Average Error: 0.2 → 0.2
Time: 1.6s
Precision: binary64
\[\frac{x}{1 + \sqrt{x + 1}} \]
↓
\[\frac{x}{\mathsf{fma}\left(1, \sqrt{x + 1}, 1\right)} \]
(FPCore (x) :precision binary64 (/ x (+ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x) :precision binary64 (/ x (fma 1.0 (sqrt (+ x 1.0)) 1.0)))
double code(double x) {
	return x / (1.0 + sqrt((x + 1.0)));
}
↓
double code(double x) {
	return x / fma(1.0, sqrt((x + 1.0)), 1.0);
}

function code(x)
	return Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0))))
end
↓
function code(x)
	return Float64(x / fma(1.0, sqrt(Float64(x + 1.0)), 1.0))
end

code[x_] := N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(x / N[(1.0 * N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Derivation

  1. Initial program 0.2

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

  2. Applied egg-rr39.8

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

  3. Applied egg-rr0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2022160 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))