Average Error: 43.2 → 0.7
Time: 1.1m
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 r2969249 = x;
        double r2969250 = exp(r2969249);
        double r2969251 = -r2969249;
        double r2969252 = exp(r2969251);
        double r2969253 = r2969250 + r2969252;
        double r2969254 = 2.0;
        double r2969255 = r2969253 / r2969254;
        double r2969256 = y;
        double r2969257 = cos(r2969256);
        double r2969258 = r2969255 * r2969257;
        double r2969259 = r2969250 - r2969252;
        double r2969260 = r2969259 / r2969254;
        double r2969261 = sin(r2969256);
        double r2969262 = r2969260 * r2969261;
        double r2969263 = /* ERROR: no complex support in C */;
        double r2969264 = /* ERROR: no complex support in C */;
        return r2969264;
}
↓
double f(double x, double y) {
        double r2969265 = x;
        double r2969266 = exp(r2969265);
        double r2969267 = -r2969265;
        double r2969268 = exp(r2969267);
        double r2969269 = r2969266 + r2969268;
        double r2969270 = 2.0;
        double r2969271 = r2969269 / r2969270;
        double r2969272 = y;
        double r2969273 = cos(r2969272);
        double r2969274 = r2969271 * r2969273;
        double r2969275 = 0.016666666666666666;
        double r2969276 = 5.0;
        double r2969277 = pow(r2969265, r2969276);
        double r2969278 = 0.3333333333333333;
        double r2969279 = r2969265 * r2969265;
        double r2969280 = fma(r2969278, r2969279, r2969270);
        double r2969281 = r2969280 * r2969265;
        double r2969282 = fma(r2969275, r2969277, r2969281);
        double r2969283 = r2969282 / r2969270;
        double r2969284 = sin(r2969272);
        double r2969285 = r2969283 * r2969284;
        double r2969286 = /* ERROR: no complex support in C */;
        double r2969287 = /* ERROR: no complex support in C */;
        return r2969287;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.2

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019119 +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)))))