Average Error: 0.2 → 0.2
Time: 8.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[1 \cdot \left(\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
1 \cdot \left(\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\right)
double f(double B, double x) {
        double r59034 = x;
        double r59035 = 1.0;
        double r59036 = B;
        double r59037 = tan(r59036);
        double r59038 = r59035 / r59037;
        double r59039 = r59034 * r59038;
        double r59040 = -r59039;
        double r59041 = sin(r59036);
        double r59042 = r59035 / r59041;
        double r59043 = r59040 + r59042;
        return r59043;
}
↓
double f(double B, double x) {
        double r59044 = 1.0;
        double r59045 = 1.0;
        double r59046 = B;
        double r59047 = sin(r59046);
        double r59048 = r59045 / r59047;
        double r59049 = x;
        double r59050 = cos(r59046);
        double r59051 = r59049 * r59050;
        double r59052 = r59051 / r59047;
        double r59053 = r59048 - r59052;
        double r59054 = r59044 * r59053;
        return r59054;
}

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. Taylor expanded around inf 0.2

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

  3. Simplified0.2

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

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

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))