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

Average Error: 0.2 → 0.2

Time: 5.1s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r96586 = x;
        double r96587 = 1.0;
        double r96588 = r96586 + r96587;
        double r96589 = sqrt(r96588);
        double r96590 = r96587 + r96589;
        double r96591 = r96586 / r96590;
        return r96591;
}

↓

double f(double x) {
        double r96592 = x;
        double r96593 = 1.0;
        double r96594 = r96592 + r96593;
        double r96595 = sqrt(r96594);
        double r96596 = r96593 + r96595;
        double r96597 = r96592 / r96596;
        return r96597;
}

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

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

4. Final simplification 0.2

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

Reproduce

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