Average Error: 29.9 → 0.1
Time: 1.0m
Precision: 64
Internal Precision: 1344
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{-x} + \left(e^{x} - 2\right) \le 0.014331332012292242:\\ \;\;\;\;(\frac{1}{12} \cdot \left({x}^{4}\right) + \left((\left({x}^{6}\right) \cdot \frac{1}{360} + \left(x \cdot x\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(e^{x} + 2\right) \cdot \left(\left(e^{x} - 2\right) \cdot e^{x}\right) + \left(e^{x} + 2\right))_*}{\left(e^{x} + 2\right) \cdot e^{x}}\\ \end{array}\]

Error

Bits error versus x

Target

Original29.9
Target0.0
Herbie0.1
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (- (exp x) 2) (exp (- x))) < 0.014331332012292242

    1. Initial program 30.4

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

    2. Initial simplification30.4

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

    3. Taylor expanded around 0 0.0

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

    4. Simplified0.0

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

    if 0.014331332012292242 < (+ (- (exp x) 2) (exp (- x)))

    1. Initial program 0.6

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

    2. Initial simplification0.6

      \[\leadsto \left(e^{x} - 2\right) - \frac{-1}{e^{x}}\]
    3. Using strategy rm

    4. Applied flip--5.0

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

    5. Applied frac-sub6.1

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

    6. Simplified5.9

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} + \left(e^{x} - 2\right) \le 0.014331332012292242:\\ \;\;\;\;(\frac{1}{12} \cdot \left({x}^{4}\right) + \left((\left({x}^{6}\right) \cdot \frac{1}{360} + \left(x \cdot x\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(e^{x} + 2\right) \cdot \left(\left(e^{x} - 2\right) \cdot e^{x}\right) + \left(e^{x} + 2\right))_*}{\left(e^{x} + 2\right) \cdot e^{x}}\\ \end{array}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))