Average Error: 58.7 → 0.4
Time: 23.9s
Precision: 64
Internal Precision: 1408
\[e^{x} - 1\]
↓
\[e^{\frac{1}{24} \cdot {x}^{2}} \cdot \left(\sqrt{e^{x}} \cdot x\right)\]

Error

Bits error versus x

Target

Original58.7
Target0.5
Herbie0.4
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.5

    \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\]

  3. Applied simplify0.5

    \[\leadsto \color{blue}{x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)}\]
  4. Using strategy rm

  5. Applied add-exp-log35.1

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

  6. Taylor expanded around 0 35.0

    \[\leadsto e^{\color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\log x + \frac{1}{2} \cdot x\right)}}\]
  7. Using strategy rm

  8. Applied exp-sum35.0

    \[\leadsto \color{blue}{e^{\frac{1}{24} \cdot {x}^{2}} \cdot e^{\log x + \frac{1}{2} \cdot x}}\]

  9. Applied simplify0.4

    \[\leadsto e^{\frac{1}{24} \cdot {x}^{2}} \cdot \color{blue}{\left(\sqrt{e^{x}} \cdot x\right)}\]

Runtime

Time bar (total: 23.9s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1 (/ x 2)) (/ (* x x) 6)))

  (- (exp x) 1))