Average Error: 58.7 → 0.4
Time: 40.0s
Precision: 64
Internal Precision: 1344
\[e^{x} - 1\]
↓
\[{\left(e^{x}\right)}^{\left(x \cdot \frac{1}{24} + \frac{1}{2}\right)} \cdot x\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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-log35.5

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

  6. Taylor expanded around 0 35.4

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

  7. Taylor expanded around -inf 62.9

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

  8. Applied simplify0.4

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

  9. Applied simplify0.4

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

Runtime

Time bar (total: 40.0s)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))