Average Error: 0.0 → 0.0
Time: 6.0s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \cos y\right) + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\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(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \cos y\right) + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r41764 = x;
        double r41765 = exp(r41764);
        double r41766 = -r41764;
        double r41767 = exp(r41766);
        double r41768 = r41765 + r41767;
        double r41769 = 2.0;
        double r41770 = r41768 / r41769;
        double r41771 = y;
        double r41772 = cos(r41771);
        double r41773 = r41770 * r41772;
        double r41774 = r41765 - r41767;
        double r41775 = r41774 / r41769;
        double r41776 = sin(r41771);
        double r41777 = r41775 * r41776;
        double r41778 = /* ERROR: no complex support in C */;
        double r41779 = /* ERROR: no complex support in C */;
        return r41779;
}
↓
double f(double x, double y) {
        double r41780 = x;
        double r41781 = exp(r41780);
        double r41782 = -r41780;
        double r41783 = exp(r41782);
        double r41784 = r41781 + r41783;
        double r41785 = 2.0;
        double r41786 = r41784 / r41785;
        double r41787 = sqrt(r41786);
        double r41788 = y;
        double r41789 = cos(r41788);
        double r41790 = r41787 * r41789;
        double r41791 = r41787 * r41790;
        double r41792 = r41781 - r41783;
        double r41793 = r41792 / r41785;
        double r41794 = sin(r41788);
        double r41795 = r41793 * r41794;
        double r41796 = /* ERROR: no complex support in C */;
        double r41797 = /* ERROR: no complex support in C */;
        return r41797;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020083 
(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)))))