Average Error: 0.2 → 0.2
Time: 4.9s
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 r8007 = x;
        double r8008 = 1.0;
        double r8009 = B;
        double r8010 = tan(r8009);
        double r8011 = r8008 / r8010;
        double r8012 = r8007 * r8011;
        double r8013 = -r8012;
        double r8014 = sin(r8009);
        double r8015 = r8008 / r8014;
        double r8016 = r8013 + r8015;
        return r8016;
}
↓
double f(double B, double x) {
        double r8017 = 1.0;
        double r8018 = 1.0;
        double r8019 = x;
        double r8020 = B;
        double r8021 = cos(r8020);
        double r8022 = r8019 * r8021;
        double r8023 = r8018 - r8022;
        double r8024 = r8017 * r8023;
        double r8025 = sin(r8020);
        double r8026 = r8024 / r8025;
        return r8026;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))