Average Error: 0.0 → 0.0
Time: 5.8s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, e^{-x}\right)}{2} \cdot \cos y\]
\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(\sqrt{e^{x}}, \sqrt{e^{x}}, e^{-x}\right)}{2} \cdot \cos y
double f(double x, double y) {
        double r41912 = x;
        double r41913 = exp(r41912);
        double r41914 = -r41912;
        double r41915 = exp(r41914);
        double r41916 = r41913 + r41915;
        double r41917 = 2.0;
        double r41918 = r41916 / r41917;
        double r41919 = y;
        double r41920 = cos(r41919);
        double r41921 = r41918 * r41920;
        double r41922 = r41913 - r41915;
        double r41923 = r41922 / r41917;
        double r41924 = sin(r41919);
        double r41925 = r41923 * r41924;
        double r41926 = /* ERROR: no complex support in C */;
        double r41927 = /* ERROR: no complex support in C */;
        return r41927;
}
↓
double f(double x, double y) {
        double r41928 = x;
        double r41929 = exp(r41928);
        double r41930 = sqrt(r41929);
        double r41931 = -r41928;
        double r41932 = exp(r41931);
        double r41933 = fma(r41930, r41930, r41932);
        double r41934 = 2.0;
        double r41935 = r41933 / r41934;
        double r41936 = y;
        double r41937 = cos(r41936);
        double r41938 = r41935 * r41937;
        return r41938;
}

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. Using strategy rm

  4. Applied add-sqr-sqrt0.0

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

  5. Applied fma-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020089 +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)))))