Average Error: 0.2 → 0.2
Time: 24.8s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - \frac{\cos B}{\sin B} \cdot x\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\sin B} - \frac{\cos B}{\sin B} \cdot x
double f(double B, double x) {
        double r689377 = x;
        double r689378 = 1.0;
        double r689379 = B;
        double r689380 = tan(r689379);
        double r689381 = r689378 / r689380;
        double r689382 = r689377 * r689381;
        double r689383 = -r689382;
        double r689384 = sin(r689379);
        double r689385 = r689378 / r689384;
        double r689386 = r689383 + r689385;
        return r689386;
}
↓
double f(double B, double x) {
        double r689387 = 1.0;
        double r689388 = B;
        double r689389 = sin(r689388);
        double r689390 = r689387 / r689389;
        double r689391 = cos(r689388);
        double r689392 = r689391 / r689389;
        double r689393 = x;
        double r689394 = r689392 * r689393;
        double r689395 = r689390 - r689394;
        return r689395;
}

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 *-un-lft-identity0.2

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

  6. Applied times-frac0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))