Average Error: 0.2 → 0.2
Time: 22.6s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - \left(\cos B \cdot x\right) \cdot 1}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 - \left(\cos B \cdot x\right) \cdot 1}{\sin B}
double f(double B, double x) {
        double r2228752 = x;
        double r2228753 = 1.0;
        double r2228754 = B;
        double r2228755 = tan(r2228754);
        double r2228756 = r2228753 / r2228755;
        double r2228757 = r2228752 * r2228756;
        double r2228758 = -r2228757;
        double r2228759 = sin(r2228754);
        double r2228760 = r2228753 / r2228759;
        double r2228761 = r2228758 + r2228760;
        return r2228761;
}
↓
double f(double B, double x) {
        double r2228762 = 1.0;
        double r2228763 = B;
        double r2228764 = cos(r2228763);
        double r2228765 = x;
        double r2228766 = r2228764 * r2228765;
        double r2228767 = r2228766 * r2228762;
        double r2228768 = r2228762 - r2228767;
        double r2228769 = sin(r2228763);
        double r2228770 = r2228768 / r2228769;
        return r2228770;
}

Error

Bits error versus B

Bits error versus x

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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

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

  3. Taylor expanded around inf 0.2

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

  4. Simplified0.2

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

  6. Applied associate-*l/0.2

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

  7. Applied sub-div0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))