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

Percentage Accurate: 99.7% → 99.7%

Time: 7.1s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{1 + \sqrt{x + 1}} \end{array} \]

FPCore

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

C

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

Java

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)))

Julia

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + \sqrt{x + 1}} \end{array} \]

FPCore

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

C

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

Java

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)))

Julia

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Step-by-step derivation

   1. clear-num 99.5%

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

   2. associate-/r/ 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + \sqrt{1 + x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

2. Final simplification 99.7%

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

Alternative 3: 68.0% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + \left(1 + x \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ 1.0 (+ 1.0 (* x 0.5)))))

C

double code(double x) {
	return x / (1.0 + (1.0 + (x * 0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (1.0d0 + (1.0d0 + (x * 0.5d0)))
end function

Java

public static double code(double x) {
	return x / (1.0 + (1.0 + (x * 0.5)));
}

Python

def code(x):
	return x / (1.0 + (1.0 + (x * 0.5)))

Julia

function code(x)
	return Float64(x / Float64(1.0 + Float64(1.0 + Float64(x * 0.5))))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 + (1.0 + (x * 0.5)));
end

Wolfram

code[x_] := N[(x / N[(1.0 + N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 70.8%

   \[\leadsto \frac{x}{1 + \color{blue}{\left(1 + 0.5 \cdot x\right)}} \]
3. Step-by-step derivation

   1. +-commutative 70.8%

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

4. Simplified 70.8%

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

5. Final simplification 70.8%

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

Alternative 4: 68.0% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x \cdot 0.5 + 2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ (* x 0.5) 2.0)))

C

double code(double x) {
	return x / ((x * 0.5) + 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / ((x * 0.5d0) + 2.0d0)
end function

Java

public static double code(double x) {
	return x / ((x * 0.5) + 2.0);
}

Python

def code(x):
	return x / ((x * 0.5) + 2.0)

Julia

function code(x)
	return Float64(x / Float64(Float64(x * 0.5) + 2.0))
end

MATLAB

function tmp = code(x)
	tmp = x / ((x * 0.5) + 2.0);
end

Wolfram

code[x_] := N[(x / N[(N[(x * 0.5), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 70.8%

   \[\leadsto \frac{x}{1 + \color{blue}{\left(1 + 0.5 \cdot x\right)}} \]
3. Step-by-step derivation

   1. +-commutative 70.8%

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

4. Simplified 70.8%

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

5. Taylor expanded in x around 0 70.8%

   \[\leadsto \frac{x}{\color{blue}{2 + 0.5 \cdot x}} \]

6. Final simplification 70.8%

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

Alternative 5: 67.3% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.5))

C

double code(double x) {
	return x * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.5d0
end function

Java

public static double code(double x) {
	return x * 0.5;
}

Python

def code(x):
	return x * 0.5

Julia

function code(x)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.5;
end

Wolfram

code[x_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 70.4%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

3. Final simplification 70.4%

   \[\leadsto x \cdot 0.5 \]

Alternative 6: 4.9% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 70.8%

   \[\leadsto \frac{x}{1 + \color{blue}{\left(1 + 0.5 \cdot x\right)}} \]
3. Step-by-step derivation

   1. +-commutative 70.8%

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

4. Simplified 70.8%

   \[\leadsto \frac{x}{1 + \color{blue}{\left(0.5 \cdot x + 1\right)}} \]
5. Step-by-step derivation

   1. clear-num 70.6%

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

   2. associate-/r/ 70.8%

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

   3. +-commutative 70.8%

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

   4. associate-+l+ 70.8%

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

   5. *-commutative 70.8%

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

   6. metadata-eval 70.8%

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

6. Applied egg-rr 70.8%

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

7. Taylor expanded in x around inf 4.9%

   \[\leadsto \color{blue}{2} \]

8. Final simplification 4.9%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023331 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))