Average Error: 0.0 → 0.0
Time: 4.0s
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 r24230 = x;
        double r24231 = exp(r24230);
        double r24232 = -r24230;
        double r24233 = exp(r24232);
        double r24234 = r24231 + r24233;
        double r24235 = 2.0;
        double r24236 = r24234 / r24235;
        double r24237 = y;
        double r24238 = cos(r24237);
        double r24239 = r24236 * r24238;
        double r24240 = r24231 - r24233;
        double r24241 = r24240 / r24235;
        double r24242 = sin(r24237);
        double r24243 = r24241 * r24242;
        double r24244 = /* ERROR: no complex support in C */;
        double r24245 = /* ERROR: no complex support in C */;
        return r24245;
}
↓
double f(double x, double y) {
        double r24246 = x;
        double r24247 = exp(r24246);
        double r24248 = -r24246;
        double r24249 = exp(r24248);
        double r24250 = r24247 + r24249;
        double r24251 = 2.0;
        double r24252 = r24250 / r24251;
        double r24253 = y;
        double r24254 = cos(r24253);
        double r24255 = r24252 * r24254;
        double r24256 = r24247 - r24249;
        double r24257 = r24256 / r24251;
        double r24258 = sin(r24253);
        double r24259 = r24257 * r24258;
        double r24260 = /* ERROR: no complex support in C */;
        double r24261 = /* ERROR: no complex support in C */;
        return r24261;
}

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 2019354 +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)))))