Average Error: 43.5 → 0.8
Time: 36.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{\frac{1}{60} \cdot {x}^{5} + x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right)}{2} i\right))\]
\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 + \sin y \cdot \frac{\frac{1}{60} \cdot {x}^{5} + x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right)}{2} i\right))
double f(double x, double y) {
        double r2675178 = x;
        double r2675179 = exp(r2675178);
        double r2675180 = -r2675178;
        double r2675181 = exp(r2675180);
        double r2675182 = r2675179 + r2675181;
        double r2675183 = 2.0;
        double r2675184 = r2675182 / r2675183;
        double r2675185 = y;
        double r2675186 = cos(r2675185);
        double r2675187 = r2675184 * r2675186;
        double r2675188 = r2675179 - r2675181;
        double r2675189 = r2675188 / r2675183;
        double r2675190 = sin(r2675185);
        double r2675191 = r2675189 * r2675190;
        double r2675192 = /* ERROR: no complex support in C */;
        double r2675193 = /* ERROR: no complex support in C */;
        return r2675193;
}
↓
double f(double x, double y) {
        double r2675194 = x;
        double r2675195 = exp(r2675194);
        double r2675196 = -r2675194;
        double r2675197 = exp(r2675196);
        double r2675198 = r2675195 + r2675197;
        double r2675199 = 2.0;
        double r2675200 = r2675198 / r2675199;
        double r2675201 = y;
        double r2675202 = cos(r2675201);
        double r2675203 = r2675200 * r2675202;
        double r2675204 = sin(r2675201);
        double r2675205 = 0.016666666666666666;
        double r2675206 = 5.0;
        double r2675207 = pow(r2675194, r2675206);
        double r2675208 = r2675205 * r2675207;
        double r2675209 = 0.3333333333333333;
        double r2675210 = r2675194 * r2675194;
        double r2675211 = r2675209 * r2675210;
        double r2675212 = 2.0;
        double r2675213 = r2675211 + r2675212;
        double r2675214 = r2675194 * r2675213;
        double r2675215 = r2675208 + r2675214;
        double r2675216 = r2675215 / r2675199;
        double r2675217 = r2675204 * r2675216;
        double r2675218 = /* ERROR: no complex support in C */;
        double r2675219 = /* ERROR: no complex support in C */;
        return r2675219;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.5

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

  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

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

  5. Applied *-commutative0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019171 
(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)))))