ln(1 + x)

Average Error: 39.2 → 0.3

Time: 3.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.00000000056375149:\\ \;\;\;\;\log 1 + \left(1 \cdot x + x \cdot \left(\frac{x}{1} \cdot \frac{\frac{-1}{2}}{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \frac{1}{2} \cdot \log \left(1 + x\right)\\ \end{array}\]

C

double code(double x) {
	return ((double) log(((double) (1.0 + x))));
}

↓

double code(double x) {
	double VAR;
	if ((((double) (1.0 + x)) <= 1.0000000005637515)) {
		VAR = ((double) (((double) log(1.0)) + ((double) (((double) (1.0 * x)) + ((double) (x * ((double) (((double) (x / 1.0)) * ((double) (-0.5 / 1.0))))))))));
	} else {
		VAR = ((double) (((double) log(((double) sqrt(((double) (1.0 + x)))))) + ((double) (0.5 * ((double) log(((double) (1.0 + x))))))));
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	39.2
Target	0.3
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (+ 1.0 x) < 1.00000000056375149

   1. Initial program 59.4

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

   2. Taylor expanded around 0 0.3

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

   3. Simplified 0.3

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

   5. Applied sub-neg 0.3

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

   6. Applied distribute-lft-in 0.3

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

   7. Simplified 0.3

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

   if 1.00000000056375149 < (+ 1.0 x)

   1. Initial program 0.4

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

   3. Applied add-sqr-sqrt 0.4

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

   4. Applied log-prod 0.4

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

   6. Applied pow1/2 0.4

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

   7. Applied log-pow 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.00000000056375149:\\ \;\;\;\;\log 1 + \left(1 \cdot x + x \cdot \left(\frac{x}{1} \cdot \frac{\frac{-1}{2}}{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \frac{1}{2} \cdot \log \left(1 + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "ln(1 + x)"
  :precision binary64

  :herbie-target
  (if (== (+ 1.0 x) 1.0) x (/ (* x (log (+ 1.0 x))) (- (+ 1.0 x) 1.0)))

  (log (+ 1.0 x)))