Average Error: 0.2 → 0.2
Time: 22.8s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right) + \frac{x \cdot \cos B}{\sin B} \cdot \left(\left(-1\right) + 1\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right) + \frac{x \cdot \cos B}{\sin B} \cdot \left(\left(-1\right) + 1\right)
double f(double B, double x) {
        double r25097 = x;
        double r25098 = 1.0;
        double r25099 = B;
        double r25100 = tan(r25099);
        double r25101 = r25098 / r25100;
        double r25102 = r25097 * r25101;
        double r25103 = -r25102;
        double r25104 = sin(r25099);
        double r25105 = r25098 / r25104;
        double r25106 = r25103 + r25105;
        return r25106;
}
↓
double f(double B, double x) {
        double r25107 = 1.0;
        double r25108 = -r25107;
        double r25109 = x;
        double r25110 = B;
        double r25111 = cos(r25110);
        double r25112 = r25109 * r25111;
        double r25113 = sin(r25110);
        double r25114 = r25112 / r25113;
        double r25115 = r25107 / r25113;
        double r25116 = fma(r25108, r25114, r25115);
        double r25117 = r25108 + r25107;
        double r25118 = r25114 * r25117;
        double r25119 = r25116 + r25118;
        return r25119;
}

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. Simplified0.2

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

  4. Applied associate-*r/0.2

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

  6. Applied tan-quot0.2

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

  7. Applied associate-/r/0.2

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

  8. Applied add-sqr-sqrt32.9

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

  9. Applied prod-diff32.9

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\sin B}}, \sqrt{\frac{1}{\sin B}}, -\cos B \cdot \frac{x \cdot 1}{\sin B}\right) + \mathsf{fma}\left(-\cos B, \frac{x \cdot 1}{\sin B}, \cos B \cdot \frac{x \cdot 1}{\sin B}\right)}\]

  10. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right)} + \mathsf{fma}\left(-\cos B, \frac{x \cdot 1}{\sin B}, \cos B \cdot \frac{x \cdot 1}{\sin B}\right)\]

  11. Simplified0.2

    \[\leadsto \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right) + \color{blue}{\frac{x \cdot \cos B}{\sin B} \cdot \left(\left(-1\right) + 1\right)}\]

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))