VandenBroeck and Keller, Equation (23)

Average Error: 13.9 → 1.5

Time: 15.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \le -1.3553598376870535 \cdot 10^{154}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{{\left(e^{0.5}\right)}^{\left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)} + 1 \cdot \frac{{\left(e^{0.5}\right)}^{\left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}}{F \cdot F}} \cdot \frac{1}{\sin B}\\ \mathbf{elif}\;F \le 1.96833105963148191 \cdot 10^{142}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B} \cdot \frac{F}{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\left(\frac{1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B} \cdot \frac{F}{{\left(e^{0.5}\right)}^{\left(\log 1 - \log F \cdot -2\right)} + 1 \cdot \frac{{\left(e^{0.5}\right)}^{\left(\log 1 - \log F \cdot -2\right)}}{F \cdot F}}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -1.3553598376870535e154

   1. Initial program 41.5

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

   2. Simplified 36.4

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

   4. Applied div-inv 36.4

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

   5. Applied associate-*r* 36.4

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

   6. Simplified 36.4

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

   8. Applied distribute-frac-neg 36.4

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

   9. Applied pow-neg 36.4

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

   10. Applied un-div-inv 36.4

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

   11. Taylor expanded around -inf 3.9

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

   12. Simplified 4.2

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

   if -1.3553598376870535e154 < F < 1.96833105963148191e142

   1. Initial program 2.2

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

   2. Simplified 0.4

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

   4. Applied div-inv 0.4

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

   5. Applied associate-*r* 0.4

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

   6. Simplified 0.4

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

   8. Applied distribute-frac-neg 0.4

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

   9. Applied pow-neg 0.4

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

   10. Applied un-div-inv 0.3

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

   if 1.96833105963148191e142 < F

   1. Initial program 40.0

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

   2. Simplified 34.5

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

   4. Applied div-inv 34.5

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

   5. Applied associate-*r* 34.5

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

   6. Simplified 34.5

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

   8. Applied distribute-frac-neg 34.5

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

   9. Applied pow-neg 34.5

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

   10. Applied un-div-inv 34.5

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

   11. Taylor expanded around inf 3.9

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

   12. Simplified 4.2

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -1.3553598376870535 \cdot 10^{154}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{{\left(e^{0.5}\right)}^{\left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)} + 1 \cdot \frac{{\left(e^{0.5}\right)}^{\left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}}{F \cdot F}} \cdot \frac{1}{\sin B}\\ \mathbf{elif}\;F \le 1.96833105963148191 \cdot 10^{142}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B} \cdot \frac{F}{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\left(\frac{1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B} \cdot \frac{F}{{\left(e^{0.5}\right)}^{\left(\log 1 - \log F \cdot -2\right)} + 1 \cdot \frac{{\left(e^{0.5}\right)}^{\left(\log 1 - \log F \cdot -2\right)}}{F \cdot F}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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))))))