sqrtexp (problem 3.4.4)

Average Error: 41.1 → 0.4

Time: 7.1s

Precision: binary64

Math

\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9.9722389174514045 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(-1 \cdot 1\right) + {\left(e^{2}\right)}^{\left(2 \cdot x\right)}}{e^{2 \cdot x} + 1}}{\log \left(e^{e^{x} - 1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + x \cdot 0.5\right) + 2}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -9.9722389174514045e-5

   1. Initial program 0.0

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
   2. Using strategy rm

   3. Applied add-log-exp 0.0

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - \color{blue}{\log \left(e^{1}\right)}}}\]

   4. Applied add-log-exp 0.1

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\log \left(e^{e^{x}}\right)} - \log \left(e^{1}\right)}}\]

   5. Applied diff-log 0.1

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\log \left(\frac{e^{e^{x}}}{e^{1}}\right)}}}\]

   6. Simplified 0.1

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\log \color{blue}{\left(e^{e^{x} - 1}\right)}}}\]
   7. Using strategy rm

   8. Applied flip-- 0.1

      \[\leadsto \sqrt{\frac{\color{blue}{\frac{e^{2 \cdot x} \cdot e^{2 \cdot x} - 1 \cdot 1}{e^{2 \cdot x} + 1}}}{\log \left(e^{e^{x} - 1}\right)}}\]

   9. Simplified 0.1

      \[\leadsto \sqrt{\frac{\frac{\color{blue}{\left(-1 \cdot 1\right) + {\left(e^{2}\right)}^{\left(2 \cdot x\right)}}}{e^{2 \cdot x} + 1}}{\log \left(e^{e^{x} - 1}\right)}}\]

   if -9.9722389174514045e-5 < x

   1. Initial program 61.6

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

   2. Taylor expanded around 0 0.6

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
   3. Using strategy rm

   4. Applied associate-+r+ 0.6

      \[\leadsto \sqrt{\color{blue}{\left(0.5 \cdot {x}^{2} + 1 \cdot x\right) + 2}}\]

   5. Simplified 0.6

      \[\leadsto \sqrt{\color{blue}{x \cdot \left(1 + x \cdot 0.5\right)} + 2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.9722389174514045 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(-1 \cdot 1\right) + {\left(e^{2}\right)}^{\left(2 \cdot x\right)}}{e^{2 \cdot x} + 1}}{\log \left(e^{e^{x} - 1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + x \cdot 0.5\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020168 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))