VandenBroeck and Keller, Equation (23)

Average Error: 13.8 → 1.4

Time: 12.8s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \le -3.6079474982776699 \cdot 10^{144}:\\ \;\;\;\;\frac{F \cdot {\left(e^{-0.5}\right)}^{\left(\log \left(\frac{-1}{F}\right) \cdot -2\right)} - 1 \cdot \frac{{\left(e^{-0.5}\right)}^{\left(\log \left(\frac{-1}{F}\right) \cdot -2\right)}}{F}}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \le 1.3731501411501404 \cdot 10^{154}:\\ \;\;\;\;\frac{F}{\sin B \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - 1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{-0.5}\right)}^{\left(-2 \cdot \left(-\log F\right)\right)} \cdot F - \frac{1}{\frac{F}{{\left(e^{-0.5}\right)}^{\left(-2 \cdot \left(-\log F\right)\right)}}}}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -3.6079474982776699e144

   1. Initial program 38.1

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

   2. Simplified 38.1

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

   4. Applied associate-*l/ 31.9

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

   5. Taylor expanded around -inf 3.7

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

   6. Simplified 3.7

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

   if -3.6079474982776699e144 < F < 1.3731501411501404e154

   1. Initial program 2.8

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

   2. Simplified 2.8

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

   4. Applied associate-*l/ 0.3

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

   6. Applied associate-*r/ 0.2

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

   8. Applied pow-neg 0.3

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

   9. Applied un-div-inv 0.2

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

   10. Applied associate-/l/ 0.3

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

   11. Taylor expanded around inf 0.3

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

   if 1.3731501411501404e154 < F

   1. Initial program 40.2

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

   2. Simplified 40.2

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

   4. Applied associate-*l/ 35.3

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

   5. Taylor expanded around inf 3.8

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

   6. Simplified 3.8

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -3.6079474982776699 \cdot 10^{144}:\\ \;\;\;\;\frac{F \cdot {\left(e^{-0.5}\right)}^{\left(\log \left(\frac{-1}{F}\right) \cdot -2\right)} - 1 \cdot \frac{{\left(e^{-0.5}\right)}^{\left(\log \left(\frac{-1}{F}\right) \cdot -2\right)}}{F}}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \le 1.3731501411501404 \cdot 10^{154}:\\ \;\;\;\;\frac{F}{\sin B \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - 1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{-0.5}\right)}^{\left(-2 \cdot \left(-\log F\right)\right)} \cdot F - \frac{1}{\frac{F}{{\left(e^{-0.5}\right)}^{\left(-2 \cdot \left(-\log F\right)\right)}}}}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))))))