VandenBroeck and Keller, Equation (23)

Average Error: 13.2 → 0.2

Time: 31.1s

Precision: 64

Math

\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;F \le -18032.337247481762:\\ \;\;\;\;\frac{\frac{1}{F \cdot F} - 1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 53320.07745021123:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{\cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{F \cdot F}}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

C

double f(double F, double B, double x) {
        double r613614 = x;
        double r613615 = 1.0;
        double r613616 = B;
        double r613617 = tan(r613616);
        double r613618 = r613615 / r613617;
        double r613619 = r613614 * r613618;
        double r613620 = -r613619;
        double r613621 = F;
        double r613622 = sin(r613616);
        double r613623 = r613621 / r613622;
        double r613624 = r613621 * r613621;
        double r613625 = 2.0;
        double r613626 = r613624 + r613625;
        double r613627 = r613625 * r613614;
        double r613628 = r613626 + r613627;
        double r613629 = r613615 / r613625;
        double r613630 = -r613629;
        double r613631 = pow(r613628, r613630);
        double r613632 = r613623 * r613631;
        double r613633 = r613620 + r613632;
        return r613633;
}

↓

double f(double F, double B, double x) {
        double r613634 = F;
        double r613635 = -18032.337247481762;
        bool r613636 = r613634 <= r613635;
        double r613637 = 1.0;
        double r613638 = r613634 * r613634;
        double r613639 = r613637 / r613638;
        double r613640 = r613639 - r613637;
        double r613641 = B;
        double r613642 = sin(r613641);
        double r613643 = r613640 / r613642;
        double r613644 = x;
        double r613645 = tan(r613641);
        double r613646 = r613644 / r613645;
        double r613647 = r613643 - r613646;
        double r613648 = 53320.07745021123;
        bool r613649 = r613634 <= r613648;
        double r613650 = 2.0;
        double r613651 = r613638 + r613650;
        double r613652 = r613650 * r613644;
        double r613653 = r613651 + r613652;
        double r613654 = -0.5;
        double r613655 = pow(r613653, r613654);
        double r613656 = r613634 / r613642;
        double r613657 = r613655 * r613656;
        double r613658 = cos(r613641);
        double r613659 = r613658 * r613644;
        double r613660 = r613659 / r613642;
        double r613661 = r613657 - r613660;
        double r613662 = r613637 - r613639;
        double r613663 = r613662 / r613642;
        double r613664 = r613663 - r613646;
        double r613665 = r613649 ? r613661 : r613664;
        double r613666 = r613636 ? r613647 : r613665;
        return r613666;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -18032.337247481762

   1. Initial program 24.5

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

   2. Simplified 24.5

      \[\leadsto \color{blue}{{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}}\]
   3. Using strategy rm

   4. Applied associate-*r/ 19.5

      \[\leadsto \color{blue}{\frac{{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot F}{\sin B}} - \frac{x}{\tan B}\]

   5. Taylor expanded around -inf 0.2

      \[\leadsto \frac{\color{blue}{\frac{1}{{F}^{2}} - 1}}{\sin B} - \frac{x}{\tan B}\]

   6. Simplified 0.2

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

   if -18032.337247481762 < F < 53320.07745021123

   1. Initial program 0.4

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

   2. Simplified 0.3

      \[\leadsto \color{blue}{{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}}\]

   3. Taylor expanded around inf 0.3

      \[\leadsto {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \color{blue}{\frac{x \cdot \cos B}{\sin B}}\]

   if 53320.07745021123 < F

   1. Initial program 24.1

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

   2. Simplified 24.1

      \[\leadsto \color{blue}{{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}}\]
   3. Using strategy rm

   4. Applied associate-*r/ 18.7

      \[\leadsto \color{blue}{\frac{{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot F}{\sin B}} - \frac{x}{\tan B}\]

   5. Taylor expanded around inf 0.2

      \[\leadsto \frac{\color{blue}{1 - \frac{1}{{F}^{2}}}}{\sin B} - \frac{x}{\tan B}\]

   6. Simplified 0.2

      \[\leadsto \frac{\color{blue}{1 - \frac{1}{F \cdot F}}}{\sin B} - \frac{x}{\tan B}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -18032.337247481762:\\ \;\;\;\;\frac{\frac{1}{F \cdot F} - 1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 53320.07745021123:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{\cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{F \cdot F}}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))