ln(1 + x)

Average Error: 39.4 → 0.3

Time: 11.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.000000154584223865938952258147764950991:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{2}}{{1}^{2}}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(\left|\sqrt[3]{1 + x}\right| \cdot \sqrt{\sqrt[3]{1 + x}}\right)\\ \end{array}\]

C

double f(double x) {
        double r75046 = 1.0;
        double r75047 = x;
        double r75048 = r75046 + r75047;
        double r75049 = log(r75048);
        return r75049;
}

↓

double f(double x) {
        double r75050 = 1.0;
        double r75051 = x;
        double r75052 = r75050 + r75051;
        double r75053 = 1.0000001545842239;
        bool r75054 = r75052 <= r75053;
        double r75055 = 2.0;
        double r75056 = pow(r75051, r75055);
        double r75057 = pow(r75050, r75055);
        double r75058 = r75056 / r75057;
        double r75059 = -0.5;
        double r75060 = log(r75050);
        double r75061 = fma(r75050, r75051, r75060);
        double r75062 = fma(r75058, r75059, r75061);
        double r75063 = sqrt(r75052);
        double r75064 = log(r75063);
        double r75065 = cbrt(r75052);
        double r75066 = fabs(r75065);
        double r75067 = sqrt(r75065);
        double r75068 = r75066 * r75067;
        double r75069 = log(r75068);
        double r75070 = r75064 + r75069;
        double r75071 = r75054 ? r75062 : r75070;
        return r75071;
}

Error

Bits error versus x

Target

Original	39.4
Target	0.2
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) < 1.0000001545842239

   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

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

   if 1.0000001545842239 < (+ 1.0 x)

   1. Initial program 0.2

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

   3. Applied add-sqr-sqrt 0.2

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

   4. Applied log-prod 0.2

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

   6. Applied add-cube-cbrt 0.2

      \[\leadsto \log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{\color{blue}{\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right) \cdot \sqrt[3]{1 + x}}}\right)\]

   7. Applied sqrt-prod 0.2

      \[\leadsto \log \left(\sqrt{1 + x}\right) + \log \color{blue}{\left(\sqrt{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt{\sqrt[3]{1 + x}}\right)}\]

   8. Simplified 0.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.000000154584223865938952258147764950991:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{2}}{{1}^{2}}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(\left|\sqrt[3]{1 + x}\right| \cdot \sqrt{\sqrt[3]{1 + x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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)))