Average Error: 0.0 → 0.0
Time: 4.5s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \cos y\right)\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \cos y\right)
double f(double x, double y) {
        double r40071 = x;
        double r40072 = exp(r40071);
        double r40073 = -r40071;
        double r40074 = exp(r40073);
        double r40075 = r40072 + r40074;
        double r40076 = 2.0;
        double r40077 = r40075 / r40076;
        double r40078 = y;
        double r40079 = cos(r40078);
        double r40080 = r40077 * r40079;
        double r40081 = r40072 - r40074;
        double r40082 = r40081 / r40076;
        double r40083 = sin(r40078);
        double r40084 = r40082 * r40083;
        double r40085 = /* ERROR: no complex support in C */;
        double r40086 = /* ERROR: no complex support in C */;
        return r40086;
}
↓
double f(double x, double y) {
        double r40087 = x;
        double r40088 = exp(r40087);
        double r40089 = -r40087;
        double r40090 = exp(r40089);
        double r40091 = r40088 + r40090;
        double r40092 = 2.0;
        double r40093 = r40091 / r40092;
        double r40094 = sqrt(r40093);
        double r40095 = y;
        double r40096 = cos(r40095);
        double r40097 = r40094 * r40096;
        double r40098 = r40094 * r40097;
        return r40098;
}

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

    \[\leadsto \color{blue}{\frac{e^{x} + e^{-x}}{2} \cdot \cos y}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.0

    \[\leadsto \color{blue}{\left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \sqrt{\frac{e^{x} + e^{-x}}{2}}\right)} \cdot \cos y\]

  5. Applied associate-*l*0.0

    \[\leadsto \color{blue}{\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \cos y\right)}\]

  6. Final simplification0.0

    \[\leadsto \sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \left(\sqrt{\frac{e^{x} + e^{-x}}{2}} \cdot \cos y\right)\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  :precision binary64
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))