Average Error: 0.0 → 0.0
Time: 2.7s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r25017 = x;
        double r25018 = exp(r25017);
        double r25019 = -r25017;
        double r25020 = exp(r25019);
        double r25021 = r25018 + r25020;
        double r25022 = 2.0;
        double r25023 = r25021 / r25022;
        double r25024 = y;
        double r25025 = cos(r25024);
        double r25026 = r25023 * r25025;
        double r25027 = r25018 - r25020;
        double r25028 = r25027 / r25022;
        double r25029 = sin(r25024);
        double r25030 = r25028 * r25029;
        double r25031 = /* ERROR: no complex support in C */;
        double r25032 = /* ERROR: no complex support in C */;
        return r25032;
}
↓
double f(double x, double y) {
        double r25033 = x;
        double r25034 = exp(r25033);
        double r25035 = -r25033;
        double r25036 = exp(r25035);
        double r25037 = r25034 + r25036;
        double r25038 = 2.0;
        double r25039 = r25037 / r25038;
        double r25040 = y;
        double r25041 = cos(r25040);
        double r25042 = r25039 * r25041;
        double r25043 = r25034 - r25036;
        double r25044 = r25043 / r25038;
        double r25045 = sin(r25040);
        double r25046 = r25044 * r25045;
        double r25047 = /* ERROR: no complex support in C */;
        double r25048 = /* ERROR: no complex support in C */;
        return r25048;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019362 
(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)))))