Average Error: 0.2 → 0.3
Time: 20.8s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - \left(\left(x \cdot \cos B\right) \cdot \frac{1}{\sin B}\right) \cdot 1\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\sin B} - \left(\left(x \cdot \cos B\right) \cdot \frac{1}{\sin B}\right) \cdot 1
double f(double B, double x) {
        double r2343306 = x;
        double r2343307 = 1.0;
        double r2343308 = B;
        double r2343309 = tan(r2343308);
        double r2343310 = r2343307 / r2343309;
        double r2343311 = r2343306 * r2343310;
        double r2343312 = -r2343311;
        double r2343313 = sin(r2343308);
        double r2343314 = r2343307 / r2343313;
        double r2343315 = r2343312 + r2343314;
        return r2343315;
}
↓
double f(double B, double x) {
        double r2343316 = 1.0;
        double r2343317 = B;
        double r2343318 = sin(r2343317);
        double r2343319 = r2343316 / r2343318;
        double r2343320 = x;
        double r2343321 = cos(r2343317);
        double r2343322 = r2343320 * r2343321;
        double r2343323 = 1.0;
        double r2343324 = r2343323 / r2343318;
        double r2343325 = r2343322 * r2343324;
        double r2343326 = r2343325 * r2343316;
        double r2343327 = r2343319 - r2343326;
        return r2343327;
}

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

    \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot 1}{\tan B}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.2

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

  5. Applied times-frac0.2

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

  6. Simplified0.2

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

  7. Taylor expanded around inf 0.2

    \[\leadsto \frac{1}{\sin B} - \color{blue}{1 \cdot \frac{x \cdot \cos B}{\sin B}}\]
  8. Using strategy rm

  9. Applied div-inv0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))