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

Average Error: 0.2 → 0.2

Time: 10.2s

Precision: binary64

Cost: 20416

Math

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

↓

\[\begin{array}{l} t_0 := 1 + \sqrt{1 + x}\\ \frac{x}{\frac{t_0 \cdot t_0}{t_0}} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (sqrt (+ 1.0 x))))) (/ x (/ (* t_0 t_0) t_0))))

C

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

↓

double code(double x) {
	double t_0 = 1.0 + sqrt((1.0 + x));
	return x / ((t_0 * t_0) / t_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
    real(8) :: t_0
    t_0 = 1.0d0 + sqrt((1.0d0 + x))
    code = x / ((t_0 * t_0) / t_0)
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 t_0 = 1.0 + Math.sqrt((1.0 + x));
	return x / ((t_0 * t_0) / t_0);
}

Python

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

↓

def code(x):
	t_0 = 1.0 + math.sqrt((1.0 + x))
	return x / ((t_0 * t_0) / t_0)

Julia

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

↓

function code(x)
	t_0 = Float64(1.0 + sqrt(Float64(1.0 + x)))
	return Float64(x / Float64(Float64(t_0 * t_0) / t_0))
end

MATLAB

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

↓

function tmp = code(x)
	t_0 = 1.0 + sqrt((1.0 + x));
	tmp = x / ((t_0 * t_0) / t_0);
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[(1.0 + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(x / N[(N[(t$95$0 * t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.2

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

2. Applied egg-rr 0.2

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

3. Applied egg-rr 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	6848

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

Alternative 2
Error	20.9
Cost	448

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

Alternative 3
Error	20.8
Cost	448

\[\frac{x + x}{x + 4} \]

Alternative 4
Error	21.3
Cost	192

\[0.5 \cdot x \]

Alternative 5
Error	60.9
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023075 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))