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

↓

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

Derivation

Initial program 13.0
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
Applied simplify12.9
\[\leadsto \color{blue}{{\left(\left(2 + 2 \cdot x\right) + F \cdot F\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}}\]
Using strategy rm
Applied div-inv12.9
\[\leadsto {\left(\left(2 + 2 \cdot x\right) + F \cdot F\right)}^{\frac{-1}{2}} \cdot \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} - \frac{x}{\tan B}\]
Applied associate-*r*10.2
\[\leadsto \color{blue}{\left({\left(\left(2 + 2 \cdot x\right) + F \cdot F\right)}^{\frac{-1}{2}} \cdot F\right) \cdot \frac{1}{\sin B}} - \frac{x}{\tan B}\]
Using strategy rm
Applied add-sqr-sqrt10.3
\[\leadsto \left({\color{blue}{\left(\sqrt{\left(2 + 2 \cdot x\right) + F \cdot F} \cdot \sqrt{\left(2 + 2 \cdot x\right) + F \cdot F}\right)}}^{\frac{-1}{2}} \cdot F\right) \cdot \frac{1}{\sin B} - \frac{x}{\tan B}\]
Applied unpow-prod-down10.3
\[\leadsto \left(\color{blue}{\left({\left(\sqrt{\left(2 + 2 \cdot x\right) + F \cdot F}\right)}^{\frac{-1}{2}} \cdot {\left(\sqrt{\left(2 + 2 \cdot x\right) + F \cdot F}\right)}^{\frac{-1}{2}}\right)} \cdot F\right) \cdot \frac{1}{\sin B} - \frac{x}{\tan B}\]

Runtime

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(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))))))

Error

Derivation

Runtime