = 0.3333333333333333;
        double r39953 = x;
        double r39954 = 3.0;
        double r39955 = pow(r39953, r39954);
        double r39956 = r39952 * r39955;
        double r39957 = 0.016666666666666666;
        double r39958 = 5.0;
        double r39959 = pow(r39953, r39958);
        double r39960 = r39957 * r39959;
        double r39961 = 2.0;
        double r39962 = r39961 * r39953;
        double r39963 = r39960 + r39962;
        double r39964 = r39956 + r39963;
        double r39965 = 2.0;
        double r39966 = r39964 / r39965;
        double r39967 = y;
        double r39968 = sin(r39967);
        double r39969 = r39966 * r39968;
        return r39969;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.9

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

  2. Simplified43.9

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

  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 2020043 
(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)))))