Average Error: 0.2 → 0.2
Time: 14.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - 1 \cdot \left(x \cdot \cos B\right)}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 - 1 \cdot \left(x \cdot \cos B\right)}{\sin B}
double f(double B, double x) {
        double r45114 = x;
        double r45115 = 1.0;
        double r45116 = B;
        double r45117 = tan(r45116);
        double r45118 = r45115 / r45117;
        double r45119 = r45114 * r45118;
        double r45120 = -r45119;
        double r45121 = sin(r45116);
        double r45122 = r45115 / r45121;
        double r45123 = r45120 + r45122;
        return r45123;
}
↓
double f(double B, double x) {
        double r45124 = 1.0;
        double r45125 = x;
        double r45126 = B;
        double r45127 = cos(r45126);
        double r45128 = r45125 * r45127;
        double r45129 = r45124 * r45128;
        double r45130 = r45124 - r45129;
        double r45131 = sin(r45126);
        double r45132 = r45130 / r45131;
        return r45132;
}

Error

Bits error versus B

Bits error versus x

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Your Program's Arguments

Results

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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/0.2

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

  5. Taylor expanded around inf 0.2

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

  7. Applied associate-*r/0.2

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

  8. Applied sub-div0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019305 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))