Average Error: 0.0 → 0.0
Time: 11.4s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\left(\cos y \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}\right) \cdot \sqrt[3]{{\left(\sqrt{\frac{e^{x} + e^{-x}}{2}}\right)}^{3}}\]
\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[3]{{\left(\sqrt{\frac{e^{x} + e^{-x}}{2}}\right)}^{3}}
double f(double x, double y) {
        double r32624 = x;
        double r32625 = exp(r32624);
        double r32626 = -r32624;
        double r32627 = exp(r32626);
        double r32628 = r32625 + r32627;
        double r32629 = 2.0;
        double r32630 = r32628 / r32629;
        double r32631 = y;
        double r32632 = cos(r32631);
        double r32633 = r32630 * r32632;
        double r32634 = r32625 - r32627;
        double r32635 = r32634 / r32629;
        double r32636 = sin(r32631);
        double r32637 = r32635 * r32636;
        double r32638 = /* ERROR: no complex support in C */;
        double r32639 = /* ERROR: no complex support in C */;
        return r32639;
}
↓
double f(double x, double y) {
        double r32640 = y;
        double r32641 = cos(r32640);
        double r32642 = x;
        double r32643 = exp(r32642);
        double r32644 = -r32642;
        double r32645 = exp(r32644);
        double r32646 = r32643 + r32645;
        double r32647 = 2.0;
        double r32648 = r32646 / r32647;
        double r32649 = sqrt(r32648);
        double r32650 = r32641 * r32649;
        double r32651 = 3.0;
        double r32652 = pow(r32649, r32651);
        double r32653 = cbrt(r32652);
        double r32654 = r32650 * r32653;
        return r32654;
}

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. Using strategy rm

  7. Applied add-cbrt-cube0.0

    \[\leadsto \left(\cos y \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}\right) \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}\right) \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}}}\]

  8. Simplified0.0

    \[\leadsto \left(\cos y \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}\right) \cdot \sqrt[3]{\color{blue}{{\left(\sqrt{\frac{e^{x} + e^{-x}}{2}}\right)}^{3}}}\]

  9. Final simplification0.0

    \[\leadsto \left(\cos y \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}\right) \cdot \sqrt[3]{{\left(\sqrt{\frac{e^{x} + e^{-x}}{2}}\right)}^{3}}\]

Reproduce

herbie shell --seed 2020045 +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)))))