Average Error: 39.1 → 0.2

Time: 26.8s

Precision: 64

Internal Precision: 1344

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

↓

\[\begin{array}{l} \mathbf{if}\;\log \left(1 + x\right) \le 5.108126122833254 \cdot 10^{-08}:\\ \;\;\;\;x - \left(\frac{1}{2} - x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \left(\log \left(\sqrt{\sqrt{1 + x}}\right) + \log \left(\sqrt{\sqrt{1 + x}}\right)\right)\\ \end{array}\]

Error

The X axis uses an exponential scale — Bits error versus `x`

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In
Out

Enter valid numbers for all inputs

Target

Original	39.1
Target	0.2
Herbie	0.2

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

Derivation

Split input into 2 regimes
if (log (+ 1 x)) < 5.108126122833254e-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. Applied simplify0.2
  \[\leadsto \color{blue}{x - \left(\frac{1}{2} - x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)}\]
if 5.108126122833254e-08 < (log (+ 1 x))
1. Initial program 0.2
  \[\log \left(1 + x\right)\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.2
  \[\leadsto \log \color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x}\right)}\]
4. Applied log-prod0.2
  \[\leadsto \color{blue}{\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)}\]
5. Using strategy rm
6. Applied add-sqr-sqrt0.2
  \[\leadsto \log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{\color{blue}{\sqrt{1 + x} \cdot \sqrt{1 + x}}}\right)\]
7. Applied sqrt-prod0.2
  \[\leadsto \log \left(\sqrt{1 + x}\right) + \log \color{blue}{\left(\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}\right)}\]
8. Applied log-prod0.2
  \[\leadsto \log \left(\sqrt{1 + x}\right) + \color{blue}{\left(\log \left(\sqrt{\sqrt{1 + x}}\right) + \log \left(\sqrt{\sqrt{1 + x}}\right)\right)}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed 2018207 
(FPCore (x)
  :name "ln(1 + x)"

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

  (log (+ 1 x)))

Error

Try it out

Target

Derivation

`if (log (+ 1 x)) < 5.108126122833254e-08`

`if 5.108126122833254e-08 < (log (+ 1 x))`

Runtime