ln(1 + x)

Average Error: 39.0 → 0.3

Time: 20.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.000000061006203200264508268446661531925:\\ \;\;\;\;\frac{x \cdot \left({1}^{3} + {\left(\frac{\frac{-1}{2}}{\frac{1 \cdot 1}{x}}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{\frac{-1}{2}}{\frac{1 \cdot 1}{x}} \cdot \frac{\frac{-1}{2}}{\frac{1 \cdot 1}{x}} - 1 \cdot \frac{\frac{-1}{2}}{\frac{1 \cdot 1}{x}}\right)} + \log 1\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array}\]

C

double f(double x) {
        double r83462 = 1.0;
        double r83463 = x;
        double r83464 = r83462 + r83463;
        double r83465 = log(r83464);
        return r83465;
}

↓

double f(double x) {
        double r83466 = 1.0;
        double r83467 = x;
        double r83468 = r83466 + r83467;
        double r83469 = 1.0000000610062032;
        bool r83470 = r83468 <= r83469;
        double r83471 = 3.0;
        double r83472 = pow(r83466, r83471);
        double r83473 = -0.5;
        double r83474 = r83466 * r83466;
        double r83475 = r83474 / r83467;
        double r83476 = r83473 / r83475;
        double r83477 = pow(r83476, r83471);
        double r83478 = r83472 + r83477;
        double r83479 = r83467 * r83478;
        double r83480 = r83476 * r83476;
        double r83481 = r83466 * r83476;
        double r83482 = r83480 - r83481;
        double r83483 = r83474 + r83482;
        double r83484 = r83479 / r83483;
        double r83485 = log(r83466);
        double r83486 = r83484 + r83485;
        double r83487 = log(r83468);
        double r83488 = r83470 ? r83486 : r83487;
        return r83488;
}

Error

Bits error versus x

Target

Original	39.0
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) < 1.0000000610062032

   1. Initial program 59.1

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

   2. Taylor expanded around 0 0.4

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

   3. Simplified 0.4

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

   5. Applied flip3-+ 0.4

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

   6. Applied associate-*r/ 0.4

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

   if 1.0000000610062032 < (+ 1.0 x)

   1. Initial program 0.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.000000061006203200264508268446661531925:\\ \;\;\;\;\frac{x \cdot \left({1}^{3} + {\left(\frac{\frac{-1}{2}}{\frac{1 \cdot 1}{x}}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{\frac{-1}{2}}{\frac{1 \cdot 1}{x}} \cdot \frac{\frac{-1}{2}}{\frac{1 \cdot 1}{x}} - 1 \cdot \frac{\frac{-1}{2}}{\frac{1 \cdot 1}{x}}\right)} + \log 1\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "ln(1 + x)"
  :precision binary64

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

  (log (+ 1 x)))