Average Error: 39.2 → 0.3
Time: 22.3s
Precision: 64
Internal Precision: 1344
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(x + 1\right) \le 1.7057343258869295 \cdot 10^{-08}:\\ \;\;\;\;x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - \frac{1}{3} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if (log (+ 1 x)) < 1.7057343258869295e-08

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if 1.7057343258869295e-08 < (log (+ 1 x))

    1. Initial program 0.3

      \[\log \left(1 + x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + 1\right) \le 1.7057343258869295 \cdot 10^{-08}:\\ \;\;\;\;x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - \frac{1}{3} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Runtime

Time bar (total: 22.3s)Debug logProfile

herbie shell --seed 2018215 
(FPCore (x)
  :name "ln(1 + x)"

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

  (log (+ 1 x)))