exp2 (problem 3.3.7)

Average Error: 29.9 → 0.2

Time: 5.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.030068988456940456:\\ \;\;\;\;\frac{\left(e^{x} + 2\right) \cdot \left(1 + e^{x} \cdot \left(e^{x} - 2\right)\right)}{\left(e^{x} + 2\right) \cdot e^{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original	29.9
Target	0.1
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if x < -0.030068988456940456

   1. Initial program 1.0

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

   3. Applied exp-neg 1.1

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

   4. Applied flip-- 1.2

      \[\leadsto \color{blue}{\frac{e^{x} \cdot e^{x} - 2 \cdot 2}{e^{x} + 2}} + \frac{1}{e^{x}}\]

   5. Applied frac-add 1.0

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

   6. Simplified 1.0

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

   if -0.030068988456940456 < x

   1. Initial program 30.1

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

   2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.030068988456940456:\\ \;\;\;\;\frac{\left(e^{x} + 2\right) \cdot \left(1 + e^{x} \cdot \left(e^{x} - 2\right)\right)}{\left(e^{x} + 2\right) \cdot e^{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020124 
(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 (- x))))