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Derivation

1. Split input into 3 regimes

2. if F < -6.020899605642303e+97

   1. Initial program 33.1

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

   2. Applied simplify33.1

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

   4. Applied tan-quot33.1

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

   5. Applied associate-/r/33.1

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

   6. Taylor expanded around -inf 16.2

      \[\leadsto \color{blue}{\left(\frac{1}{{F}^{3}} - \frac{1}{F}\right)} \cdot \frac{F}{\sin B} - \frac{x}{\sin B} \cdot \cos B\]

   7. Applied simplify0.2

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

   if -6.020899605642303e+97 < F < 80861792.86785662

   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. Applied simplify0.7

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

   4. Applied distribute-frac-neg0.7

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

   5. Applied pow-neg0.7

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

   6. Applied frac-times0.3

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

   7. Applied simplify0.3

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

   if 80861792.86785662 < F

   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. Applied simplify24.4

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

   3. Taylor expanded around inf 11.7

      \[\leadsto \color{blue}{\left(\frac{1}{F} - \frac{1}{{F}^{3}}\right)} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\]

   4. Applied simplify0.2

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2020178 
(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))))))