Average Error: 0.2 → 0.3
Time: 5.0s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\frac{\sin B}{1 \cdot \left(1 - x \cdot \cos B\right)}}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\frac{\sin B}{1 \cdot \left(1 - x \cdot \cos B\right)}}
double f(double B, double x) {
        double r9208 = x;
        double r9209 = 1.0;
        double r9210 = B;
        double r9211 = tan(r9210);
        double r9212 = r9209 / r9211;
        double r9213 = r9208 * r9212;
        double r9214 = -r9213;
        double r9215 = sin(r9210);
        double r9216 = r9209 / r9215;
        double r9217 = r9214 + r9216;
        return r9217;
}
↓
double f(double B, double x) {
        double r9218 = 1.0;
        double r9219 = B;
        double r9220 = sin(r9219);
        double r9221 = 1.0;
        double r9222 = x;
        double r9223 = cos(r9219);
        double r9224 = r9222 * r9223;
        double r9225 = r9218 - r9224;
        double r9226 = r9221 * r9225;
        double r9227 = r9220 / r9226;
        double r9228 = r9218 / r9227;
        return r9228;
}

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

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

  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

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

  6. Applied associate-*l/0.2

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

  8. Applied clear-num0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))