Average Error: 0.2 → 0.2
Time: 35.1s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\left(\frac{1}{\sin B} - \frac{\cos B}{\frac{\sin B}{x}}\right) + \left(\frac{\cos B}{\frac{\sin B}{x}} - \frac{\cos B}{\frac{\sin B}{x}}\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\left(\frac{1}{\sin B} - \frac{\cos B}{\frac{\sin B}{x}}\right) + \left(\frac{\cos B}{\frac{\sin B}{x}} - \frac{\cos B}{\frac{\sin B}{x}}\right)
double f(double B, double x) {
        double r1406649 = x;
        double r1406650 = 1.0;
        double r1406651 = B;
        double r1406652 = tan(r1406651);
        double r1406653 = r1406650 / r1406652;
        double r1406654 = r1406649 * r1406653;
        double r1406655 = -r1406654;
        double r1406656 = sin(r1406651);
        double r1406657 = r1406650 / r1406656;
        double r1406658 = r1406655 + r1406657;
        return r1406658;
}
↓
double f(double B, double x) {
        double r1406659 = 1.0;
        double r1406660 = B;
        double r1406661 = sin(r1406660);
        double r1406662 = r1406659 / r1406661;
        double r1406663 = cos(r1406660);
        double r1406664 = x;
        double r1406665 = r1406661 / r1406664;
        double r1406666 = r1406663 / r1406665;
        double r1406667 = r1406662 - r1406666;
        double r1406668 = r1406666 - r1406666;
        double r1406669 = r1406667 + r1406668;
        return r1406669;
}

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. Using strategy rm

  4. Applied tan-quot0.2

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

  5. Applied associate-/r/0.2

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

  6. Applied *-un-lft-identity0.2

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

  7. Applied add-sqr-sqrt0.2

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

  8. Applied times-frac0.2

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

  9. Applied prod-diff0.2

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

  10. Simplified0.2

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))