Average Error: 0.0 → 0.0
Time: 16.6s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} + e^{-x}}{2} \cdot \cos y\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{e^{x} + e^{-x}}{2} \cdot \cos y
double f(double x, double y) {
        double r32012 = x;
        double r32013 = exp(r32012);
        double r32014 = -r32012;
        double r32015 = exp(r32014);
        double r32016 = r32013 + r32015;
        double r32017 = 2.0;
        double r32018 = r32016 / r32017;
        double r32019 = y;
        double r32020 = cos(r32019);
        double r32021 = r32018 * r32020;
        double r32022 = r32013 - r32015;
        double r32023 = r32022 / r32017;
        double r32024 = sin(r32019);
        double r32025 = r32023 * r32024;
        double r32026 = /* ERROR: no complex support in C */;
        double r32027 = /* ERROR: no complex support in C */;
        return r32027;
}
↓
double f(double x, double y) {
        double r32028 = x;
        double r32029 = exp(r32028);
        double r32030 = -r32028;
        double r32031 = exp(r32030);
        double r32032 = r32029 + r32031;
        double r32033 = 2.0;
        double r32034 = r32032 / r32033;
        double r32035 = y;
        double r32036 = cos(r32035);
        double r32037 = r32034 * r32036;
        return r32037;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +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)))))