Average Error: 0.2 → 0.2
Time: 12.0s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[-\frac{1 \cdot \left(x \cdot \cos B - 1\right)}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
-\frac{1 \cdot \left(x \cdot \cos B - 1\right)}{\sin B}
double f(double B, double x) {
        double r34541 = x;
        double r34542 = 1.0;
        double r34543 = B;
        double r34544 = tan(r34543);
        double r34545 = r34542 / r34544;
        double r34546 = r34541 * r34545;
        double r34547 = -r34546;
        double r34548 = sin(r34543);
        double r34549 = r34542 / r34548;
        double r34550 = r34547 + r34549;
        return r34550;
}
↓
double f(double B, double x) {
        double r34551 = 1.0;
        double r34552 = x;
        double r34553 = B;
        double r34554 = cos(r34553);
        double r34555 = r34552 * r34554;
        double r34556 = 1.0;
        double r34557 = r34555 - r34556;
        double r34558 = r34551 * r34557;
        double r34559 = sin(r34553);
        double r34560 = r34558 / r34559;
        double r34561 = -r34560;
        return r34561;
}

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. Using strategy rm

  3. Applied tan-quot0.2

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

  4. Applied associate-/r/0.2

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

  5. Applied associate-*r*0.2

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

  6. Simplified0.2

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

  7. Taylor expanded around inf 0.2

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

  8. Simplified0.2

    \[\leadsto \left(-\color{blue}{\frac{1 \cdot x}{\frac{\sin B}{\cos B}}}\right) + \frac{1}{\sin B}\]
  9. Using strategy rm

  10. Applied neg-sub00.2

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

  11. Applied associate-+l-0.2

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

  12. Simplified0.2

    \[\leadsto 0 - \color{blue}{\frac{1}{\sin B} \cdot \left(x \cdot \cos B - 1\right)}\]
  13. Using strategy rm

  14. Applied associate-*l/0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2019350 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))