exp2 (problem 3.3.7)

Average Error: 30.1 → 0.1

Time: 4.4s

Precision: binary64

Math

\[\left(e^{x} - 2\right) + e^{-x}\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \le 1.4521224485864 \cdot 10^{-5}:\\ \;\;\;\;x \cdot x + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot 2 + e^{x} \cdot \left(\left({\left(e^{x}\right)}^{3} - {2}^{3}\right) + \left(e^{x} + 2\right)\right)}{{\left(e^{x}\right)}^{3} + e^{x} \cdot \left(2 \cdot \left(e^{x} + 2\right)\right)}\\ \end{array}\]

Error

Bits error versus x

Target

Original	30.1
Target	0.0
Herbie	0.1

\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

1. Split input into 2 regimes

2. if (+ (- (exp x) 2.0) (exp (neg x))) < 1.4521224485864e-5

   1. Initial program 30.6

      \[\left(e^{x} - 2\right) + e^{-x}\]

   2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}\]

   3. Simplified 0.0

      \[\leadsto \color{blue}{x \cdot x + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}\]

   if 1.4521224485864e-5 < (+ (- (exp x) 2.0) (exp (neg x)))

   1. Initial program 3.1

      \[\left(e^{x} - 2\right) + e^{-x}\]
   2. Using strategy rm

   3. Applied exp-neg 2.9

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

   4. Applied flip3-- 5.1

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{3} - {2}^{3}}{e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)}} + \frac{1}{e^{x}}\]

   5. Applied frac-add 6.2

      \[\leadsto \color{blue}{\frac{\left({\left(e^{x}\right)}^{3} - {2}^{3}\right) \cdot e^{x} + \left(e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)\right) \cdot 1}{\left(e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)\right) \cdot e^{x}}}\]

   6. Simplified 5.9

      \[\leadsto \frac{\color{blue}{2 \cdot 2 + e^{x} \cdot \left(\left({\left(e^{x}\right)}^{3} - {2}^{3}\right) + \left(e^{x} + 2\right)\right)}}{\left(e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)\right) \cdot e^{x}}\]

   7. Simplified 5.9

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \le 1.4521224485864 \cdot 10^{-5}:\\ \;\;\;\;x \cdot x + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot 2 + e^{x} \cdot \left(\left({\left(e^{x}\right)}^{3} - {2}^{3}\right) + \left(e^{x} + 2\right)\right)}{{\left(e^{x}\right)}^{3} + e^{x} \cdot \left(2 \cdot \left(e^{x} + 2\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (neg x))))