Average Error: 0.0 → 0.0
Time: 24.5s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} \cdot \cos y + \frac{\frac{\cos y}{\sqrt{e^{x}}}}{\sqrt{e^{x}}}}{2}\]
double f(double x, double y) {
        double r468962 = x;
        double r468963 = exp(r468962);
        double r468964 = -r468962;
        double r468965 = exp(r468964);
        double r468966 = r468963 + r468965;
        double r468967 = 2.0;
        double r468968 = r468966 / r468967;
        double r468969 = y;
        double r468970 = cos(r468969);
        double r468971 = r468968 * r468970;
        double r468972 = r468963 - r468965;
        double r468973 = r468972 / r468967;
        double r468974 = sin(r468969);
        double r468975 = r468973 * r468974;
        double r468976 = /* ERROR: no complex support in C */;
        double r468977 = /* ERROR: no complex support in C */;
        return r468977;
}
↓
double f(double x, double y) {
        double r468978 = x;
        double r468979 = exp(r468978);
        double r468980 = y;
        double r468981 = cos(r468980);
        double r468982 = r468979 * r468981;
        double r468983 = sqrt(r468979);
        double r468984 = r468981 / r468983;
        double r468985 = r468984 / r468983;
        double r468986 = r468982 + r468985;
        double r468987 = 2.0;
        double r468988 = r468986 / r468987;
        return r468988;
}


\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} \cdot \cos y + \frac{\frac{\cos y}{\sqrt{e^{x}}}}{\sqrt{e^{x}}}}{2}

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

    \[\leadsto \color{blue}{\frac{\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}}{2}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.0

    \[\leadsto \frac{\frac{\cos y}{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}}} + \cos y \cdot e^{x}}{2}\]

  5. Applied associate-/r*0.0

    \[\leadsto \frac{\color{blue}{\frac{\frac{\cos y}{\sqrt{e^{x}}}}{\sqrt{e^{x}}}} + \cos y \cdot e^{x}}{2}\]

  6. Final simplification0.0

    \[\leadsto \frac{e^{x} \cdot \cos y + \frac{\frac{\cos y}{\sqrt{e^{x}}}}{\sqrt{e^{x}}}}{2}\]

Reproduce

herbie shell --seed 2019102 
(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)))))