Average Error: 0.2 → 0.2
Time: 6.9m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - \frac{\cos B}{\frac{\sin B}{x}}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\sin B} - \frac{\cos B}{\frac{\sin B}{x}}
double f(double B, double x) {
        double r15066552 = x;
        double r15066553 = 1.0;
        double r15066554 = B;
        double r15066555 = tan(r15066554);
        double r15066556 = r15066553 / r15066555;
        double r15066557 = r15066552 * r15066556;
        double r15066558 = -r15066557;
        double r15066559 = sin(r15066554);
        double r15066560 = r15066553 / r15066559;
        double r15066561 = r15066558 + r15066560;
        return r15066561;
}
↓
double f(double B, double x) {
        double r15066562 = 1.0;
        double r15066563 = B;
        double r15066564 = sin(r15066563);
        double r15066565 = r15066562 / r15066564;
        double r15066566 = cos(r15066563);
        double r15066567 = x;
        double r15066568 = r15066564 / r15066567;
        double r15066569 = r15066566 / r15066568;
        double r15066570 = r15066565 - r15066569;
        return r15066570;
}

Error

Bits error versus B

Bits error versus x

Try it out

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. Using strategy rm

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

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

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

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

  9. Applied times-frac0.2

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

  10. Applied prod-diff0.2

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

  11. Simplified0.2

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

  12. Simplified0.2

    \[\leadsto \left(\frac{1}{\sin B} - \frac{\cos B}{\frac{\sin B}{x}}\right) + \color{blue}{0}\]

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))