Average Error: 43.4 → 0.7
Time: 41.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{60} \cdot {x}^{5} + \left(2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)}{2} \cdot \sin y i\right))\]
\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 {x}^{5} + \left(2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1642982 = x;
        double r1642983 = exp(r1642982);
        double r1642984 = -r1642982;
        double r1642985 = exp(r1642984);
        double r1642986 = r1642983 + r1642985;
        double r1642987 = 2.0;
        double r1642988 = r1642986 / r1642987;
        double r1642989 = y;
        double r1642990 = cos(r1642989);
        double r1642991 = r1642988 * r1642990;
        double r1642992 = r1642983 - r1642985;
        double r1642993 = r1642992 / r1642987;
        double r1642994 = sin(r1642989);
        double r1642995 = r1642993 * r1642994;
        double r1642996 = /* ERROR: no complex support in C */;
        double r1642997 = /* ERROR: no complex support in C */;
        return r1642997;
}
↓
double f(double x, double y) {
        double r1642998 = x;
        double r1642999 = exp(r1642998);
        double r1643000 = -r1642998;
        double r1643001 = exp(r1643000);
        double r1643002 = r1642999 + r1643001;
        double r1643003 = 2.0;
        double r1643004 = r1643002 / r1643003;
        double r1643005 = y;
        double r1643006 = cos(r1643005);
        double r1643007 = r1643004 * r1643006;
        double r1643008 = 0.016666666666666666;
        double r1643009 = 5.0;
        double r1643010 = pow(r1642998, r1643009);
        double r1643011 = r1643008 * r1643010;
        double r1643012 = r1643003 * r1642998;
        double r1643013 = 0.3333333333333333;
        double r1643014 = r1642998 * r1642998;
        double r1643015 = r1643013 * r1643014;
        double r1643016 = r1643015 * r1642998;
        double r1643017 = r1643012 + r1643016;
        double r1643018 = r1643011 + r1643017;
        double r1643019 = r1643018 / r1643003;
        double r1643020 = sin(r1643005);
        double r1643021 = r1643019 * r1643020;
        double r1643022 = /* ERROR: no complex support in C */;
        double r1643023 = /* ERROR: no complex support in C */;
        return r1643023;
}

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 0.7

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

  3. Simplified0.7

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

  5. Applied distribute-lft-in0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019139 
(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)))))