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

Percentage Accurate: 99.7% → 99.7%

Time: 4.4s

Alternatives: 5

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

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

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]

Derivation

1. Initial program 99.8%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto \frac{x}{1 + \sqrt{x + 1}} \]
4. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{x \cdot 0.5 + 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.45e-5) (/ x (+ (* x 0.5) 2.0)) (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 1.45e-5) {
		tmp = x / ((x * 0.5) + 2.0);
	} else {
		tmp = sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.45d-5) then
        tmp = x / ((x * 0.5d0) + 2.0d0)
    else
        tmp = sqrt((x + 1.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.45e-5) {
		tmp = x / ((x * 0.5) + 2.0);
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.45e-5:
		tmp = x / ((x * 0.5) + 2.0)
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45e-5)
		tmp = Float64(x / Float64(Float64(x * 0.5) + 2.0));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.45e-5)
		tmp = x / ((x * 0.5) + 2.0);
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.45e-5], N[(x / N[(N[(x * 0.5), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e-5

   1. Initial program 100.0%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.8%

      \[\leadsto \frac{x}{\color{blue}{2 + 0.5 \cdot x}} \]
   4. Step-by-step derivation

      1. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   if 1.45e-5 < x

   1. Initial program 99.2%

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

      1. flip-+ 98.6%

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

      2. frac-2neg 98.6%

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

      3. metadata-eval 98.6%

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

      4. add-sqr-sqrt 99.4%

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

      5. +-commutative 99.4%

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

      6. associate--r+ 99.4%

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

      7. metadata-eval 99.4%

         \[\leadsto \frac{x}{\frac{-\left(\color{blue}{0} - x\right)}{-\left(1 - \sqrt{x + 1}\right)}} \]

      8. neg-sub0 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. remove-double-neg 99.4%

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

      2. neg-sub0 99.4%

         \[\leadsto \frac{x}{\frac{x}{\color{blue}{0 - \left(1 - \sqrt{x + 1}\right)}}} \]

      3. associate--r- 99.4%

         \[\leadsto \frac{x}{\frac{x}{\color{blue}{\left(0 - 1\right) + \sqrt{x + 1}}}} \]

      4. metadata-eval 99.4%

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

   6. Simplified 99.4%

      \[\leadsto \frac{x}{\color{blue}{\frac{x}{-1 + \sqrt{x + 1}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 99.4%

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

      2. *-inverses 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. +-commutative 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{x \cdot 0.5 + 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.8%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 74.0%

   \[\leadsto \frac{x}{\color{blue}{2 + 0.5 \cdot x}} \]
4. Step-by-step derivation

   1. +-commutative 74.0%

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

5. Simplified 74.0%

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

6. Final simplification 74.0%

   \[\leadsto \frac{x}{x \cdot 0.5 + 2} \]
7. Add Preprocessing

Alternative 4: 67.3% 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 99.8%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 73.4%

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

4. Final simplification 73.4%

   \[\leadsto \frac{x}{2} \]
5. Add Preprocessing

Alternative 5: 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 99.8%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 74.0%

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

   1. +-commutative 74.0%

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

5. Simplified 74.0%

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

6. Taylor expanded in x around inf 4.8%

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

7. Final simplification 4.8%

   \[\leadsto 2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))