Average Error: 0.2 → 0.2
Time: 5.1s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 \cdot \left(1 - x \cdot \cos B\right)}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 \cdot \left(1 - x \cdot \cos B\right)}{\sin B}
double f(double B, double x) {
        double r43010 = x;
        double r43011 = 1.0;
        double r43012 = B;
        double r43013 = tan(r43012);
        double r43014 = r43011 / r43013;
        double r43015 = r43010 * r43014;
        double r43016 = -r43015;
        double r43017 = sin(r43012);
        double r43018 = r43011 / r43017;
        double r43019 = r43016 + r43018;
        return r43019;
}
↓
double f(double B, double x) {
        double r43020 = 1.0;
        double r43021 = 1.0;
        double r43022 = x;
        double r43023 = B;
        double r43024 = cos(r43023);
        double r43025 = r43022 * r43024;
        double r43026 = r43021 - r43025;
        double r43027 = r43020 * r43026;
        double r43028 = sin(r43023);
        double r43029 = r43027 / r43028;
        return r43029;
}

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. Simplified0.3

    \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \left(1 - x \cdot \cos B\right)}\]
  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))