Error

Derivation

1. Split input into 3 regimes

2. if F < -7.0895910786336465e+19

   1. Initial program 25.6

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

   2. Applied simplify25.6

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

   4. Applied pow-neg25.6

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

   5. Applied frac-times20.2

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

   6. Applied simplify20.2

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

   7. Taylor expanded around -inf 0.1

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

   if -7.0895910786336465e+19 < F < 32590182.281621672

   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

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

   4. Applied add-sqr-sqrt0.3

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

   5. Applied unpow-prod-down0.3

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

   6. Applied associate-*l*0.3

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

   if 32590182.281621672 < F

   1. Initial program 24.0

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

   2. Applied simplify24.0

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

   4. Applied pow-neg24.0

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

   5. Applied frac-times18.9

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

   6. Applied simplify18.9

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

   7. Taylor expanded around inf 0.1

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

Runtime

Time bar (total: 49.6s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' 
(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))))))