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Derivation

1. Split input into 3 regimes

2. if F < -3.592454986189758e+86

   1. Initial program 30.8

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

   2. Initial simplification30.7

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

   3. Taylor expanded around -inf 0.1

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

   if -3.592454986189758e+86 < F < 3.5113251116085693e-16

   1. Initial program 0.8

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

   2. Initial simplification0.7

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

   4. Applied pow-neg0.7

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

   5. Applied associate-*l/0.7

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

   6. Simplified0.7

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

   if 3.5113251116085693e-16 < F

   1. Initial program 23.5

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

   2. Initial simplification23.4

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

   4. Applied neg-sub023.4

      \[\leadsto {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\color{blue}{\left(0 - \frac{1}{2}\right)}} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\]

   5. Applied pow-sub23.4

      \[\leadsto \color{blue}{\frac{{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{0}}{{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\]

   6. Applied frac-times18.1

      \[\leadsto \color{blue}{\frac{{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{0} \cdot F}{{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B}} - \frac{x}{\tan B}\]

   7. Simplified18.1

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

   8. Taylor expanded around inf 1.8

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

4. Final simplification0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -3.592454986189758 \cdot 10^{+86}:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 3.5113251116085693 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right) - \frac{x}{\tan B}\\ \end{array}\]

Runtime

Time bar (total: 31.9s)Debug logProfile

herbie shell --seed 2018227 
(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))))))