Numeric.Log:$clog1p from log-domain-0.10.2.1, B

Average Error: 0.2 → 0.0

Time: 15.6s

Precision: 64

Math

\[\frac{x}{1 + \sqrt{x + 1}}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 2.653971370452547882012916288428881643354 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{\sqrt[3]{{\left(1 + \sqrt{x + 1}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{\sqrt{x + 1} + 1}\\ \end{array}\]

C

double f(double x) {
        double r86092 = x;
        double r86093 = 1.0;
        double r86094 = r86092 + r86093;
        double r86095 = sqrt(r86094);
        double r86096 = r86093 + r86095;
        double r86097 = r86092 / r86096;
        return r86097;
}

↓

double f(double x) {
        double r86098 = x;
        double r86099 = 2.653971370452548e-23;
        bool r86100 = r86098 <= r86099;
        double r86101 = 1.0;
        double r86102 = r86098 + r86101;
        double r86103 = sqrt(r86102);
        double r86104 = r86101 + r86103;
        double r86105 = 3.0;
        double r86106 = pow(r86104, r86105);
        double r86107 = cbrt(r86106);
        double r86108 = r86098 / r86107;
        double r86109 = sqrt(r86098);
        double r86110 = r86103 + r86101;
        double r86111 = r86109 / r86110;
        double r86112 = r86109 * r86111;
        double r86113 = r86100 ? r86108 : r86112;
        return r86113;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < 2.653971370452548e-23

   1. Initial program 0.0

      \[\frac{x}{1 + \sqrt{x + 1}}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 0.0

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

   4. Simplified 0.0

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

   if 2.653971370452548e-23 < x

   1. Initial program 0.5

      \[\frac{x}{1 + \sqrt{x + 1}}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 19.8

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

   4. Simplified 19.8

      \[\leadsto \frac{x}{\sqrt[3]{\color{blue}{{\left(1 + \sqrt{x + 1}\right)}^{3}}}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 19.8

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

   7. Applied unpow-prod-down 19.8

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

   8. Applied cbrt-prod 19.8

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

   9. Applied add-sqr-sqrt 19.7

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

   10. Applied times-frac 19.7

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

   11. Simplified 19.7

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

   12. Simplified 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.653971370452547882012916288428881643354 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{\sqrt[3]{{\left(1 + \sqrt{x + 1}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{\sqrt{x + 1} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))