Average Error: 0.2 → 0.2
Time: 16.5s
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 r39952 = x;
        double r39953 = 1.0;
        double r39954 = B;
        double r39955 = tan(r39954);
        double r39956 = r39953 / r39955;
        double r39957 = r39952 * r39956;
        double r39958 = -r39957;
        double r39959 = sin(r39954);
        double r39960 = r39953 / r39959;
        double r39961 = r39958 + r39960;
        return r39961;
}
↓
double f(double B, double x) {
        double r39962 = 1.0;
        double r39963 = 1.0;
        double r39964 = x;
        double r39965 = B;
        double r39966 = cos(r39965);
        double r39967 = r39964 * r39966;
        double r39968 = r39963 - r39967;
        double r39969 = sin(r39965);
        double r39970 = r39968 / r39969;
        double r39971 = r39962 * r39970;
        return r39971;
}

Error

Bits error versus B

Bits error versus x

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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. Using strategy rm

  4. Applied associate-*r/0.1

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

  5. Taylor expanded around inf 0.2

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

  6. Simplified0.2

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

  8. Applied sub-div0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))