Average Error: 0.2 → 0.2
Time: 20.1s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[1 \cdot \left(\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
1 \cdot \left(\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\right)
double f(double B, double x) {
        double r41657 = x;
        double r41658 = 1.0;
        double r41659 = B;
        double r41660 = tan(r41659);
        double r41661 = r41658 / r41660;
        double r41662 = r41657 * r41661;
        double r41663 = -r41662;
        double r41664 = sin(r41659);
        double r41665 = r41658 / r41664;
        double r41666 = r41663 + r41665;
        return r41666;
}
↓
double f(double B, double x) {
        double r41667 = 1.0;
        double r41668 = 1.0;
        double r41669 = B;
        double r41670 = sin(r41669);
        double r41671 = r41668 / r41670;
        double r41672 = x;
        double r41673 = r41672 / r41670;
        double r41674 = cos(r41669);
        double r41675 = r41673 * r41674;
        double r41676 = r41671 - r41675;
        double r41677 = r41667 * r41676;
        return r41677;
}

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

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

  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

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

  6. Applied associate-/l*0.2

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

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

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

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

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

  10. Applied distribute-lft-out--0.2

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))