Average Error: 0.0 → 0.0
Time: 13.6s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\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 r1715021 = x;
        double r1715022 = exp(r1715021);
        double r1715023 = -r1715021;
        double r1715024 = exp(r1715023);
        double r1715025 = r1715022 + r1715024;
        double r1715026 = 2.0;
        double r1715027 = r1715025 / r1715026;
        double r1715028 = y;
        double r1715029 = cos(r1715028);
        double r1715030 = r1715027 * r1715029;
        double r1715031 = r1715022 - r1715024;
        double r1715032 = r1715031 / r1715026;
        double r1715033 = sin(r1715028);
        double r1715034 = r1715032 * r1715033;
        double r1715035 = /* ERROR: no complex support in C */;
        double r1715036 = /* ERROR: no complex support in C */;
        return r1715036;
}
↓
double f(double x, double y) {
        double r1715037 = x;
        double r1715038 = exp(r1715037);
        double r1715039 = -r1715037;
        double r1715040 = exp(r1715039);
        double r1715041 = r1715038 + r1715040;
        double r1715042 = 2.0;
        double r1715043 = r1715041 / r1715042;
        double r1715044 = y;
        double r1715045 = cos(r1715044);
        double r1715046 = r1715043 * r1715045;
        double r1715047 = r1715038 - r1715040;
        double r1715048 = r1715047 / r1715042;
        double r1715049 = sin(r1715044);
        double r1715050 = r1715048 * r1715049;
        double r1715051 = /* ERROR: no complex support in C */;
        double r1715052 = /* ERROR: no complex support in C */;
        return r1715052;
}

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 2019163 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))