Average Error: 43.4 → 0.7
Time: 31.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{x \cdot \left(\log \left(e^{\frac{1}{3} \cdot \left(x \cdot x\right)}\right) + 2\right) + {x}^{5} \cdot \frac{1}{60}}{2} i\right))\]
\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 + \sin y \cdot \frac{x \cdot \left(\log \left(e^{\frac{1}{3} \cdot \left(x \cdot x\right)}\right) + 2\right) + {x}^{5} \cdot \frac{1}{60}}{2} i\right))
double f(double x, double y) {
        double r2200950 = x;
        double r2200951 = exp(r2200950);
        double r2200952 = -r2200950;
        double r2200953 = exp(r2200952);
        double r2200954 = r2200951 + r2200953;
        double r2200955 = 2.0;
        double r2200956 = r2200954 / r2200955;
        double r2200957 = y;
        double r2200958 = cos(r2200957);
        double r2200959 = r2200956 * r2200958;
        double r2200960 = r2200951 - r2200953;
        double r2200961 = r2200960 / r2200955;
        double r2200962 = sin(r2200957);
        double r2200963 = r2200961 * r2200962;
        double r2200964 = /* ERROR: no complex support in C */;
        double r2200965 = /* ERROR: no complex support in C */;
        return r2200965;
}
↓
double f(double x, double y) {
        double r2200966 = x;
        double r2200967 = exp(r2200966);
        double r2200968 = -r2200966;
        double r2200969 = exp(r2200968);
        double r2200970 = r2200967 + r2200969;
        double r2200971 = 2.0;
        double r2200972 = r2200970 / r2200971;
        double r2200973 = y;
        double r2200974 = cos(r2200973);
        double r2200975 = r2200972 * r2200974;
        double r2200976 = sin(r2200973);
        double r2200977 = 0.3333333333333333;
        double r2200978 = r2200966 * r2200966;
        double r2200979 = r2200977 * r2200978;
        double r2200980 = exp(r2200979);
        double r2200981 = log(r2200980);
        double r2200982 = 2.0;
        double r2200983 = r2200981 + r2200982;
        double r2200984 = r2200966 * r2200983;
        double r2200985 = 5.0;
        double r2200986 = pow(r2200966, r2200985);
        double r2200987 = 0.016666666666666666;
        double r2200988 = r2200986 * r2200987;
        double r2200989 = r2200984 + r2200988;
        double r2200990 = r2200989 / r2200971;
        double r2200991 = r2200976 * r2200990;
        double r2200992 = /* ERROR: no complex support in C */;
        double r2200993 = /* ERROR: no complex support in C */;
        return r2200993;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  5. Applied add-log-exp0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019192 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))