Average Error: 0.0 → 0.0
Time: 35.7s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{1}{2} \cdot \left(\left(e^{x} + \frac{1}{e^{x}}\right) \cdot \cos y\right)\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{1}{2} \cdot \left(\left(e^{x} + \frac{1}{e^{x}}\right) \cdot \cos y\right)
double f(double x, double y) {
        double r2185210 = x;
        double r2185211 = exp(r2185210);
        double r2185212 = -r2185210;
        double r2185213 = exp(r2185212);
        double r2185214 = r2185211 + r2185213;
        double r2185215 = 2.0;
        double r2185216 = r2185214 / r2185215;
        double r2185217 = y;
        double r2185218 = cos(r2185217);
        double r2185219 = r2185216 * r2185218;
        double r2185220 = r2185211 - r2185213;
        double r2185221 = r2185220 / r2185215;
        double r2185222 = sin(r2185217);
        double r2185223 = r2185221 * r2185222;
        double r2185224 = /* ERROR: no complex support in C */;
        double r2185225 = /* ERROR: no complex support in C */;
        return r2185225;
}
↓
double f(double x, double y) {
        double r2185226 = 0.5;
        double r2185227 = x;
        double r2185228 = exp(r2185227);
        double r2185229 = 1.0;
        double r2185230 = r2185229 / r2185228;
        double r2185231 = r2185228 + r2185230;
        double r2185232 = y;
        double r2185233 = cos(r2185232);
        double r2185234 = r2185231 * r2185233;
        double r2185235 = r2185226 * r2185234;
        return r2185235;
}

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

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}\right)}\]
  3. Using strategy rm

  4. Applied div-inv0.0

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

  5. Applied distribute-lft-out0.0

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

  6. Final simplification0.0

    \[\leadsto \frac{1}{2} \cdot \left(\left(e^{x} + \frac{1}{e^{x}}\right) \cdot \cos y\right)\]

Reproduce

herbie shell --seed 2019158 
(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)))))