Average Error: 0.2 → 0.2
Time: 5.0s
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 r8428 = x;
        double r8429 = 1.0;
        double r8430 = B;
        double r8431 = tan(r8430);
        double r8432 = r8429 / r8431;
        double r8433 = r8428 * r8432;
        double r8434 = -r8433;
        double r8435 = sin(r8430);
        double r8436 = r8429 / r8435;
        double r8437 = r8434 + r8436;
        return r8437;
}
↓
double f(double B, double x) {
        double r8438 = 1.0;
        double r8439 = 1.0;
        double r8440 = x;
        double r8441 = B;
        double r8442 = cos(r8441);
        double r8443 = r8440 * r8442;
        double r8444 = r8439 - r8443;
        double r8445 = r8438 * r8444;
        double r8446 = sin(r8441);
        double r8447 = r8445 / r8446;
        return r8447;
}

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 2020027 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))