Average Error: 0.0 → 0.0
Time: 3.3s
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 r19086 = x;
        double r19087 = exp(r19086);
        double r19088 = -r19086;
        double r19089 = exp(r19088);
        double r19090 = r19087 + r19089;
        double r19091 = 2.0;
        double r19092 = r19090 / r19091;
        double r19093 = y;
        double r19094 = cos(r19093);
        double r19095 = r19092 * r19094;
        double r19096 = r19087 - r19089;
        double r19097 = r19096 / r19091;
        double r19098 = sin(r19093);
        double r19099 = r19097 * r19098;
        double r19100 = /* ERROR: no complex support in C */;
        double r19101 = /* ERROR: no complex support in C */;
        return r19101;
}
↓
double f(double x, double y) {
        double r19102 = x;
        double r19103 = exp(r19102);
        double r19104 = -r19102;
        double r19105 = exp(r19104);
        double r19106 = r19103 + r19105;
        double r19107 = 2.0;
        double r19108 = r19106 / r19107;
        double r19109 = y;
        double r19110 = cos(r19109);
        double r19111 = r19108 * r19110;
        return r19111;
}

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 2020018 +o rules:numerics
(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)))))