Average Error: 0.2 → 0.2
Time: 24.2s
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 r912096 = x;
        double r912097 = 1.0;
        double r912098 = B;
        double r912099 = tan(r912098);
        double r912100 = r912097 / r912099;
        double r912101 = r912096 * r912100;
        double r912102 = -r912101;
        double r912103 = sin(r912098);
        double r912104 = r912097 / r912103;
        double r912105 = r912102 + r912104;
        return r912105;
}
↓
double f(double B, double x) {
        double r912106 = 1.0;
        double r912107 = B;
        double r912108 = cos(r912107);
        double r912109 = x;
        double r912110 = r912108 * r912109;
        double r912111 = r912110 * r912106;
        double r912112 = r912106 - r912111;
        double r912113 = sin(r912107);
        double r912114 = r912112 / r912113;
        return r912114;
}

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

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

  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))))