Average Error: 0.0 → 0.0
Time: 7.4s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\left(\cos y\right), \left(e^{x}\right), \left(\frac{\cos y}{e^{x}}\right)\right)}{2}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\mathsf{fma}\left(\left(\cos y\right), \left(e^{x}\right), \left(\frac{\cos y}{e^{x}}\right)\right)}{2}
double f(double x, double y) {
        double r65210 = x;
        double r65211 = exp(r65210);
        double r65212 = -r65210;
        double r65213 = exp(r65212);
        double r65214 = r65211 + r65213;
        double r65215 = 2.0;
        double r65216 = r65214 / r65215;
        double r65217 = y;
        double r65218 = cos(r65217);
        double r65219 = r65216 * r65218;
        double r65220 = r65211 - r65213;
        double r65221 = r65220 / r65215;
        double r65222 = sin(r65217);
        double r65223 = r65221 * r65222;
        double r65224 = /* ERROR: no complex support in C */;
        double r65225 = /* ERROR: no complex support in C */;
        return r65225;
}
↓
double f(double x, double y) {
        double r65226 = y;
        double r65227 = cos(r65226);
        double r65228 = x;
        double r65229 = exp(r65228);
        double r65230 = r65227 / r65229;
        double r65231 = fma(r65227, r65229, r65230);
        double r65232 = 2.0;
        double r65233 = r65231 / r65232;
        return r65233;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos y\right), \left(e^{x}\right), \left(\frac{\cos y}{e^{x}}\right)\right)}{2}}\]

  3. Final simplification0.0

    \[\leadsto \frac{\mathsf{fma}\left(\left(\cos y\right), \left(e^{x}\right), \left(\frac{\cos y}{e^{x}}\right)\right)}{2}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))