Average Error: 43.9 → 0.7
Time: 15.2s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\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}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r57003 = x;
        double r57004 = exp(r57003);
        double r57005 = -r57003;
        double r57006 = exp(r57005);
        double r57007 = r57004 + r57006;
        double r57008 = 2.0;
        double r57009 = r57007 / r57008;
        double r57010 = y;
        double r57011 = cos(r57010);
        double r57012 = r57009 * r57011;
        double r57013 = r57004 - r57006;
        double r57014 = r57013 / r57008;
        double r57015 = sin(r57010);
        double r57016 = r57014 * r57015;
        double r57017 = /* ERROR: no complex support in C */;
        double r57018 = /* ERROR: no complex support in C */;
        return r57018;
}
↓
double f(double x, double y) {
        double r57019 = x;
        double r57020 = exp(r57019);
        double r57021 = -r57019;
        double r57022 = exp(r57021);
        double r57023 = r57020 + r57022;
        double r57024 = 2.0;
        double r57025 = r57023 / r57024;
        double r57026 = y;
        double r57027 = cos(r57026);
        double r57028 = r57025 * r57027;
        double r57029 = 0.3333333333333333;
        double r57030 = 3.0;
        double r57031 = pow(r57019, r57030);
        double r57032 = r57029 * r57031;
        double r57033 = 0.016666666666666666;
        double r57034 = 5.0;
        double r57035 = pow(r57019, r57034);
        double r57036 = r57033 * r57035;
        double r57037 = 2.0;
        double r57038 = r57037 * r57019;
        double r57039 = r57036 + r57038;
        double r57040 = r57032 + r57039;
        double r57041 = r57040 / r57024;
        double r57042 = sin(r57026);
        double r57043 = r57041 * r57042;
        double r57044 = /* ERROR: no complex support in C */;
        double r57045 = /* ERROR: no complex support in C */;
        return r57045;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.9

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

  2. Taylor expanded around 0 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2020001 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))