Error

Derivation

1. Split input into 2 regimes

2. if F < -3.708582369031604e+154 or 2.0342863373936442e+151 < F

   1. Initial program 40.3

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

   2. Simplified40.3

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

   4. Applied add-sqr-sqrt40.3

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

   5. Applied unpow-prod-down40.3

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

   7. Applied pow-prod-up40.3

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

   8. Simplified40.3

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

   9. Taylor expanded around inf 35.4

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

   10. Simplified35.4

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

   if -3.708582369031604e+154 < F < 2.0342863373936442e+151

   1. Initial program 2.5

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

   2. Simplified2.4

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

   4. Applied add-sqr-sqrt2.4

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

   5. Applied unpow-prod-down2.5

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

4. Final simplification12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -3.708582369031604 \cdot 10^{+154} \lor \neg \left(F \le 2.0342863373936442 \cdot 10^{+151}\right):\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;(\left({\left(\sqrt{(2 \cdot x + \left((F \cdot F + 2)_*\right))_*}\right)}^{\frac{-1}{2}} \cdot {\left(\sqrt{(2 \cdot x + \left((F \cdot F + 2)_*\right))_*}\right)}^{\frac{-1}{2}}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(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))))))

Details

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