VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 22.8s

Precision: 64

Math

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

↓

\[\frac{\frac{\tan B}{\sin B} - x}{\tan B}\]

C

double f(double B, double x) {
        double r1252297 = x;
        double r1252298 = 1.0;
        double r1252299 = B;
        double r1252300 = tan(r1252299);
        double r1252301 = r1252298 / r1252300;
        double r1252302 = r1252297 * r1252301;
        double r1252303 = -r1252302;
        double r1252304 = sin(r1252299);
        double r1252305 = r1252298 / r1252304;
        double r1252306 = r1252303 + r1252305;
        return r1252306;
}

↓

double f(double B, double x) {
        double r1252307 = B;
        double r1252308 = tan(r1252307);
        double r1252309 = sin(r1252307);
        double r1252310 = r1252308 / r1252309;
        double r1252311 = x;
        double r1252312 = r1252310 - r1252311;
        double r1252313 = r1252312 / r1252308;
        return r1252313;
}

Error

Bits error versus B

Bits error versus x

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

   \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}}\]
3. Using strategy rm

4. Applied frac-sub 10.4

   \[\leadsto \color{blue}{\frac{1 \cdot \tan B - \sin B \cdot x}{\sin B \cdot \tan B}}\]

5. Simplified 10.4

   \[\leadsto \frac{\color{blue}{\tan B - x \cdot \sin B}}{\sin B \cdot \tan B}\]
6. Using strategy rm

7. Applied associate-/r* 0.2

   \[\leadsto \color{blue}{\frac{\frac{\tan B - x \cdot \sin B}{\sin B}}{\tan B}}\]

8. Simplified 0.2

   \[\leadsto \frac{\color{blue}{\frac{\tan B}{\sin B} - x}}{\tan B}\]

9. Final simplification 0.2

   \[\leadsto \frac{\frac{\tan B}{\sin B} - x}{\tan B}\]

Reproduce

herbie shell --seed 2019133 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))