Average Error: 43.9 → 0.8
Time: 30.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r50086 = x;
        double r50087 = exp(r50086);
        double r50088 = -r50086;
        double r50089 = exp(r50088);
        double r50090 = r50087 + r50089;
        double r50091 = 2.0;
        double r50092 = r50090 / r50091;
        double r50093 = y;
        double r50094 = cos(r50093);
        double r50095 = r50092 * r50094;
        double r50096 = r50087 - r50089;
        double r50097 = r50096 / r50091;
        double r50098 = sin(r50093);
        double r50099 = r50097 * r50098;
        double r50100 = /* ERROR: no complex support in C */;
        double r50101 = /* ERROR: no complex support in C */;
        return r50101;
}
↓
double f(double x, double y) {
        double r50102 = 0.3333333333333333;
        double r50103 = x;
        double r50104 = 3.0;
        double r50105 = pow(r50103, r50104);
        double r50106 = r50102 * r50105;
        double r50107 = 0.016666666666666666;
        double r50108 = 5.0;
        double r50109 = pow(r50103, r50108);
        double r50110 = r50107 * r50109;
        double r50111 = 2.0;
        double r50112 = r50111 * r50103;
        double r50113 = r50110 + r50112;
        double r50114 = r50106 + r50113;
        double r50115 = 2.0;
        double r50116 = r50114 / r50115;
        double r50117 = y;
        double r50118 = sin(r50117);
        double r50119 = r50116 * r50118;
        return r50119;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.9

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

  2. Simplified43.9

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.8

    \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2} \cdot \sin y\]

  4. Final simplification0.8

    \[\leadsto \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]

Reproduce

herbie shell --seed 2019305 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))