ln(1 + x)

Average Error: 39.1 → 0.3

Time: 19.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.000000397479964719948952733830083161592:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{\frac{-1}{2}}{1 \cdot 1}, x, 1\right), \log 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(\left|\sqrt[3]{1 + x}\right| \cdot \sqrt{\sqrt[3]{1 + x}}\right)\\ \end{array}\]

C

double f(double x) {
        double r102956 = 1.0;
        double r102957 = x;
        double r102958 = r102956 + r102957;
        double r102959 = log(r102958);
        return r102959;
}

↓

double f(double x) {
        double r102960 = 1.0;
        double r102961 = x;
        double r102962 = r102960 + r102961;
        double r102963 = 1.0000003974799647;
        bool r102964 = r102962 <= r102963;
        double r102965 = -0.5;
        double r102966 = r102960 * r102960;
        double r102967 = r102965 / r102966;
        double r102968 = fma(r102967, r102961, r102960);
        double r102969 = log(r102960);
        double r102970 = fma(r102961, r102968, r102969);
        double r102971 = sqrt(r102962);
        double r102972 = log(r102971);
        double r102973 = cbrt(r102962);
        double r102974 = fabs(r102973);
        double r102975 = sqrt(r102973);
        double r102976 = r102974 * r102975;
        double r102977 = log(r102976);
        double r102978 = r102972 + r102977;
        double r102979 = r102964 ? r102970 : r102978;
        return r102979;
}

Error

Bits error versus x

Target

Original	39.1
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) < 1.0000003974799647

   1. Initial program 59.0

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

   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

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

   if 1.0000003974799647 < (+ 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.000000397479964719948952733830083161592:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{\frac{-1}{2}}{1 \cdot 1}, x, 1\right), \log 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(\left|\sqrt[3]{1 + x}\right| \cdot \sqrt{\sqrt[3]{1 + x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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)))