Average Error: 0.0 → 0.0
Time: 4.7s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\left(e^{x} + e^{-x}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\left(e^{x} + e^{-x}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)
double f(double x, double y) {
        double r25183 = x;
        double r25184 = exp(r25183);
        double r25185 = -r25183;
        double r25186 = exp(r25185);
        double r25187 = r25184 + r25186;
        double r25188 = 2.0;
        double r25189 = r25187 / r25188;
        double r25190 = y;
        double r25191 = cos(r25190);
        double r25192 = r25189 * r25191;
        double r25193 = r25184 - r25186;
        double r25194 = r25193 / r25188;
        double r25195 = sin(r25190);
        double r25196 = r25194 * r25195;
        double r25197 = /* ERROR: no complex support in C */;
        double r25198 = /* ERROR: no complex support in C */;
        return r25198;
}
↓
double f(double x, double y) {
        double r25199 = x;
        double r25200 = exp(r25199);
        double r25201 = -r25199;
        double r25202 = exp(r25201);
        double r25203 = r25200 + r25202;
        double r25204 = 1.0;
        double r25205 = 2.0;
        double r25206 = r25204 / r25205;
        double r25207 = y;
        double r25208 = cos(r25207);
        double r25209 = r25206 * r25208;
        double r25210 = r25203 * r25209;
        return r25210;
}

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. Using strategy rm

  4. Applied div-inv0.0

    \[\leadsto \color{blue}{\left(\left(e^{x} + e^{-x}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\]

  5. Applied associate-*l*0.0

    \[\leadsto \color{blue}{\left(e^{x} + e^{-x}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}\]

  6. Final simplification0.0

    \[\leadsto \left(e^{x} + e^{-x}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\]

Reproduce

herbie shell --seed 2019356 
(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)))))