VandenBroeck and Keller, Equation (23)

Average Error: 13.8 → 1.4

Time: 10.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \le -1.329634943152444 \cdot 10^{154}:\\ \;\;\;\;\frac{F}{\left({\left(e^{0.5}\right)}^{\left(\log 1 + \log \left(\frac{-1}{F}\right) \cdot -2\right)} + 1 \cdot \frac{{\left(e^{0.5}\right)}^{\left(\log 1 + \log \left(\frac{-1}{F}\right) \cdot -2\right)}}{F \cdot F}\right) \cdot \sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \le 2.0158552208199639 \cdot 10^{146}:\\ \;\;\;\;\frac{F}{\sin B \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\left(\frac{1}{2}\right)}} - 1 \cdot \left(\frac{x}{\sin B} \cdot \cos B\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \left({\left(e^{-0.5}\right)}^{\left(\log 1 + 2 \cdot \log F\right)} - 1 \cdot \frac{{\left(e^{-0.5}\right)}^{\left(\log 1 + 2 \cdot \log F\right)}}{F \cdot F}\right)}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -1.329634943152444e154

   1. Initial program 42.5

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

   2. Simplified 36.6

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

   4. Applied associate-*r/ 36.6

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

   5. Simplified 36.6

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

   7. Applied distribute-frac-neg 36.6

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

   8. Applied pow-neg 36.6

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

   9. Applied un-div-inv 36.6

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

   10. Applied associate-/l/ 36.6

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

   11. Simplified 36.6

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

   12. Taylor expanded around -inf 3.9

       \[\leadsto \frac{F}{\color{blue}{\left(1 \cdot \frac{e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}}{{F}^{2}} + e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}\right)} \cdot \sin B} - x \cdot \frac{1}{\tan B}\]

   13. Simplified 4.2

       \[\leadsto \frac{F}{\color{blue}{\left({\left(e^{0.5}\right)}^{\left(\log 1 + \log \left(\frac{-1}{F}\right) \cdot -2\right)} + 1 \cdot \frac{{\left(e^{0.5}\right)}^{\left(\log 1 + \log \left(\frac{-1}{F}\right) \cdot -2\right)}}{F \cdot F}\right)} \cdot \sin B} - x \cdot \frac{1}{\tan B}\]

   if -1.329634943152444e154 < F < 2.0158552208199639e146

   1. Initial program 2.4

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

   2. Simplified 0.4

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

   4. Applied associate-*r/ 0.3

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

   5. Simplified 0.3

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

   7. Applied distribute-frac-neg 0.3

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

   8. Applied pow-neg 0.4

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

   9. Applied un-div-inv 0.3

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

   10. Applied associate-/l/ 0.4

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

   11. Simplified 0.4

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

   12. Taylor expanded around inf 0.3

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

   13. Simplified 0.3

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

   if 2.0158552208199639e146 < F

   1. Initial program 41.1

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

   2. Simplified 35.0

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

   4. Applied associate-*r/ 35.0

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

   5. Simplified 35.0

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

   6. Taylor expanded around inf 4.0

      \[\leadsto \frac{F \cdot \color{blue}{\left(e^{-0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{1}{F}\right)\right)} - 1 \cdot \frac{e^{-0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{1}{F}\right)\right)}}{{F}^{2}}\right)}}{\sin B} - x \cdot \frac{1}{\tan B}\]

   7. Simplified 4.0

      \[\leadsto \frac{F \cdot \color{blue}{\left({\left(e^{-0.5}\right)}^{\left(\log 1 + 2 \cdot \log F\right)} - 1 \cdot \frac{{\left(e^{-0.5}\right)}^{\left(\log 1 + 2 \cdot \log F\right)}}{F \cdot F}\right)}}{\sin B} - x \cdot \frac{1}{\tan B}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -1.329634943152444 \cdot 10^{154}:\\ \;\;\;\;\frac{F}{\left({\left(e^{0.5}\right)}^{\left(\log 1 + \log \left(\frac{-1}{F}\right) \cdot -2\right)} + 1 \cdot \frac{{\left(e^{0.5}\right)}^{\left(\log 1 + \log \left(\frac{-1}{F}\right) \cdot -2\right)}}{F \cdot F}\right) \cdot \sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \le 2.0158552208199639 \cdot 10^{146}:\\ \;\;\;\;\frac{F}{\sin B \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\left(\frac{1}{2}\right)}} - 1 \cdot \left(\frac{x}{\sin B} \cdot \cos B\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \left({\left(e^{-0.5}\right)}^{\left(\log 1 + 2 \cdot \log F\right)} - 1 \cdot \frac{{\left(e^{-0.5}\right)}^{\left(\log 1 + 2 \cdot \log F\right)}}{F \cdot F}\right)}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (neg (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (neg (/ 1.0 2.0))))))