Average Error: 0.0 → 0.0
Time: 16.5s
Precision: 64
Internal Precision: 128
\[e^{-\left(1 - x \cdot x\right)}\]
↓
\[e^{-\frac{1 - {x}^{4}}{1 + x \cdot x}}\]

Error

Bits error versus x

Derivation

  1. Initial program 0.0

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

  3. Applied flip--0.0

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

  4. Taylor expanded around 0 0.0

    \[\leadsto e^{-\frac{1 \cdot 1 - \color{blue}{{x}^{4}}}{1 + x \cdot x}}\]

  5. Final simplification0.0

    \[\leadsto e^{-\frac{1 - {x}^{4}}{1 + x \cdot x}}\]

Reproduce

herbie shell --seed 2019053 
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))