Error

Derivation

1. Split input into 3 regimes

2. if F < -12772389.54087001

   1. Initial program 24.2

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

   2. Simplified19.1

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

   4. Applied sqr-pow19.2

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

   5. Applied associate-*l*19.2

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

   6. Taylor expanded around -inf 0.1

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

   7. Simplified0.1

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

   if -12772389.54087001 < F < 5856533.636759231

   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

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

   4. Applied associate-/l*0.3

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

   if 5856533.636759231 < F

   1. Initial program 23.8

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

   2. Simplified18.4

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

   4. Applied sqr-pow18.5

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

   5. Applied associate-*l*18.5

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

   6. Taylor expanded around inf 0.1

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

   7. Simplified0.1

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -12772389.54087001:\\ \;\;\;\;\frac{\frac{1}{F \cdot F} - 1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 5856533.636759231:\\ \;\;\;\;\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{\sin B}}{F \cdot F}\right) - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(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))))))