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

Percentage Accurate: 99.7% → 99.7%

Time: 4.7s

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

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

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 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, 0.5, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25e-5) (* x (/ 1.0 (fma x 0.5 2.0))) (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 1.25e-5) {
		tmp = x * (1.0 / fma(x, 0.5, 2.0));
	} else {
		tmp = sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25e-5)
		tmp = Float64(x * Float64(1.0 / fma(x, 0.5, 2.0)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.25e-5], N[(x * N[(1.0 / N[(x * 0.5 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25000000000000006e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. +-commutative 99.2%

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

   5. Simplified 99.2%

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

      1. div-inv 99.3%

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

      2. *-commutative 99.3%

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

      3. *-commutative 99.3%

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

      4. fma-def 99.3%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, 0.5, 2\right)}} \cdot x \]

   7. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x, 0.5, 2\right)} \cdot x} \]

   if 1.25000000000000006e-5 < x

   1. Initial program 99.1%

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

      1. flip-+ 98.7%

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

      2. metadata-eval 98.7%

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

      3. add-sqr-sqrt 99.5%

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

      4. +-commutative 99.5%

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

      5. associate--r+ 99.5%

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

      6. metadata-eval 99.5%

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

      7. neg-sub0 99.5%

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

      8. associate-/r/ 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. remove-double-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. *-inverses 99.5%

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

      4. metadata-eval 99.5%

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

      5. neg-mul-1 99.5%

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

      6. neg-sub0 99.5%

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

      7. associate--r- 99.5%

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

      8. metadata-eval 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, 0.5, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{1 + \left(1 + x \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25e-5) (/ x (+ 1.0 (+ 1.0 (* x 0.5)))) (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 1.25e-5) {
		tmp = x / (1.0 + (1.0 + (x * 0.5)));
	} 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 <= 1.25d-5) then
        tmp = x / (1.0d0 + (1.0d0 + (x * 0.5d0)))
    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 <= 1.25e-5) {
		tmp = x / (1.0 + (1.0 + (x * 0.5)));
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25e-5:
		tmp = x / (1.0 + (1.0 + (x * 0.5)))
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25e-5)
		tmp = Float64(x / Float64(1.0 + Float64(1.0 + Float64(x * 0.5))));
	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 <= 1.25e-5)
		tmp = x / (1.0 + (1.0 + (x * 0.5)));
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25e-5], N[(x / N[(1.0 + N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25000000000000006e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. +-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 1.25000000000000006e-5 < x

   1. Initial program 99.1%

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

      1. flip-+ 98.7%

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

      2. metadata-eval 98.7%

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

      3. add-sqr-sqrt 99.5%

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

      4. +-commutative 99.5%

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

      5. associate--r+ 99.5%

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

      6. metadata-eval 99.5%

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

      7. neg-sub0 99.5%

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

      8. associate-/r/ 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. remove-double-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. *-inverses 99.5%

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

      4. metadata-eval 99.5%

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

      5. neg-mul-1 99.5%

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

      6. neg-sub0 99.5%

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

      7. associate--r- 99.5%

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

      8. metadata-eval 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{1 + \left(1 + x \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.7% 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. Add Preprocessing

3. Taylor expanded in x around 0 66.9%

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

   1. +-commutative 66.9%

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

5. Simplified 66.9%

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

6. Final simplification 66.9%

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

Alternative 5: 68.7% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0 66.9%

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

   1. +-commutative 66.9%

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

5. Simplified 66.9%

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

6. Final simplification 66.9%

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

Alternative 6: 68.0% 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. Add Preprocessing

3. Taylor expanded in x around 0 65.5%

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

4. Final simplification 65.5%

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

Alternative 7: 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. Add Preprocessing

3. Taylor expanded in x around 0 66.9%

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

   1. +-commutative 66.9%

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

5. Simplified 66.9%

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

6. Taylor expanded in x around inf 5.2%

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

7. Final simplification 5.2%

   \[\leadsto 2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))