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

Average Accuracy: 99.7% → 99.7%

Time: 9.9s

Precision: binary64

Cost: 6848

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 + sqrt((x + 1.0d0)))
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.sqrt((x + 1.0)));
}

Python

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

↓

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

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 + sqrt(Float64(x + 1.0))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = x / (1.0 + sqrt((x + 1.0)));
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[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

2. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	6852

\[\begin{array}{l} t_0 := \frac{x}{\left(x \cdot x\right) \cdot 0.25 + -4}\\ \mathbf{if}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;t_0 \cdot \left(x \cdot 0.5\right) + t_0 \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

Alternative 2
Accuracy	68.2%
Cost	576

\[x \cdot \frac{1}{x \cdot 0.5 + 2} \]

Alternative 3
Accuracy	68.2%
Cost	576

\[\frac{x}{1 + \left(1 + x \cdot 0.5\right)} \]

Alternative 4
Accuracy	68.0%
Cost	448

\[\frac{-1}{-0.5 + \frac{-2}{x}} \]

Alternative 5
Accuracy	68.2%
Cost	448

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

Alternative 6
Accuracy	67.6%
Cost	192

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

Alternative 7
Accuracy	4.9%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))