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

Percentage Accurate: 99.7% → 99.7%

Time: 3.4s

Alternatives: 7

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 99.4%

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

   2. associate-/r/ 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00019:\\ \;\;\;\;x \cdot \left(0.5 + \left(x \cdot \left(x \cdot 0.0625\right) + x \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00019)
   (* x (+ 0.5 (+ (* x (* x 0.0625)) (* x -0.125))))
   (+ (sqrt (+ 1.0 x)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.00019) {
		tmp = x * (0.5 + ((x * (x * 0.0625)) + (x * -0.125)));
	} else {
		tmp = sqrt((1.0 + x)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00019d0) then
        tmp = x * (0.5d0 + ((x * (x * 0.0625d0)) + (x * (-0.125d0))))
    else
        tmp = sqrt((1.0d0 + x)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.00019) {
		tmp = x * (0.5 + ((x * (x * 0.0625)) + (x * -0.125)));
	} else {
		tmp = Math.sqrt((1.0 + x)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00019:
		tmp = x * (0.5 + ((x * (x * 0.0625)) + (x * -0.125)))
	else:
		tmp = math.sqrt((1.0 + x)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00019)
		tmp = Float64(x * Float64(0.5 + Float64(Float64(x * Float64(x * 0.0625)) + Float64(x * -0.125))));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00019)
		tmp = x * (0.5 + ((x * (x * 0.0625)) + (x * -0.125)));
	else
		tmp = sqrt((1.0 + x)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00019], N[(x * N[(0.5 + N[(N[(x * N[(x * 0.0625), $MachinePrecision]), $MachinePrecision] + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9000000000000001e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.4%

      \[\leadsto \color{blue}{-0.125 \cdot {x}^{2} + \left(0.5 \cdot x + 0.0625 \cdot {x}^{3}\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 99.4%

         \[\leadsto \color{blue}{\left(-0.125 \cdot {x}^{2} + 0.5 \cdot x\right) + 0.0625 \cdot {x}^{3}} \]

      2. +-commutative 99.4%

         \[\leadsto \color{blue}{\left(0.5 \cdot x + -0.125 \cdot {x}^{2}\right)} + 0.0625 \cdot {x}^{3} \]

      3. associate-+l+ 99.4%

         \[\leadsto \color{blue}{0.5 \cdot x + \left(-0.125 \cdot {x}^{2} + 0.0625 \cdot {x}^{3}\right)} \]

      4. fma-def 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, -0.125 \cdot {x}^{2} + 0.0625 \cdot {x}^{3}\right)} \]

      5. *-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(0.5, x, \color{blue}{{x}^{2} \cdot -0.125} + 0.0625 \cdot {x}^{3}\right) \]

      6. *-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(0.5, x, {x}^{2} \cdot -0.125 + \color{blue}{{x}^{3} \cdot 0.0625}\right) \]

      7. unpow3 99.4%

         \[\leadsto \mathsf{fma}\left(0.5, x, {x}^{2} \cdot -0.125 + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot 0.0625\right) \]

      8. unpow2 99.4%

         \[\leadsto \mathsf{fma}\left(0.5, x, {x}^{2} \cdot -0.125 + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot 0.0625\right) \]

      9. associate-*l* 99.4%

         \[\leadsto \mathsf{fma}\left(0.5, x, {x}^{2} \cdot -0.125 + \color{blue}{{x}^{2} \cdot \left(x \cdot 0.0625\right)}\right) \]

      10. distribute-lft-out 99.4%

          \[\leadsto \mathsf{fma}\left(0.5, x, \color{blue}{{x}^{2} \cdot \left(-0.125 + x \cdot 0.0625\right)}\right) \]

      11. unpow2 99.4%

          \[\leadsto \mathsf{fma}\left(0.5, x, \color{blue}{\left(x \cdot x\right)} \cdot \left(-0.125 + x \cdot 0.0625\right)\right) \]

   4. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, \left(x \cdot x\right) \cdot \left(-0.125 + x \cdot 0.0625\right)\right)} \]
   5. Step-by-step derivation

