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

Average Error: 0.2 → 0.0

Time: 13.7s

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 r89087 = x;
        double r89088 = 1.0;
        double r89089 = r89087 + r89088;
        double r89090 = sqrt(r89089);
        double r89091 = r89088 + r89090;
        double r89092 = r89087 / r89091;
        return r89092;
}

↓

double f(double x) {
        double r89093 = x;
        double r89094 = 2.653971370452548e-23;
        bool r89095 = r89093 <= r89094;
        double r89096 = 1.0;
        double r89097 = r89093 + r89096;
        double r89098 = sqrt(r89097);
        double r89099 = r89096 + r89098;
        double r89100 = 3.0;
        double r89101 = pow(r89099, r89100);
        double r89102 = cbrt(r89101);
        double r89103 = r89093 / r89102;
        double r89104 = sqrt(r89093);
        double r89105 = r89098 + r89096;
        double r89106 = r89104 / r89105;
        double r89107 = r89104 * r89106;
        double r89108 = r89095 ? r89103 : r89107;
        return r89108;
}

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