Average Error: 0.0 → 0.0
Time: 15.4s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\left(\cos y \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}\right) \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\left(\cos y \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}\right) \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}
double f(double x, double y) {
        double r26815 = x;
        double r26816 = exp(r26815);
        double r26817 = -r26815;
        double r26818 = exp(r26817);
        double r26819 = r26816 + r26818;
        double r26820 = 2.0;
        double r26821 = r26819 / r26820;
        double r26822 = y;
        double r26823 = cos(r26822);
        double r26824 = r26821 * r26823;
        double r26825 = r26816 - r26818;
        double r26826 = r26825 / r26820;
        double r26827 = sin(r26822);
        double r26828 = r26826 * r26827;
        double r26829 = /* ERROR: no complex support in C */;
        double r26830 = /* ERROR: no complex support in C */;
        return r26830;
}
↓
double f(double x, double y) {
        double r26831 = y;
        double r26832 = cos(r26831);
        double r26833 = x;
        double r26834 = exp(r26833);
        double r26835 = -r26833;
        double r26836 = exp(r26835);
        double r26837 = r26834 + r26836;
        double r26838 = 2.0;
        double r26839 = r26837 / r26838;
        double r26840 = sqrt(r26839);
        double r26841 = r26832 * r26840;
        double r26842 = r26841 * r26840;
        return r26842;
}

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

    \[\leadsto \color{blue}{\cos y \cdot \frac{e^{x} + e^{-x}}{2}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-*r*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +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)))))