      1. fma-udef 99.4%

         \[\leadsto \color{blue}{0.5 \cdot x + \left(x \cdot x\right) \cdot \left(-0.125 + x \cdot 0.0625\right)} \]

      2. *-commutative 99.4%

         \[\leadsto \color{blue}{x \cdot 0.5} + \left(x \cdot x\right) \cdot \left(-0.125 + x \cdot 0.0625\right) \]

      3. associate-*l* 99.4%

         \[\leadsto x \cdot 0.5 + \color{blue}{x \cdot \left(x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)} \]

      4. distribute-lft-out 99.4%

         \[\leadsto \color{blue}{x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)} \]

      5. +-commutative 99.4%

         \[\leadsto x \cdot \left(0.5 + x \cdot \color{blue}{\left(x \cdot 0.0625 + -0.125\right)}\right) \]

      6. fma-def 99.4%

         \[\leadsto x \cdot \left(0.5 + x \cdot \color{blue}{\mathsf{fma}\left(x, 0.0625, -0.125\right)}\right) \]

   6. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + x \cdot \mathsf{fma}\left(x, 0.0625, -0.125\right)\right)} \]
   7. Step-by-step derivation

      1. fma-udef 99.4%

         \[\leadsto x \cdot \left(0.5 + x \cdot \color{blue}{\left(x \cdot 0.0625 + -0.125\right)}\right) \]

      2. distribute-rgt-in 99.4%

         \[\leadsto x \cdot \left(0.5 + \color{blue}{\left(\left(x \cdot 0.0625\right) \cdot x + -0.125 \cdot x\right)}\right) \]

   8. Applied egg-rr 99.4%

      \[\leadsto x \cdot \left(0.5 + \color{blue}{\left(\left(x \cdot 0.0625\right) \cdot x + -0.125 \cdot x\right)}\right) \]

   if 1.9000000000000001e-4 < x

   1. Initial program 99.1%

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

      1. flip-+ 99.1%

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

      2. metadata-eval 99.1%

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

      3. add-sqr-sqrt 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--r+ 99.8%

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

      6. metadata-eval 99.8%

         \[\leadsto \frac{x}{\frac{\color{blue}{0} - x}{1 - \sqrt{x + 1}}} \]

      7. neg-sub0 99.8%

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

      8. associate-/r/ 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. *-inverses 99.8%

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

      4. metadata-eval 99.8%

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

      5. neg-mul-1 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-in 99.8%

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

      9. remove-double-neg 99.8%

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

      10. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00019:\\ \;\;\;\;x \cdot \left(0.5 + \left(x \cdot \left(x \cdot 0.0625\right) + x \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + -1\\ \end{array} \]

Alternative 3: 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 4: 68.3% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1}{x \cdot 0.5 + 2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (/ 1.0 (+ (* x 0.5) 2.0))))

C

double code(double x) {
	return x * (1.0 / ((x * 0.5) + 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 / ((x * 0.5d0) + 2.0d0))
end function

Java

public static double code(double x) {
	return x * (1.0 / ((x * 0.5) + 2.0));
}

Python

def code(x):
	return x * (1.0 / ((x * 0.5) + 2.0))

Julia

function code(x)
	return Float64(x * Float64(1.0 / Float64(Float64(x * 0.5) + 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 / ((x * 0.5) + 2.0));
end

Wolfram

code[x_] := N[(x * N[(1.0 / N[(N[(x * 0.5), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. clear-num 99.4%

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

   2. associate-/r/ 99.7%

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

3. Applied egg-rr 99.7%

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

4. Taylor expanded in x around 0 67.3%

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

5. Final simplification 67.3%

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

Alternative 5: 68.3% 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 67.3%

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

3. Final simplification 67.3%

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

Alternative 6: 67.7% accurate, 35.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ x 2.0))

C

double code(double x) {
	return x / 2.0;
}

Fortran

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

Java

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

Python

def code(x):
	return x / 2.0

Julia

function code(x)
	return Float64(x / 2.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 66.4%

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

3. Final simplification 66.4%

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

Alternative 7: 4.8% 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 67.3%

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

3. Taylor expanded in x around inf 5.0%

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

4. Final simplification 5.0%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023215 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))