Average Error: 0.2 → 0.3
Time: 30.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\left(1 - \cos B \cdot x\right) \cdot \frac{1}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\left(1 - \cos B \cdot x\right) \cdot \frac{1}{\sin B}
double f(double B, double x) {
        double r889546 = x;
        double r889547 = 1.0;
        double r889548 = B;
        double r889549 = tan(r889548);
        double r889550 = r889547 / r889549;
        double r889551 = r889546 * r889550;
        double r889552 = -r889551;
        double r889553 = sin(r889548);
        double r889554 = r889547 / r889553;
        double r889555 = r889552 + r889554;
        return r889555;
}
↓
double f(double B, double x) {
        double r889556 = 1.0;
        double r889557 = B;
        double r889558 = cos(r889557);
        double r889559 = x;
        double r889560 = r889558 * r889559;
        double r889561 = r889556 - r889560;
        double r889562 = 1.0;
        double r889563 = sin(r889557);
        double r889564 = r889562 / r889563;
        double r889565 = r889561 * r889564;
        return r889565;
}

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. Taylor expanded around inf 0.2

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

  4. Simplified0.2

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

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

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

  7. Applied times-frac0.3

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

  8. Applied *-un-lft-identity0.3

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

  9. Applied *-un-lft-identity0.3

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

  10. Applied times-frac0.3

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

  11. Applied distribute-rgt-out--0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))