Average Error: 0.0 → 0.0
Time: 32.6s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} + e^{-x}}{2} \cdot \cos y\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{e^{x} + e^{-x}}{2} \cdot \cos y
double f(double x, double y) {
        double r19144 = x;
        double r19145 = exp(r19144);
        double r19146 = -r19144;
        double r19147 = exp(r19146);
        double r19148 = r19145 + r19147;
        double r19149 = 2.0;
        double r19150 = r19148 / r19149;
        double r19151 = y;
        double r19152 = cos(r19151);
        double r19153 = r19150 * r19152;
        double r19154 = r19145 - r19147;
        double r19155 = r19154 / r19149;
        double r19156 = sin(r19151);
        double r19157 = r19155 * r19156;
        double r19158 = /* ERROR: no complex support in C */;
        double r19159 = /* ERROR: no complex support in C */;
        return r19159;
}
↓
double f(double x, double y) {
        double r19160 = x;
        double r19161 = exp(r19160);
        double r19162 = -r19160;
        double r19163 = exp(r19162);
        double r19164 = r19161 + r19163;
        double r19165 = 2.0;
        double r19166 = r19164 / r19165;
        double r19167 = y;
        double r19168 = cos(r19167);
        double r19169 = r19166 * r19168;
        return r19169;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

    \[\leadsto \frac{e^{x} + e^{-x}}{2} \cdot \cos y\]

Reproduce

herbie shell --seed 2019298 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  :precision binary64
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))