Average Error: 43.1 → 0.8
Time: 35.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r48042 = x;
        double r48043 = exp(r48042);
        double r48044 = -r48042;
        double r48045 = exp(r48044);
        double r48046 = r48043 + r48045;
        double r48047 = 2.0;
        double r48048 = r48046 / r48047;
        double r48049 = y;
        double r48050 = cos(r48049);
        double r48051 = r48048 * r48050;
        double r48052 = r48043 - r48045;
        double r48053 = r48052 / r48047;
        double r48054 = sin(r48049);
        double r48055 = r48053 * r48054;
        double r48056 = /* ERROR: no complex support in C */;
        double r48057 = /* ERROR: no complex support in C */;
        return r48057;
}
↓
double f(double x, double y) {
        double r48058 = 0.3333333333333333;
        double r48059 = x;
        double r48060 = 3.0;
        double r48061 = pow(r48059, r48060);
        double r48062 = 0.016666666666666666;
        double r48063 = 5.0;
        double r48064 = pow(r48059, r48063);
        double r48065 = 2.0;
        double r48066 = r48065 * r48059;
        double r48067 = fma(r48062, r48064, r48066);
        double r48068 = fma(r48058, r48061, r48067);
        double r48069 = 2.0;
        double r48070 = r48068 / r48069;
        double r48071 = y;
        double r48072 = sin(r48071);
        double r48073 = r48070 * r48072;
        return r48073;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.1

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

  2. Simplified43.1

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

  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. Simplified0.8

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

  5. Final simplification0.8

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(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)))))