Average Error: 0.0 → 0.0
Time: 20.9s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(e^{x}, \cos y, \frac{\frac{1}{\sqrt{e^{x}}}}{\sqrt{\sqrt{e^{x}}}} \cdot \frac{\cos y}{\sqrt{\sqrt{e^{x}}}}\right)}{2}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\mathsf{fma}\left(e^{x}, \cos y, \frac{\frac{1}{\sqrt{e^{x}}}}{\sqrt{\sqrt{e^{x}}}} \cdot \frac{\cos y}{\sqrt{\sqrt{e^{x}}}}\right)}{2}
double f(double x, double y) {
        double r1902693 = x;
        double r1902694 = exp(r1902693);
        double r1902695 = -r1902693;
        double r1902696 = exp(r1902695);
        double r1902697 = r1902694 + r1902696;
        double r1902698 = 2.0;
        double r1902699 = r1902697 / r1902698;
        double r1902700 = y;
        double r1902701 = cos(r1902700);
        double r1902702 = r1902699 * r1902701;
        double r1902703 = r1902694 - r1902696;
        double r1902704 = r1902703 / r1902698;
        double r1902705 = sin(r1902700);
        double r1902706 = r1902704 * r1902705;
        double r1902707 = /* ERROR: no complex support in C */;
        double r1902708 = /* ERROR: no complex support in C */;
        return r1902708;
}
↓
double f(double x, double y) {
        double r1902709 = x;
        double r1902710 = exp(r1902709);
        double r1902711 = y;
        double r1902712 = cos(r1902711);
        double r1902713 = 1.0;
        double r1902714 = sqrt(r1902710);
        double r1902715 = r1902713 / r1902714;
        double r1902716 = sqrt(r1902714);
        double r1902717 = r1902715 / r1902716;
        double r1902718 = r1902712 / r1902716;
        double r1902719 = r1902717 * r1902718;
        double r1902720 = fma(r1902710, r1902712, r1902719);
        double r1902721 = 2.0;
        double r1902722 = r1902720 / r1902721;
        return r1902722;
}

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

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(e^{x}, \cos y, \frac{\cos y}{e^{x}}\right)}{2}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-/r*0.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Applied sqrt-prod0.0

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

  9. Applied div-inv0.0

    \[\leadsto \frac{\mathsf{fma}\left(e^{x}, \cos y, \frac{\color{blue}{\cos y \cdot \frac{1}{\sqrt{e^{x}}}}}{\sqrt{\sqrt{e^{x}}} \cdot \sqrt{\sqrt{e^{x}}}}\right)}{2}\]

  10. Applied times-frac0.0

    \[\leadsto \frac{\mathsf{fma}\left(e^{x}, \cos y, \color{blue}{\frac{\cos y}{\sqrt{\sqrt{e^{x}}}} \cdot \frac{\frac{1}{\sqrt{e^{x}}}}{\sqrt{\sqrt{e^{x}}}}}\right)}{2}\]

  11. Final simplification0.0

    \[\leadsto \frac{\mathsf{fma}\left(e^{x}, \cos y, \frac{\frac{1}{\sqrt{e^{x}}}}{\sqrt{\sqrt{e^{x}}}} \cdot \frac{\cos y}{\sqrt{\sqrt{e^{x}}}}\right)}{2}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))