Average Error: 0.0 → 0.0
Time: 6.5s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \left(\sqrt{\frac{e^{\log \left(e^{x} + e^{-x}\right)}}{2}} \cdot \cos y\right)\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \left(\sqrt{\frac{e^{\log \left(e^{x} + e^{-x}\right)}}{2}} \cdot \cos y\right)
double f(double x, double y) {
        double r30272 = x;
        double r30273 = exp(r30272);
        double r30274 = -r30272;
        double r30275 = exp(r30274);
        double r30276 = r30273 + r30275;
        double r30277 = 2.0;
        double r30278 = r30276 / r30277;
        double r30279 = y;
        double r30280 = cos(r30279);
        double r30281 = r30278 * r30280;
        double r30282 = r30273 - r30275;
        double r30283 = r30282 / r30277;
        double r30284 = sin(r30279);
        double r30285 = r30283 * r30284;
        double r30286 = /* ERROR: no complex support in C */;
        double r30287 = /* ERROR: no complex support in C */;
        return r30287;
}
↓
double f(double x, double y) {
        double r30288 = x;
        double r30289 = exp(r30288);
        double r30290 = -r30288;
        double r30291 = exp(r30290);
        double r30292 = r30289 + r30291;
        double r30293 = 2.0;
        double r30294 = r30292 / r30293;
        double r30295 = sqrt(r30294);
        double r30296 = log(r30292);
        double r30297 = exp(r30296);
        double r30298 = r30297 / r30293;
        double r30299 = sqrt(r30298);
        double r30300 = y;
        double r30301 = cos(r30300);
        double r30302 = r30299 * r30301;
        double r30303 = r30295 * r30302;
        return r30303;
}

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

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

  5. Applied associate-*l*0.0

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

  7. Applied add-exp-log0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019352 +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)))))