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

Average Error: 0.2 → 0.2

Time: 6.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\frac{x}{\mathsf{fma}\left(\sqrt{1}, \sqrt{1}, \sqrt{x + 1}\right)}\]

C

double f(double x) {
        double r118291 = x;
        double r118292 = 1.0;
        double r118293 = r118291 + r118292;
        double r118294 = sqrt(r118293);
        double r118295 = r118292 + r118294;
        double r118296 = r118291 / r118295;
        return r118296;
}

↓

double f(double x) {
        double r118297 = x;
        double r118298 = 1.0;
        double r118299 = sqrt(r118298);
        double r118300 = r118297 + r118298;
        double r118301 = sqrt(r118300);
        double r118302 = fma(r118299, r118299, r118301);
        double r118303 = r118297 / r118302;
        return r118303;
}

Error

Bits error versus x

Derivation

1. Initial program 0.2

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

3. Applied add-sqr-sqrt 0.2

   \[\leadsto \frac{x}{\color{blue}{\sqrt{1} \cdot \sqrt{1}} + \sqrt{x + 1}}\]

4. Applied fma-def 0.2

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\sqrt{1}, \sqrt{1}, \sqrt{x + 1}\right)}}\]

5. Final simplification 0.2

   \[\leadsto \frac{x}{\mathsf{fma}\left(\sqrt{1}, \sqrt{1}, \sqrt{x + 1}\right)}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))