ln(1 + x)

Average Error: 39.0 → 0.4

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.00000000019218915:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \frac{1}{2} \cdot \log \left(1 + x\right)\\ \end{array}\]

C

double f(double x) {
        double r51662 = 1.0;
        double r51663 = x;
        double r51664 = r51662 + r51663;
        double r51665 = log(r51664);
        return r51665;
}

↓

double f(double x) {
        double r51666 = 1.0;
        double r51667 = x;
        double r51668 = r51666 + r51667;
        double r51669 = 1.0000000001921892;
        bool r51670 = r51668 <= r51669;
        double r51671 = log(r51666);
        double r51672 = 0.5;
        double r51673 = 2.0;
        double r51674 = pow(r51667, r51673);
        double r51675 = pow(r51666, r51673);
        double r51676 = r51674 / r51675;
        double r51677 = r51672 * r51676;
        double r51678 = r51671 - r51677;
        double r51679 = fma(r51667, r51666, r51678);
        double r51680 = sqrt(r51668);
        double r51681 = log(r51680);
        double r51682 = log(r51668);
        double r51683 = r51672 * r51682;
        double r51684 = r51681 + r51683;
        double r51685 = r51670 ? r51679 : r51684;
        return r51685;
}

Error

Bits error versus x

Target

Original	39.0
Target	0.2
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (+ 1.0 x) < 1.0000000001921892

   1. Initial program 59.3

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

   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

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

   if 1.0000000001921892 < (+ 1.0 x)

   1. Initial program 0.5

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

   3. Applied add-sqr-sqrt 0.6

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

   4. Applied log-prod 0.5

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

   6. Applied pow1/2 0.5

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

   7. Applied log-pow 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.00000000019218915:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \frac{1}{2} \cdot \log \left(1 + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(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)))