Average Error: 0.2 → 0.2
Time: 24.3s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - 1 \cdot \frac{x \cdot \cos B}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\sin B} - 1 \cdot \frac{x \cdot \cos B}{\sin B}
double f(double B, double x) {
        double r30370 = x;
        double r30371 = 1.0;
        double r30372 = B;
        double r30373 = tan(r30372);
        double r30374 = r30371 / r30373;
        double r30375 = r30370 * r30374;
        double r30376 = -r30375;
        double r30377 = sin(r30372);
        double r30378 = r30371 / r30377;
        double r30379 = r30376 + r30378;
        return r30379;
}
↓
double f(double B, double x) {
        double r30380 = 1.0;
        double r30381 = B;
        double r30382 = sin(r30381);
        double r30383 = r30380 / r30382;
        double r30384 = x;
        double r30385 = cos(r30381);
        double r30386 = r30384 * r30385;
        double r30387 = r30386 / r30382;
        double r30388 = r30380 * r30387;
        double r30389 = r30383 - r30388;
        return r30389;
}

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. Simplified0.2

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

  3. Taylor expanded around inf 0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))