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

↓

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

double f(double B, double x) {
        double r546117 = x;
        double r546118 = 1.0;
        double r546119 = B;
        double r546120 = tan(r546119);
        double r546121 = r546118 / r546120;
        double r546122 = r546117 * r546121;
        double r546123 = -r546122;
        double r546124 = sin(r546119);
        double r546125 = r546118 / r546124;
        double r546126 = r546123 + r546125;
        return r546126;
}

↓

double f(double B, double x) {
        double r546127 = 1.0;
        double r546128 = B;
        double r546129 = sin(r546128);
        double r546130 = r546127 / r546129;
        double r546131 = cos(r546128);
        double r546132 = r546131 / r546129;
        double r546133 = x;
        double r546134 = r546132 * r546133;
        double r546135 = r546130 - r546134;
        return r546135;
}

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

↓

\frac{1}{\sin B} - \frac{\cos B}{\sin B} \cdot x

Derivation

Initial program 0.2
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
Simplified0.2
\[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}}\]
Taylor expanded around inf 0.2
\[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}}\]
Using strategy rm
Applied *-un-lft-identity0.2
\[\leadsto \frac{1}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{1 \cdot \sin B}}\]
Applied times-frac0.2
\[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{1} \cdot \frac{\cos B}{\sin B}}\]
Simplified0.2
\[\leadsto \frac{1}{\sin B} - \color{blue}{x} \cdot \frac{\cos B}{\sin B}\]
Final simplification0.2
\[\leadsto \frac{1}{\sin B} - \frac{\cos B}{\sin B} \cdot x\]

Reproduce

herbie shell --seed 2019101 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))

Error

Derivation

Reproduce