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

Percentage Accurate: 99.7% → 99.7%

Time: 8.7s

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

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

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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0003:\\ \;\;\;\;\frac{1}{0.5 + \left(x \cdot -0.125 + 2 \cdot \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0003)
   (/ 1.0 (+ 0.5 (+ (* x -0.125) (* 2.0 (/ 1.0 x)))))
   (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0003) {
		tmp = 1.0 / (0.5 + ((x * -0.125) + (2.0 * (1.0 / x))));
	} 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 <= 0.0003d0) then
        tmp = 1.0d0 / (0.5d0 + ((x * (-0.125d0)) + (2.0d0 * (1.0d0 / x))))
    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 <= 0.0003) {
		tmp = 1.0 / (0.5 + ((x * -0.125) + (2.0 * (1.0 / x))));
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0003:
		tmp = 1.0 / (0.5 + ((x * -0.125) + (2.0 * (1.0 / x))))
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0003)
		tmp = Float64(1.0 / Float64(0.5 + Float64(Float64(x * -0.125) + Float64(2.0 * Float64(1.0 / x)))));
	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 <= 0.0003)
		tmp = 1.0 / (0.5 + ((x * -0.125) + (2.0 * (1.0 / x))));
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0003], N[(1.0 / N[(0.5 + N[(N[(x * -0.125), $MachinePrecision] + N[(2.0 * N[(1.0 / x), $MachinePrecision]), $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 < 2.99999999999999974e-4

   1. Initial program 99.9%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.9%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 98.7%

      \[\leadsto \frac{1}{\color{blue}{0.5 + \left(-0.125 \cdot x + 2 \cdot \frac{1}{x}\right)}} \]

   if 2.99999999999999974e-4 < x

   1. Initial program 99.3%

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

      1. clear-num 99.1%

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

      2. associate-/r/ 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. associate-*l/ 99.3%

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

      2. associate-/l* 99.1%

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

      3. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. clear-num 99.3%

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

      2. frac-2neg 99.3%

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

      3. neg-sub0 99.3%

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

      4. metadata-eval 99.3%

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

      5. associate--r+ 99.2%

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

      6. metadata-eval 99.2%

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

      7. +-commutative 99.2%

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

      8. add-sqr-sqrt 99.6%

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

      9. distribute-neg-in 99.6%

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

      10. metadata-eval 99.6%

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

      11. sub-neg 99.6%

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

      12. +-commutative 99.6%

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

      13. flip-+ 99.9%

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

      14. +-commutative 99.9%

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

      15. +-commutative 99.9%

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

      16. add-sqr-sqrt 99.8%

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

      17. hypot-1-def 99.8%

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

   8. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{x}\right) + -1} \]
   9. Step-by-step derivation

      1. hypot-undefine 99.8%

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

      2. metadata-eval 99.8%

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

      3. rem-square-sqrt 99.9%

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

      4. +-commutative 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0003:\\ \;\;\;\;\frac{1}{0.5 + \left(x \cdot -0.125 + 2 \cdot \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.0% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.7%

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

   2. associate-/l* 99.5%

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

   3. +-commutative 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0 66.4%

   \[\leadsto \frac{1}{\color{blue}{0.5 + 2 \cdot \frac{1}{x}}} \]
8. Step-by-step derivation

   1. associate-*r/ 66.4%

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

   2. metadata-eval 66.4%

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

9. Simplified 66.4%

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

10. Final simplification 66.4%

    \[\leadsto \frac{1}{0.5 + \frac{2}{x}} \]
11. Add Preprocessing

Alternative 4: 67.2% 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 66.6%

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

   1. +-commutative 66.6%

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

5. Simplified 66.6%

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

6. Final simplification 66.6%

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

Alternative 5: 66.5% 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 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. Add Preprocessing

3. Taylor expanded in x around 0 66.5%

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

   1. +-commutative 66.5%

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

5. Simplified 66.5%

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

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 2024034 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))