Average Error: 0.0 → 0.0
Time: 14.5s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \left(\cos y \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}\right) + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Re(\left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \left(\cos y \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}\right) + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r26256 = x;
        double r26257 = exp(r26256);
        double r26258 = -r26256;
        double r26259 = exp(r26258);
        double r26260 = r26257 + r26259;
        double r26261 = 2.0;
        double r26262 = r26260 / r26261;
        double r26263 = y;
        double r26264 = cos(r26263);
        double r26265 = r26262 * r26264;
        double r26266 = r26257 - r26259;
        double r26267 = r26266 / r26261;
        double r26268 = sin(r26263);
        double r26269 = r26267 * r26268;
        double r26270 = /* ERROR: no complex support in C */;
        double r26271 = /* ERROR: no complex support in C */;
        return r26271;
}
↓
double f(double x, double y) {
        double r26272 = x;
        double r26273 = exp(r26272);
        double r26274 = -r26272;
        double r26275 = exp(r26274);
        double r26276 = r26273 + r26275;
        double r26277 = 2.0;
        double r26278 = r26276 / r26277;
        double r26279 = sqrt(r26278);
        double r26280 = y;
        double r26281 = cos(r26280);
        double r26282 = r26281 * r26279;
        double r26283 = r26279 * r26282;
        double r26284 = r26273 - r26275;
        double r26285 = r26284 / r26277;
        double r26286 = sin(r26280);
        double r26287 = r26285 * r26286;
        double r26288 = /* ERROR: no complex support in C */;
        double r26289 = /* ERROR: no complex support in C */;
        return r26289;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.0

    \[\leadsto \Re(\left(\color{blue}{\left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}\right)} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  4. Applied associate-*l*0.0

    \[\leadsto \Re(\left(\color{blue}{\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \cos y\right)} + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  5. Simplified0.0

    \[\leadsto \Re(\left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \color{blue}{\left(\sqrt{\frac{e^{-x} + e^{x}}{2}} \cdot \cos y\right)} + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  6. Final simplification0.0

    \[\leadsto \Re(\left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \left(\cos y \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}\right) + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))