Average Error: 0.2 → 0.2
Time: 4.6s
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 r39312 = x;
        double r39313 = 1.0;
        double r39314 = B;
        double r39315 = tan(r39314);
        double r39316 = r39313 / r39315;
        double r39317 = r39312 * r39316;
        double r39318 = -r39317;
        double r39319 = sin(r39314);
        double r39320 = r39313 / r39319;
        double r39321 = r39318 + r39320;
        return r39321;
}
↓
double f(double B, double x) {
        double r39322 = 1.0;
        double r39323 = 1.0;
        double r39324 = x;
        double r39325 = B;
        double r39326 = cos(r39325);
        double r39327 = r39324 * r39326;
        double r39328 = r39323 - r39327;
        double r39329 = r39322 * r39328;
        double r39330 = sin(r39325);
        double r39331 = r39329 / r39330;
        return r39331;
}

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 2020001 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))