Average Error: 0.2 → 0.2
Time: 5.2s
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 r30090 = x;
        double r30091 = 1.0;
        double r30092 = B;
        double r30093 = tan(r30092);
        double r30094 = r30091 / r30093;
        double r30095 = r30090 * r30094;
        double r30096 = -r30095;
        double r30097 = sin(r30092);
        double r30098 = r30091 / r30097;
        double r30099 = r30096 + r30098;
        return r30099;
}
↓
double f(double B, double x) {
        double r30100 = 1.0;
        double r30101 = 1.0;
        double r30102 = x;
        double r30103 = B;
        double r30104 = cos(r30103);
        double r30105 = r30102 * r30104;
        double r30106 = r30101 - r30105;
        double r30107 = r30100 * r30106;
        double r30108 = sin(r30103);
        double r30109 = r30107 / r30108;
        return r30109;
}

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 2020034 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))