Average Error: 0.2 → 0.2
Time: 5.0s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} + \frac{1}{\sin B} \cdot \left(-x \cdot \cos B\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\sin B} + \frac{1}{\sin B} \cdot \left(-x \cdot \cos B\right)
double f(double B, double x) {
        double r8175 = x;
        double r8176 = 1.0;
        double r8177 = B;
        double r8178 = tan(r8177);
        double r8179 = r8176 / r8178;
        double r8180 = r8175 * r8179;
        double r8181 = -r8180;
        double r8182 = sin(r8177);
        double r8183 = r8176 / r8182;
        double r8184 = r8181 + r8183;
        return r8184;
}
↓
double f(double B, double x) {
        double r8185 = 1.0;
        double r8186 = B;
        double r8187 = sin(r8186);
        double r8188 = r8185 / r8187;
        double r8189 = x;
        double r8190 = cos(r8186);
        double r8191 = r8189 * r8190;
        double r8192 = -r8191;
        double r8193 = r8188 * r8192;
        double r8194 = r8188 + r8193;
        return r8194;
}

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 sub-neg0.2

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

  7. Applied distribute-lft-in0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))