Average Error: 0.0 → 0.0
Time: 6.3s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(1 \cdot \left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y\right) + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Re(\left(1 \cdot \left(\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 r48634 = x;
        double r48635 = exp(r48634);
        double r48636 = -r48634;
        double r48637 = exp(r48636);
        double r48638 = r48635 + r48637;
        double r48639 = 2.0;
        double r48640 = r48638 / r48639;
        double r48641 = y;
        double r48642 = cos(r48641);
        double r48643 = r48640 * r48642;
        double r48644 = r48635 - r48637;
        double r48645 = r48644 / r48639;
        double r48646 = sin(r48641);
        double r48647 = r48645 * r48646;
        double r48648 = /* ERROR: no complex support in C */;
        double r48649 = /* ERROR: no complex support in C */;
        return r48649;
}
↓
double f(double x, double y) {
        double r48650 = 1.0;
        double r48651 = x;
        double r48652 = exp(r48651);
        double r48653 = -r48651;
        double r48654 = exp(r48653);
        double r48655 = r48652 + r48654;
        double r48656 = 2.0;
        double r48657 = r48655 / r48656;
        double r48658 = y;
        double r48659 = cos(r48658);
        double r48660 = r48657 * r48659;
        double r48661 = r48650 * r48660;
        double r48662 = r48652 - r48654;
        double r48663 = r48662 / r48656;
        double r48664 = sin(r48658);
        double r48665 = r48663 * r48664;
        double r48666 = /* ERROR: no complex support in C */;
        double r48667 = /* ERROR: no complex support in C */;
        return r48667;
}

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 *-un-lft-identity0.0

    \[\leadsto \Re(\left(\frac{e^{x} + e^{-x}}{\color{blue}{1 \cdot 2}} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  4. Applied *-un-lft-identity0.0

    \[\leadsto \Re(\left(\frac{\color{blue}{1 \cdot \left(e^{x} + e^{-x}\right)}}{1 \cdot 2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  5. Applied times-frac0.0

    \[\leadsto \Re(\left(\color{blue}{\left(\frac{1}{1} \cdot \frac{e^{x} + e^{-x}}{2}\right)} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  6. Applied associate-*l*0.0

    \[\leadsto \Re(\left(\color{blue}{\frac{1}{1} \cdot \left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y\right)} + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  7. Final simplification0.0

    \[\leadsto \Re(\left(1 \cdot \left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y\right) + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019356 +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)))))