Average Error: 0.2 → 0.2
Time: 4.8s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 \cdot \left(1 - x \cdot \cos B\right)}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 \cdot \left(1 - x \cdot \cos B\right)}{\sin B}
double f(double B, double x) {
        double r41445 = x;
        double r41446 = 1.0;
        double r41447 = B;
        double r41448 = tan(r41447);
        double r41449 = r41446 / r41448;
        double r41450 = r41445 * r41449;
        double r41451 = -r41450;
        double r41452 = sin(r41447);
        double r41453 = r41446 / r41452;
        double r41454 = r41451 + r41453;
        return r41454;
}
↓
double f(double B, double x) {
        double r41455 = 1.0;
        double r41456 = 1.0;
        double r41457 = x;
        double r41458 = B;
        double r41459 = cos(r41458);
        double r41460 = r41457 * r41459;
        double r41461 = r41456 - r41460;
        double r41462 = r41455 * r41461;
        double r41463 = sin(r41458);
        double r41464 = r41462 / r41463;
        return r41464;
}

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 associate-*l/0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))