Average Error: 0.0 → 0.0
Time: 15.3s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\cos y, e^{x}, \frac{\cos y}{e^{x}}\right)\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\cos y, e^{x}, \frac{\cos y}{e^{x}}\right)
double f(double x, double y) {
        double r834200 = x;
        double r834201 = exp(r834200);
        double r834202 = -r834200;
        double r834203 = exp(r834202);
        double r834204 = r834201 + r834203;
        double r834205 = 2.0;
        double r834206 = r834204 / r834205;
        double r834207 = y;
        double r834208 = cos(r834207);
        double r834209 = r834206 * r834208;
        double r834210 = r834201 - r834203;
        double r834211 = r834210 / r834205;
        double r834212 = sin(r834207);
        double r834213 = r834211 * r834212;
        double r834214 = /* ERROR: no complex support in C */;
        double r834215 = /* ERROR: no complex support in C */;
        return r834215;
}
↓
double f(double x, double y) {
        double r834216 = 0.5;
        double r834217 = y;
        double r834218 = cos(r834217);
        double r834219 = x;
        double r834220 = exp(r834219);
        double r834221 = r834218 / r834220;
        double r834222 = fma(r834218, r834220, r834221);
        double r834223 = r834216 * r834222;
        return r834223;
}

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

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \mathsf{fma}\left(\cos y, e^{x}, \frac{\cos y}{e^{x}}\right)}\]

  3. Final simplification0.0

    \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\cos y, e^{x}, \frac{\cos y}{e^{x}}\right)\]

Reproduce

herbie shell --seed 2019149 +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)))))