Average Error: 0.0 → 0.7
Time: 4.6s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{12}, {x}^{4}, 2\right)\right)}{2} \cdot \cos y\]
\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(x, x, \mathsf{fma}\left(\frac{1}{12}, {x}^{4}, 2\right)\right)}{2} \cdot \cos y
double f(double x, double y) {
        double r22893 = x;
        double r22894 = exp(r22893);
        double r22895 = -r22893;
        double r22896 = exp(r22895);
        double r22897 = r22894 + r22896;
        double r22898 = 2.0;
        double r22899 = r22897 / r22898;
        double r22900 = y;
        double r22901 = cos(r22900);
        double r22902 = r22899 * r22901;
        double r22903 = r22894 - r22896;
        double r22904 = r22903 / r22898;
        double r22905 = sin(r22900);
        double r22906 = r22904 * r22905;
        double r22907 = /* ERROR: no complex support in C */;
        double r22908 = /* ERROR: no complex support in C */;
        return r22908;
}
↓
double f(double x, double y) {
        double r22909 = x;
        double r22910 = 0.08333333333333333;
        double r22911 = 4.0;
        double r22912 = pow(r22909, r22911);
        double r22913 = 2.0;
        double r22914 = fma(r22910, r22912, r22913);
        double r22915 = fma(r22909, r22909, r22914);
        double r22916 = 2.0;
        double r22917 = r22915 / r22916;
        double r22918 = y;
        double r22919 = cos(r22918);
        double r22920 = r22917 * r22919;
        return r22920;
}

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

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

  3. Taylor expanded around 0 0.7

    \[\leadsto \frac{\color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + 2\right)}}{2} \cdot \cos y\]

  4. Simplified0.7

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{12}, {x}^{4}, 2\right)\right)}}{2} \cdot \cos y\]

  5. Final simplification0.7

    \[\leadsto \frac{\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{12}, {x}^{4}, 2\right)\right)}{2} \cdot \cos y\]

Reproduce

herbie shell --seed 2020060 +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)))))