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Derivation

1. Split input into 3 regimes

2. if F < -1.7288022267860176e+23

   1. Initial program 25.6

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

   2. Simplified25.6

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

   4. Applied associate-*r/20.2

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

   6. Applied clear-num20.3

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

   8. Applied associate-/r/20.2

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

   9. Taylor expanded around -inf 0.2

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

   if -1.7288022267860176e+23 < F < 9811.311554800664

   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. Simplified0.3

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

   4. Applied add-sqr-sqrt0.4

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

   5. Applied unpow-prod-down0.4

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

   if 9811.311554800664 < F

   1. Initial program 25.5

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

   2. Simplified25.4

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

   4. Applied associate-*r/19.7

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

   5. Taylor expanded around inf 0.1

      \[\leadsto \frac{\color{blue}{1 - \frac{1}{{F}^{2}}}}{\sin B} - \frac{x}{\tan B}\]
3. Recombined 3 regimes into one program.

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -1.7288022267860176 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} - 1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \le 9811.311554800664:\\ \;\;\;\;\left({\left(\sqrt{2 \cdot x + \left(2 + F \cdot F\right)}\right)}^{\frac{-1}{2}} \cdot {\left(\sqrt{2 \cdot x + \left(2 + F \cdot F\right)}\right)}^{\frac{-1}{2}}\right) \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{{F}^{2}}}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 
(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))))))

Details

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