Average Error: 43.4 → 0.9
Time: 1.7m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_* \cdot x\right))_*}{2} \cdot \sin y i\right))\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_* \cdot x\right))_*}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r6179985 = x;
        double r6179986 = exp(r6179985);
        double r6179987 = -r6179985;
        double r6179988 = exp(r6179987);
        double r6179989 = r6179986 + r6179988;
        double r6179990 = 2.0;
        double r6179991 = r6179989 / r6179990;
        double r6179992 = y;
        double r6179993 = cos(r6179992);
        double r6179994 = r6179991 * r6179993;
        double r6179995 = r6179986 - r6179988;
        double r6179996 = r6179995 / r6179990;
        double r6179997 = sin(r6179992);
        double r6179998 = r6179996 * r6179997;
        double r6179999 = /* ERROR: no complex support in C */;
        double r6180000 = /* ERROR: no complex support in C */;
        return r6180000;
}
↓
double f(double x, double y) {
        double r6180001 = x;
        double r6180002 = exp(r6180001);
        double r6180003 = -r6180001;
        double r6180004 = exp(r6180003);
        double r6180005 = r6180002 + r6180004;
        double r6180006 = 2.0;
        double r6180007 = r6180005 / r6180006;
        double r6180008 = y;
        double r6180009 = cos(r6180008);
        double r6180010 = r6180007 * r6180009;
        double r6180011 = 0.016666666666666666;
        double r6180012 = 5.0;
        double r6180013 = pow(r6180001, r6180012);
        double r6180014 = 0.3333333333333333;
        double r6180015 = r6180001 * r6180001;
        double r6180016 = fma(r6180014, r6180015, r6180006);
        double r6180017 = r6180016 * r6180001;
        double r6180018 = fma(r6180011, r6180013, r6180017);
        double r6180019 = r6180018 / r6180006;
        double r6180020 = sin(r6180008);
        double r6180021 = r6180019 * r6180020;
        double r6180022 = /* ERROR: no complex support in C */;
        double r6180023 = /* ERROR: no complex support in C */;
        return r6180023;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

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

  2. Taylor expanded around 0 0.9

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2} \cdot \sin y i\right))\]

  3. Simplified0.9

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left(x \cdot (\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right))_*}}{2} \cdot \sin y i\right))\]

  4. Final simplification0.9

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_* \cdot x\right))_*}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))