Average Error: 0.2 → 0.2
Time: 5.3s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[1 \cdot \frac{1 - x \cdot \cos B}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
1 \cdot \frac{1 - x \cdot \cos B}{\sin B}
double f(double B, double x) {
        double r36590 = x;
        double r36591 = 1.0;
        double r36592 = B;
        double r36593 = tan(r36592);
        double r36594 = r36591 / r36593;
        double r36595 = r36590 * r36594;
        double r36596 = -r36595;
        double r36597 = sin(r36592);
        double r36598 = r36591 / r36597;
        double r36599 = r36596 + r36598;
        return r36599;
}
↓
double f(double B, double x) {
        double r36600 = 1.0;
        double r36601 = 1.0;
        double r36602 = x;
        double r36603 = B;
        double r36604 = cos(r36603);
        double r36605 = r36602 * r36604;
        double r36606 = r36601 - r36605;
        double r36607 = sin(r36603);
        double r36608 = r36606 / r36607;
        double r36609 = r36600 * r36608;
        return r36609;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{1}{\tan B}, \frac{1}{\sin B}\right)}\]

  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

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

  6. Applied div-inv0.2

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

  7. Applied associate-*l*0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))