Average Error: 39.1 → 0.2
Time: 18.3s
Precision: 64
Internal Precision: 1344
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.0001283050726086712:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt{x + 1}}\right) + \frac{1}{2} \cdot \log \left(\sqrt{x + 1}\right)\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.1
Target0.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 0.0001283050726086712

    1. Initial program 58.8

      \[\log \left(1 + x\right)\]

    2. Initial simplification58.8

      \[\leadsto \log \left(x + 1\right)\]

    3. Taylor expanded around 0 0.3

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

    4. Simplified0.3

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

    if 0.0001283050726086712 < x

    1. Initial program 0.1

      \[\log \left(1 + x\right)\]

    2. Initial simplification0.1

      \[\leadsto \log \left(x + 1\right)\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.1

      \[\leadsto \log \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}\]

    5. Applied log-prod0.1

      \[\leadsto \color{blue}{\log \left(\sqrt{x + 1}\right) + \log \left(\sqrt{x + 1}\right)}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt0.1

      \[\leadsto \log \left(\sqrt{x + 1}\right) + \log \color{blue}{\left(\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}\right)}\]

    8. Applied log-prod0.1

      \[\leadsto \log \left(\sqrt{x + 1}\right) + \color{blue}{\left(\log \left(\sqrt{\sqrt{x + 1}}\right) + \log \left(\sqrt{\sqrt{x + 1}}\right)\right)}\]
    9. Using strategy rm

    10. Applied pow1/20.1

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

    11. Applied log-pow0.1

      \[\leadsto \log \left(\sqrt{x + 1}\right) + \left(\color{blue}{\frac{1}{2} \cdot \log \left(\sqrt{x + 1}\right)} + \log \left(\sqrt{\sqrt{x + 1}}\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.0001283050726086712:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt{x + 1}}\right) + \frac{1}{2} \cdot \log \left(\sqrt{x + 1}\right)\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}\]

Runtime

Time bar (total: 18.3s)Debug logProfile

herbie shell --seed 2018251 
(FPCore (x)
  :name "ln(1 + x)"

  :herbie-target
  (if (== (+ 1 x) 1) x (/ (* x (log (+ 1 x))) (- (+ 1 x) 1)))

  (log (+ 1 x)))