Average Error: 0.2 → 0.2
Time: 2.2s
Precision: binary64
\[\frac{x}{1 + \sqrt{x + 1}} \]
↓
\[\frac{x}{\mathsf{fma}\left(1, 1, \sqrt{x + 1}\right)} \]
\frac{x}{1 + \sqrt{x + 1}}
↓
\frac{x}{\mathsf{fma}\left(1, 1, \sqrt{x + 1}\right)}

(FPCore (x) :precision binary64 (/ x (+ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x) :precision binary64 (/ x (fma 1.0 1.0 (sqrt (+ x 1.0)))))
double code(double x) {
	return x / (1.0 + sqrt(x + 1.0));
}
↓
double code(double x) {
	return x / fma(1.0, 1.0, sqrt(x + 1.0));
}

Error

Bits error versus x

Derivation

  1. Initial program 0.2

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

  2. Applied add-sqr-sqrt_binary640.2

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

  3. Applied fma-def_binary640.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2022088 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))