Average Error: 0.0 → 0.0
Time: 16.5s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{\log \left(e^{x} + e^{-x}\right)}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Re(\left(\frac{e^{\log \left(e^{x} + e^{-x}\right)}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r43128 = x;
        double r43129 = exp(r43128);
        double r43130 = -r43128;
        double r43131 = exp(r43130);
        double r43132 = r43129 + r43131;
        double r43133 = 2.0;
        double r43134 = r43132 / r43133;
        double r43135 = y;
        double r43136 = cos(r43135);
        double r43137 = r43134 * r43136;
        double r43138 = r43129 - r43131;
        double r43139 = r43138 / r43133;
        double r43140 = sin(r43135);
        double r43141 = r43139 * r43140;
        double r43142 = /* ERROR: no complex support in C */;
        double r43143 = /* ERROR: no complex support in C */;
        return r43143;
}
↓
double f(double x, double y) {
        double r43144 = x;
        double r43145 = exp(r43144);
        double r43146 = -r43144;
        double r43147 = exp(r43146);
        double r43148 = r43145 + r43147;
        double r43149 = log(r43148);
        double r43150 = exp(r43149);
        double r43151 = 2.0;
        double r43152 = r43150 / r43151;
        double r43153 = y;
        double r43154 = cos(r43153);
        double r43155 = r43152 * r43154;
        double r43156 = r43145 - r43147;
        double r43157 = r43156 / r43151;
        double r43158 = sin(r43153);
        double r43159 = r43157 * r43158;
        double r43160 = /* ERROR: no complex support in C */;
        double r43161 = /* ERROR: no complex support in C */;
        return r43161;
}

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. Using strategy rm

  3. Applied add-exp-log0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))