Average Error: 0.0 → 0.0
Time: 17.2s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1615994 = x;
        double r1615995 = exp(r1615994);
        double r1615996 = -r1615994;
        double r1615997 = exp(r1615996);
        double r1615998 = r1615995 + r1615997;
        double r1615999 = 2.0;
        double r1616000 = r1615998 / r1615999;
        double r1616001 = y;
        double r1616002 = cos(r1616001);
        double r1616003 = r1616000 * r1616002;
        double r1616004 = r1615995 - r1615997;
        double r1616005 = r1616004 / r1615999;
        double r1616006 = sin(r1616001);
        double r1616007 = r1616005 * r1616006;
        double r1616008 = /* ERROR: no complex support in C */;
        double r1616009 = /* ERROR: no complex support in C */;
        return r1616009;
}
↓
double f(double x, double y) {
        double r1616010 = x;
        double r1616011 = exp(r1616010);
        double r1616012 = -r1616010;
        double r1616013 = exp(r1616012);
        double r1616014 = r1616011 + r1616013;
        double r1616015 = 2.0;
        double r1616016 = r1616014 / r1616015;
        double r1616017 = y;
        double r1616018 = cos(r1616017);
        double r1616019 = r1616016 * r1616018;
        double r1616020 = r1616011 - r1616013;
        double r1616021 = r1616020 / r1616015;
        double r1616022 = sin(r1616017);
        double r1616023 = r1616021 * r1616022;
        double r1616024 = /* ERROR: no complex support in C */;
        double r1616025 = /* ERROR: no complex support in C */;
        return r1616025;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))