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

Average Error: 0.2 → 0.3

Time: 4.0s

Precision: binary64

Cost: 6852

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 1.654963891712827 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(x \cdot -0.125\right) + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 1.654963891712827e-8)
   (+ (* x (* x -0.125)) (* x 0.5))
   (+ (sqrt (+ x 1.0)) -1.0)))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= 1.654963891712827e-8) {
		tmp = (x * (x * -0.125)) + (x * 0.5);
	} else {
		tmp = sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= 1.654963891712827d-8) then
        tmp = (x * (x * (-0.125d0))) + (x * 0.5d0)
    else
        tmp = sqrt((x + 1.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= 1.654963891712827e-8) {
		tmp = (x * (x * -0.125)) + (x * 0.5);
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= 1.654963891712827e-8:
		tmp = (x * (x * -0.125)) + (x * 0.5)
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= 1.654963891712827e-8)
		tmp = Float64(Float64(x * Float64(x * -0.125)) + Float64(x * 0.5));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + -1.0);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.654963891712827e-8)
		tmp = (x * (x * -0.125)) + (x * 0.5);
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, 1.654963891712827e-8], N[(N[(x * N[(x * -0.125), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6549638917128268e-8

   1. Initial program 0.0

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

   2. Taylor expanded in x around 0 0.3

      \[\leadsto \color{blue}{-0.125 \cdot {x}^{2} + 0.5 \cdot x} \]

   3. Simplified 0.3

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right)} \]

   4. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\left(x \cdot -0.125\right) \cdot x + 0.5 \cdot x} \]

   if 1.6549638917128268e-8 < x

   1. Initial program 0.5

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

   2. Applied egg-rr 0.3

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

   3. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\sqrt{x + 1} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.654963891712827 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(x \cdot -0.125\right) + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	6848

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

Alternative 2
Error	20.2
Cost	448

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

Alternative 3
Error	20.7
Cost	192

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

Alternative 4
Error	60.9
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022284 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))