Average Error: 44.2 → 0.7
Time: 1.1m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{e^{x} - e^{-x}}{2.0} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{\frac{1}{60} \cdot {x}^{5} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)}{2.0} \cdot \sin y i\right))\]
\Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{e^{x} - e^{-x}}{2.0} \cdot \sin y i\right))
↓
\Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{\frac{1}{60} \cdot {x}^{5} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)}{2.0} \cdot \sin y i\right))
double f(double x, double y) {
        double r2675027 = x;
        double r2675028 = exp(r2675027);
        double r2675029 = -r2675027;
        double r2675030 = exp(r2675029);
        double r2675031 = r2675028 + r2675030;
        double r2675032 = 2.0;
        double r2675033 = r2675031 / r2675032;
        double r2675034 = y;
        double r2675035 = cos(r2675034);
        double r2675036 = r2675033 * r2675035;
        double r2675037 = r2675028 - r2675030;
        double r2675038 = r2675037 / r2675032;
        double r2675039 = sin(r2675034);
        double r2675040 = r2675038 * r2675039;
        double r2675041 = /* ERROR: no complex support in C */;
        double r2675042 = /* ERROR: no complex support in C */;
        return r2675042;
}
↓
double f(double x, double y) {
        double r2675043 = x;
        double r2675044 = exp(r2675043);
        double r2675045 = -r2675043;
        double r2675046 = exp(r2675045);
        double r2675047 = r2675044 + r2675046;
        double r2675048 = 2.0;
        double r2675049 = r2675047 / r2675048;
        double r2675050 = y;
        double r2675051 = cos(r2675050);
        double r2675052 = r2675049 * r2675051;
        double r2675053 = 0.016666666666666666;
        double r2675054 = 5.0;
        double r2675055 = pow(r2675043, r2675054);
        double r2675056 = r2675053 * r2675055;
        double r2675057 = r2675043 * r2675043;
        double r2675058 = 0.3333333333333333;
        double r2675059 = r2675057 * r2675058;
        double r2675060 = 2.0;
        double r2675061 = r2675059 + r2675060;
        double r2675062 = r2675043 * r2675061;
        double r2675063 = r2675056 + r2675062;
        double r2675064 = r2675063 / r2675048;
        double r2675065 = sin(r2675050);
        double r2675066 = r2675064 * r2675065;
        double r2675067 = /* ERROR: no complex support in C */;
        double r2675068 = /* ERROR: no complex support in C */;
        return r2675068;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 44.2

    \[\Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{e^{x} - e^{-x}}{2.0} \cdot \sin y i\right))\]

  2. Taylor expanded around 0 0.7

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2.0} \cdot \sin y i\right))\]

  3. Simplified0.7

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{\color{blue}{x \cdot \left(2 + \left(x \cdot x\right) \cdot \frac{1}{3}\right) + \frac{1}{60} \cdot {x}^{5}}}{2.0} \cdot \sin y i\right))\]

  4. Final simplification0.7

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{\frac{1}{60} \cdot {x}^{5} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)}{2.0} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019165 
(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)))))