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Derivation

1. Split input into 3 regimes

2. if F < -24900559.270813115

   1. Initial program 24.8

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

   2. Applied simplify24.7

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

   4. Applied pow-neg24.7

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

   5. Applied frac-times18.8

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

   6. Applied simplify18.8

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

   7. 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 -24900559.270813115 < F < 9861.535097597833

   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. Applied simplify0.3

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

   4. Applied tan-quot0.3

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

   5. Applied associate-/r/0.3

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

   if 9861.535097597833 < F

   1. Initial program 23.3

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

   2. Applied simplify23.3

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

   4. Applied pow-neg23.3

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

   5. Applied frac-times18.0

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

   6. Applied simplify18.0

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

   7. Taylor expanded around inf 0.2

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

Runtime

Time bar (total: 48.0s)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' 
(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))))))