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

Average Error: 0.2 → 0.2

Time: 7.4s

Precision: binary64

Cost: 13376

Math

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

↓

\[\frac{x}{1 + {\left({\left(x + 1\right)}^{0.25}\right)}^{2}} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ 1.0 (sqrt (+ x 1.0)))))

↓

(FPCore (x) :precision binary64 (/ x (+ 1.0 (pow (pow (+ x 1.0) 0.25) 2.0))))

C

double code(double x) {
	return x / (1.0 + sqrt((x + 1.0)));
}

↓

double code(double x) {
	return x / (1.0 + pow(pow((x + 1.0), 0.25), 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (1.0d0 + sqrt((x + 1.0d0)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (1.0d0 + (((x + 1.0d0) ** 0.25d0) ** 2.0d0))
end function

Java

public static double code(double x) {
	return x / (1.0 + Math.sqrt((x + 1.0)));
}

↓

public static double code(double x) {
	return x / (1.0 + Math.pow(Math.pow((x + 1.0), 0.25), 2.0));
}

Python

def code(x):
	return x / (1.0 + math.sqrt((x + 1.0)))

↓

def code(x):
	return x / (1.0 + math.pow(math.pow((x + 1.0), 0.25), 2.0))

Julia

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

↓

function code(x)
	return Float64(x / Float64(1.0 + ((Float64(x + 1.0) ^ 0.25) ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 + sqrt((x + 1.0)));
end

↓

function tmp = code(x)
	tmp = x / (1.0 + (((x + 1.0) ^ 0.25) ^ 2.0));
end

Wolfram

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[Power[N[Power[N[(x + 1.0), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Applied egg-rr 0.2

   \[\leadsto \frac{x}{1 + \color{blue}{{\left({\left(x + 1\right)}^{0.25}\right)}^{2}}} \]

3. Final simplification 0.2

   \[\leadsto \frac{x}{1 + {\left({\left(x + 1\right)}^{0.25}\right)}^{2}} \]

Alternatives

Alternative 1
Error	0.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 0.00106:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot -0.125 + x \cdot 0.5\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.0390625\right)\right) + \left(x \cdot x\right) \cdot \left(x \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

Alternative 2
Error	0.2
Cost	6848

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

Alternative 3
Error	20.0
Cost	448

\[\frac{x}{2 + x \cdot 0.5} \]

Alternative 4
Error	20.5
Cost	192

\[\frac{x}{2} \]

Alternative 5
Error	60.9
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))