Average Error: 0.0 → 0.0
Time: 7.6s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\sqrt{\frac{\sqrt{e^{x} + e^{-x}}}{\sqrt{2}}} \cdot \left(\sqrt{\frac{\sqrt{e^{x} + e^{-x}}}{\sqrt{2}}} \cdot \left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \cos y\right)\right) + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\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{\sqrt{e^{x} + e^{-x}}}{\sqrt{2}}} \cdot \left(\sqrt{\frac{\sqrt{e^{x} + e^{-x}}}{\sqrt{2}}} \cdot \left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \cos y\right)\right) + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r54672 = x;
        double r54673 = exp(r54672);
        double r54674 = -r54672;
        double r54675 = exp(r54674);
        double r54676 = r54673 + r54675;
        double r54677 = 2.0;
        double r54678 = r54676 / r54677;
        double r54679 = y;
        double r54680 = cos(r54679);
        double r54681 = r54678 * r54680;
        double r54682 = r54673 - r54675;
        double r54683 = r54682 / r54677;
        double r54684 = sin(r54679);
        double r54685 = r54683 * r54684;
        double r54686 = /* ERROR: no complex support in C */;
        double r54687 = /* ERROR: no complex support in C */;
        return r54687;
}
↓
double f(double x, double y) {
        double r54688 = x;
        double r54689 = exp(r54688);
        double r54690 = -r54688;
        double r54691 = exp(r54690);
        double r54692 = r54689 + r54691;
        double r54693 = sqrt(r54692);
        double r54694 = 2.0;
        double r54695 = sqrt(r54694);
        double r54696 = r54693 / r54695;
        double r54697 = sqrt(r54696);
        double r54698 = r54692 / r54694;
        double r54699 = sqrt(r54698);
        double r54700 = y;
        double r54701 = cos(r54700);
        double r54702 = r54699 * r54701;
        double r54703 = r54697 * r54702;
        double r54704 = r54697 * r54703;
        double r54705 = r54689 - r54691;
        double r54706 = r54705 / r54694;
        double r54707 = sin(r54700);
        double r54708 = r54706 * r54707;
        double r54709 = /* ERROR: no complex support in C */;
        double r54710 = /* ERROR: no complex support in C */;
        return r54710;
}

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. Using strategy rm

  6. Applied add-sqr-sqrt1.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Applied times-frac0.0

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

  9. Applied sqrt-prod0.0

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

  10. Applied associate-*l*0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020036 +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)))))