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

↓

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

double f(double B, double x) {
        double r2695295 = x;
        double r2695296 = 1.0;
        double r2695297 = B;
        double r2695298 = tan(r2695297);
        double r2695299 = r2695296 / r2695298;
        double r2695300 = r2695295 * r2695299;
        double r2695301 = -r2695300;
        double r2695302 = sin(r2695297);
        double r2695303 = r2695296 / r2695302;
        double r2695304 = r2695301 + r2695303;
        return r2695304;
}

↓

double f(double B, double x) {
        double r2695305 = 1.0;
        double r2695306 = B;
        double r2695307 = sin(r2695306);
        double r2695308 = r2695305 / r2695307;
        double r2695309 = cos(r2695306);
        double r2695310 = r2695309 / r2695307;
        double r2695311 = x;
        double r2695312 = r2695310 * r2695311;
        double r2695313 = r2695308 - r2695312;
        return r2695313;
}

\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