Average Error: 43.4 → 0.8
Time: 38.7s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{60} \cdot {x}^{5} + x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right)}{2} \cdot \sin y i\right))\]
\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} + x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2820205 = x;
        double r2820206 = exp(r2820205);
        double r2820207 = -r2820205;
        double r2820208 = exp(r2820207);
        double r2820209 = r2820206 + r2820208;
        double r2820210 = 2.0;
        double r2820211 = r2820209 / r2820210;
        double r2820212 = y;
        double r2820213 = cos(r2820212);
        double r2820214 = r2820211 * r2820213;
        double r2820215 = r2820206 - r2820208;
        double r2820216 = r2820215 / r2820210;
        double r2820217 = sin(r2820212);
        double r2820218 = r2820216 * r2820217;
        double r2820219 = /* ERROR: no complex support in C */;
        double r2820220 = /* ERROR: no complex support in C */;
        return r2820220;
}
↓
double f(double x, double y) {
        double r2820221 = x;
        double r2820222 = exp(r2820221);
        double r2820223 = -r2820221;
        double r2820224 = exp(r2820223);
        double r2820225 = r2820222 + r2820224;
        double r2820226 = 2.0;
        double r2820227 = r2820225 / r2820226;
        double r2820228 = y;
        double r2820229 = cos(r2820228);
        double r2820230 = r2820227 * r2820229;
        double r2820231 = 0.016666666666666666;
        double r2820232 = 5.0;
        double r2820233 = pow(r2820221, r2820232);
        double r2820234 = r2820231 * r2820233;
        double r2820235 = 0.3333333333333333;
        double r2820236 = r2820221 * r2820221;
        double r2820237 = r2820235 * r2820236;
        double r2820238 = 2.0;
        double r2820239 = r2820237 + r2820238;
        double r2820240 = r2820221 * r2820239;
        double r2820241 = r2820234 + r2820240;
        double r2820242 = r2820241 / r2820226;
        double r2820243 = sin(r2820228);
        double r2820244 = r2820242 * r2820243;
        double r2820245 = /* ERROR: no complex support in C */;
        double r2820246 = /* ERROR: no complex support in C */;
        return r2820246;
}

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 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))