Average Error: 0.2 → 0.2
Time: 20.4s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - \cos B \cdot x}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 - \cos B \cdot x}{\sin B}
double f(double B, double x) {
        double r337651 = x;
        double r337652 = 1.0;
        double r337653 = B;
        double r337654 = tan(r337653);
        double r337655 = r337652 / r337654;
        double r337656 = r337651 * r337655;
        double r337657 = -r337656;
        double r337658 = sin(r337653);
        double r337659 = r337652 / r337658;
        double r337660 = r337657 + r337659;
        return r337660;
}
↓
double f(double B, double x) {
        double r337661 = 1.0;
        double r337662 = B;
        double r337663 = cos(r337662);
        double r337664 = x;
        double r337665 = r337663 * r337664;
        double r337666 = r337661 - r337665;
        double r337667 = sin(r337662);
        double r337668 = r337666 / r337667;
        return r337668;
}

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

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

  3. Taylor expanded around inf 0.2

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

  5. Applied sub-div0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))