Average Error: 0.2 → 0.2
Time: 35.5s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} + \left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\sin B} + \left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right)
double f(double B, double x) {
        double r39633 = x;
        double r39634 = 1.0;
        double r39635 = B;
        double r39636 = tan(r39635);
        double r39637 = r39634 / r39636;
        double r39638 = r39633 * r39637;
        double r39639 = -r39638;
        double r39640 = sin(r39635);
        double r39641 = r39634 / r39640;
        double r39642 = r39639 + r39641;
        return r39642;
}
↓
double f(double B, double x) {
        double r39643 = 1.0;
        double r39644 = B;
        double r39645 = sin(r39644);
        double r39646 = r39643 / r39645;
        double r39647 = x;
        double r39648 = cos(r39644);
        double r39649 = r39647 * r39648;
        double r39650 = r39649 / r39645;
        double r39651 = r39643 * r39650;
        double r39652 = -r39651;
        double r39653 = r39646 + r39652;
        return r39653;
}

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. Using strategy rm

  3. Applied associate-*r/0.1

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

  4. Taylor expanded around inf 0.2

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

  6. Applied +-commutative0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))