Average Error: 0.0 → 0.0
Time: 6.8s
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 r85102 = x;
        double r85103 = exp(r85102);
        double r85104 = -r85102;
        double r85105 = exp(r85104);
        double r85106 = r85103 + r85105;
        double r85107 = 2.0;
        double r85108 = r85106 / r85107;
        double r85109 = y;
        double r85110 = cos(r85109);
        double r85111 = r85108 * r85110;
        double r85112 = r85103 - r85105;
        double r85113 = r85112 / r85107;
        double r85114 = sin(r85109);
        double r85115 = r85113 * r85114;
        double r85116 = /* ERROR: no complex support in C */;
        double r85117 = /* ERROR: no complex support in C */;
        return r85117;
}
↓
double f(double x, double y) {
        double r85118 = x;
        double r85119 = exp(r85118);
        double r85120 = -r85118;
        double r85121 = exp(r85120);
        double r85122 = r85119 + r85121;
        double r85123 = 2.0;
        double r85124 = r85122 / r85123;
        double r85125 = y;
        double r85126 = cos(r85125);
        double r85127 = r85124 * r85126;
        return r85127;
}

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 2019322 
(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)))))