Average Error: 0.2 → 0.2
Time: 8.3m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}
double f(double B, double x) {
        double r16661167 = x;
        double r16661168 = 1.0;
        double r16661169 = B;
        double r16661170 = tan(r16661169);
        double r16661171 = r16661168 / r16661170;
        double r16661172 = r16661167 * r16661171;
        double r16661173 = -r16661172;
        double r16661174 = sin(r16661169);
        double r16661175 = r16661168 / r16661174;
        double r16661176 = r16661173 + r16661175;
        return r16661176;
}
↓
double f(double B, double x) {
        double r16661177 = 1.0;
        double r16661178 = B;
        double r16661179 = sin(r16661178);
        double r16661180 = r16661177 / r16661179;
        double r16661181 = x;
        double r16661182 = cos(r16661178);
        double r16661183 = r16661181 * r16661182;
        double r16661184 = r16661183 / r16661179;
        double r16661185 = r16661180 - r16661184;
        return r16661185;
}

Error

Bits error versus B

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}}\]

  3. Taylor expanded around inf 0.2

    \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}}\]

  4. Taylor expanded around -inf 0.2

    \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x \cdot \cos B}{\sin B}}\]

  5. Final simplification0.2

    \[\leadsto \frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))