Error

Target

Derivation

1. Split input into 2 regimes

2. if (log (+ 1 x)) < 2.0766808099535407e-06

   1. Initial program 59.0

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

   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. Applied simplify0.2

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

   if 2.0766808099535407e-06 < (log (+ 1 x))

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt0.1

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

   4. Applied log-prod0.1

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

   6. Applied pow1/20.1

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

   7. Applied log-pow0.1

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

   9. Applied add-sqr-sqrt0.1

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

   10. Applied sqrt-prod0.1

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

   11. Applied log-prod0.1

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

Runtime

Time bar (total: 21.2s)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' 
(FPCore (x)
  :name "ln(1 + x)"

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

  (log (+ 1 x)))