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Derivation

1. Split input into 3 regimes

2. if F < -106776.04908489401

   1. Initial program 25.3

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

   2. Initial simplification25.3

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

   4. Applied add-sqr-sqrt25.3

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

   5. Applied unpow-prod-down25.3

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

   7. Applied associate-*r/19.5

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

   8. Taylor expanded around -inf 0.2

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

   if -106776.04908489401 < F < 58703.26061639259

   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. Initial simplification0.3

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

   4. Applied add-sqr-sqrt0.3

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

   5. Applied unpow-prod-down0.3

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

   7. Applied associate-*r/0.3

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

   9. Applied pow-prod-down0.3

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

   10. Simplified0.3

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

   if 58703.26061639259 < F

   1. Initial program 25.4

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

   2. Initial simplification25.3

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

   4. Applied add-sqr-sqrt25.3

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

   5. Applied unpow-prod-down25.4

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

   7. Applied associate-*r/20.7

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

   8. Taylor expanded around inf 0.1

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -106776.04908489401:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 58703.26061639259:\\ \;\;\;\;\frac{{\left(F \cdot F + \left(2 \cdot x + 2\right)\right)}^{\frac{-1}{2}} \cdot F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right) - \frac{x}{\tan B}\\ \end{array}\]

Runtime

Time bar (total: 31.1s)Debug logProfile

herbie shell --seed 2018295 
(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))))))