Average Error: 0.0 → 0.0
Time: 12.7s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\left(\cos y\right), \left(e^{x}\right), \left(\frac{\cos y}{e^{x}}\right)\right)}{2}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\mathsf{fma}\left(\left(\cos y\right), \left(e^{x}\right), \left(\frac{\cos y}{e^{x}}\right)\right)}{2}
double f(double x, double y) {
        double r695962 = x;
        double r695963 = exp(r695962);
        double r695964 = -r695962;
        double r695965 = exp(r695964);
        double r695966 = r695963 + r695965;
        double r695967 = 2.0;
        double r695968 = r695966 / r695967;
        double r695969 = y;
        double r695970 = cos(r695969);
        double r695971 = r695968 * r695970;
        double r695972 = r695963 - r695965;
        double r695973 = r695972 / r695967;
        double r695974 = sin(r695969);
        double r695975 = r695973 * r695974;
        double r695976 = /* ERROR: no complex support in C */;
        double r695977 = /* ERROR: no complex support in C */;
        return r695977;
}
↓
double f(double x, double y) {
        double r695978 = y;
        double r695979 = cos(r695978);
        double r695980 = x;
        double r695981 = exp(r695980);
        double r695982 = r695979 / r695981;
        double r695983 = fma(r695979, r695981, r695982);
        double r695984 = 2.0;
        double r695985 = r695983 / r695984;
        return r695985;
}

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

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos y\right), \left(e^{x}\right), \left(\frac{\cos y}{e^{x}}\right)\right)}{2}}\]

  3. Final simplification0.0

    \[\leadsto \frac{\mathsf{fma}\left(\left(\cos y\right), \left(e^{x}\right), \left(\frac{\cos y}{e^{x}}\right)\right)}{2}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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)))))