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

Percentage Accurate: 99.7% → 99.7%

Time: 3.0s

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. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{x \cdot 0.5 + 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 9.8e-6) (/ x (+ (* x 0.5) 2.0)) (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 9.8e-6) {
		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 <= 9.8d-6) 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 <= 9.8e-6) {
		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 <= 9.8e-6:
		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 <= 9.8e-6)
		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 <= 9.8e-6)
		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, 9.8e-6], 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 < 9.79999999999999934e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.0%

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

   3. Taylor expanded in x around 0 99.0%

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

   if 9.79999999999999934e-6 < x

   1. Initial program 99.3%

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

      1. flip-+ 99.3%

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

      2. metadata-eval 99.3%

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

      3. add-sqr-sqrt 100.0%

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

      4. +-commutative 100.0%

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

      5. associate--r+ 100.0%

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

      6. metadata-eval 100.0%

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

      7. neg-sub0 100.0%

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

      8. associate-/r/ 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. distribute-rgt-neg-out 99.9%

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

      5. unsub-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-/r* 99.9%

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

      9. *-inverses 99.9%

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

      10. metadata-eval 99.9%

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

      11. neg-mul-1 99.9%

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

      12. *-commutative 99.9%

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

      13. associate-/r* 99.9%

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

      14. *-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. neg-mul-1 99.9%

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

   5. Simplified 99.9%

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

      1. sub-neg 99.9%

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

      2. neg-mul-1 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-mul-1 99.9%

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

      5. remove-double-neg 99.9%

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

      6. add-sqr-sqrt 99.1%

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

      7. fma-def 99.1%

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

      8. pow1/2 99.1%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{{\left(x + 1\right)}^{0.5}}}, \sqrt{\sqrt{x + 1}}, -1\right) \]

      9. sqrt-pow1 99.4%

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

      10. metadata-eval 99.4%

          \[\leadsto \mathsf{fma}\left({\left(x + 1\right)}^{\color{blue}{0.25}}, \sqrt{\sqrt{x + 1}}, -1\right) \]

      11. pow1/2 99.4%

          \[\leadsto \mathsf{fma}\left({\left(x + 1\right)}^{0.25}, \sqrt{\color{blue}{{\left(x + 1\right)}^{0.5}}}, -1\right) \]

      12. sqrt-pow1 99.1%

          \[\leadsto \mathsf{fma}\left({\left(x + 1\right)}^{0.25}, \color{blue}{{\left(x + 1\right)}^{\left(\frac{0.5}{2}\right)}}, -1\right) \]

      13. metadata-eval 99.1%

          \[\leadsto \mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{\color{blue}{0.25}}, -1\right) \]

   7. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, -1\right)} \]
   8. Step-by-step derivation

      1. fma-udef 99.1%

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{0.25} \cdot {\left(x + 1\right)}^{0.25} + -1} \]

      2. +-commutative 99.1%

         \[\leadsto \color{blue}{-1 + {\left(x + 1\right)}^{0.25} \cdot {\left(x + 1\right)}^{0.25}} \]

      3. pow-sqr 99.9%

         \[\leadsto -1 + \color{blue}{{\left(x + 1\right)}^{\left(2 \cdot 0.25\right)}} \]

      4. metadata-eval 99.9%

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

      5. unpow1/2 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{x \cdot 0.5 + 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

Alternative 3: 68.6% 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. Taylor expanded in x around 0 70.5%

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

3. Taylor expanded in x around 0 70.5%

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

4. Final simplification 70.5%

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

Alternative 4: 67.9% 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 99.8%

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

2. Taylor expanded in x around 0 69.3%

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

3. Final simplification 69.3%

   \[\leadsto x \cdot 0.5 \]

Alternative 5: 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 99.8%

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

2. Taylor expanded in x around 0 70.5%

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

3. Taylor expanded in x around inf 4.7%

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

4. Final simplification 4.7%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023255 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))