Average Error: 43.6 → 0.8
Time: 2.2m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{{x}^{5} \cdot \frac{1}{60} + \left(2 + \left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x}{2} \cdot \sin y i\right))\]
\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{{x}^{5} \cdot \frac{1}{60} + \left(2 + \left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r6418063 = x;
        double r6418064 = exp(r6418063);
        double r6418065 = -r6418063;
        double r6418066 = exp(r6418065);
        double r6418067 = r6418064 + r6418066;
        double r6418068 = 2.0;
        double r6418069 = r6418067 / r6418068;
        double r6418070 = y;
        double r6418071 = cos(r6418070);
        double r6418072 = r6418069 * r6418071;
        double r6418073 = r6418064 - r6418066;
        double r6418074 = r6418073 / r6418068;
        double r6418075 = sin(r6418070);
        double r6418076 = r6418074 * r6418075;
        double r6418077 = /* ERROR: no complex support in C */;
        double r6418078 = /* ERROR: no complex support in C */;
        return r6418078;
}
↓
double f(double x, double y) {
        double r6418079 = x;
        double r6418080 = exp(r6418079);
        double r6418081 = -r6418079;
        double r6418082 = exp(r6418081);
        double r6418083 = r6418080 + r6418082;
        double r6418084 = 2.0;
        double r6418085 = r6418083 / r6418084;
        double r6418086 = y;
        double r6418087 = cos(r6418086);
        double r6418088 = r6418085 * r6418087;
        double r6418089 = 5.0;
        double r6418090 = pow(r6418079, r6418089);
        double r6418091 = 0.016666666666666666;
        double r6418092 = r6418090 * r6418091;
        double r6418093 = 0.3333333333333333;
        double r6418094 = r6418079 * r6418093;
        double r6418095 = r6418094 * r6418079;
        double r6418096 = r6418084 + r6418095;
        double r6418097 = r6418096 * r6418079;
        double r6418098 = r6418092 + r6418097;
        double r6418099 = r6418098 / r6418084;
        double r6418100 = sin(r6418086);
        double r6418101 = r6418099 * r6418100;
        double r6418102 = /* ERROR: no complex support in C */;
        double r6418103 = /* ERROR: no complex support in C */;
        return r6418103;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.6

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019124 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))