Average Error: 0.2 → 0.2
Time: 24.9s
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 r818548 = x;
        double r818549 = 1.0;
        double r818550 = B;
        double r818551 = tan(r818550);
        double r818552 = r818549 / r818551;
        double r818553 = r818548 * r818552;
        double r818554 = -r818553;
        double r818555 = sin(r818550);
        double r818556 = r818549 / r818555;
        double r818557 = r818554 + r818556;
        return r818557;
}
↓
double f(double B, double x) {
        double r818558 = 1.0;
        double r818559 = B;
        double r818560 = cos(r818559);
        double r818561 = x;
        double r818562 = r818560 * r818561;
        double r818563 = r818558 - r818562;
        double r818564 = sin(r818559);
        double r818565 = r818563 / r818564;
        return r818565;
}

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 2019153 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))