Average Error: 0.2 → 0.2
Time: 7.0m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - \frac{\cos B}{\frac{\sin B}{x}}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\sin B} - \frac{\cos B}{\frac{\sin B}{x}}
double f(double B, double x) {
        double r16073328 = x;
        double r16073329 = 1.0;
        double r16073330 = B;
        double r16073331 = tan(r16073330);
        double r16073332 = r16073329 / r16073331;
        double r16073333 = r16073328 * r16073332;
        double r16073334 = -r16073333;
        double r16073335 = sin(r16073330);
        double r16073336 = r16073329 / r16073335;
        double r16073337 = r16073334 + r16073336;
        return r16073337;
}
↓
double f(double B, double x) {
        double r16073338 = 1.0;
        double r16073339 = B;
        double r16073340 = sin(r16073339);
        double r16073341 = r16073338 / r16073340;
        double r16073342 = cos(r16073339);
        double r16073343 = x;
        double r16073344 = r16073340 / r16073343;
        double r16073345 = r16073342 / r16073344;
        double r16073346 = r16073341 - r16073345;
        return r16073346;
}

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

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

  4. Applied tan-quot0.2

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

  5. Applied associate-/r/0.2

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

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

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

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

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

  9. Applied times-frac0.2

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

  10. Applied prod-diff0.2

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

  11. Simplified0.2

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

  12. Simplified0.2

    \[\leadsto \left(\frac{1}{\sin B} - \frac{\cos B}{\frac{\sin B}{x}}\right) + \color{blue}{0}\]

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))