Average Error: 38.6 → 0.3
Time: 3.5s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.000000780384010967694052851584274321795:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \left(\log \left(\sqrt{\sqrt{1 + x}}\right) + \log \left(\sqrt{\sqrt{1 + x}}\right)\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.000000780384010967694052851584274321795:\\
\;\;\;\;\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) + \left(\log \left(\sqrt{\sqrt{1 + x}}\right) + \log \left(\sqrt{\sqrt{1 + x}}\right)\right)\\

\end{array}
double f(double x) {
        double r81456 = 1.0;
        double r81457 = x;
        double r81458 = r81456 + r81457;
        double r81459 = log(r81458);
        return r81459;
}
↓
double f(double x) {
        double r81460 = 1.0;
        double r81461 = x;
        double r81462 = r81460 + r81461;
        double r81463 = 1.000000780384011;
        bool r81464 = r81462 <= r81463;
        double r81465 = log(r81460);
        double r81466 = 0.5;
        double r81467 = 2.0;
        double r81468 = pow(r81461, r81467);
        double r81469 = pow(r81460, r81467);
        double r81470 = r81468 / r81469;
        double r81471 = r81466 * r81470;
        double r81472 = r81465 - r81471;
        double r81473 = fma(r81461, r81460, r81472);
        double r81474 = sqrt(r81462);
        double r81475 = log(r81474);
        double r81476 = sqrt(r81474);
        double r81477 = log(r81476);
        double r81478 = r81477 + r81477;
        double r81479 = r81475 + r81478;
        double r81480 = r81464 ? r81473 : r81479;
        return r81480;
}

Error

Bits error versus x

Target

Original38.6
Target0.3
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ 1.0 x) < 1.000000780384011

    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. Simplified0.4

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

    if 1.000000780384011 < (+ 1.0 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 add-sqr-sqrt0.1

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

    7. Applied sqrt-prod0.1

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

    8. Applied log-prod0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.000000780384010967694052851584274321795:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \left(\log \left(\sqrt{\sqrt{1 + x}}\right) + \log \left(\sqrt{\sqrt{1 + x}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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)))