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

Percentage Accurate: 99.7% → 99.7%

Time: 4.5s

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 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e-5) (/ x (+ (* x 0.5) 2.0)) (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 1e-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 <= 1d-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 <= 1e-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 <= 1e-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 <= 1e-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 <= 1e-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, 1e-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 < 1.00000000000000008e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   if 1.00000000000000008e-5 < x

   1. Initial program 99.3%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{x}{-x} \cdot \left(1 - \sqrt{x + 1}\right)} \]
   5. 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. neg-sub0 99.8%

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

      7. associate--r- 99.8%

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

      8. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.7%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{x}{x \cdot 0.5 + 2}\\ \mathbf{else}:\\ \;\;\;\;-1 + \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.0) (/ x (+ (* x 0.5) 2.0)) (+ -1.0 (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = x / ((x * 0.5) + 2.0);
	} else {
		tmp = -1.0 + sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.0d0) then
        tmp = x / ((x * 0.5d0) + 2.0d0)
    else
        tmp = (-1.0d0) + sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = x / ((x * 0.5) + 2.0);
	} else {
		tmp = -1.0 + Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.0:
		tmp = x / ((x * 0.5) + 2.0)
	else:
		tmp = -1.0 + math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(x / Float64(Float64(x * 0.5) + 2.0));
	else
		tmp = Float64(-1.0 + sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.0)
		tmp = x / ((x * 0.5) + 2.0);
	else
		tmp = -1.0 + sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.0], N[(x / N[(N[(x * 0.5), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.4%

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

      1. +-commutative 99.4%

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

   5. Simplified 99.4%

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

   if 4 < x

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. flip-+ 99.3%

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

      3. add-sqr-sqrt 100.0%

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

      4. add-exp-log 90.5%

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

      5. +-commutative 90.5%

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

      6. log1p-udef 90.5%

         \[\leadsto \frac{x}{\frac{e^{\color{blue}{\mathsf{log1p}\left(x\right)}} - 1 \cdot 1}{\sqrt{x + 1} - 1}} \]

      7. metadata-eval 90.5%

         \[\leadsto \frac{x}{\frac{e^{\mathsf{log1p}\left(x\right)} - \color{blue}{1}}{\sqrt{x + 1} - 1}} \]

      8. expm1-udef 90.5%

         \[\leadsto \frac{x}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}}{\sqrt{x + 1} - 1}} \]

      9. expm1-log1p-u 100.0%

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

      10. pow1/2 100.0%

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

      11. pow-to-exp 92.0%

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

      12. +-commutative 92.0%

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

      13. log1p-udef 92.0%

          \[\leadsto \frac{x}{\frac{x}{e^{\color{blue}{\mathsf{log1p}\left(x\right)} \cdot 0.5} - 1}} \]

      14. expm1-def 92.0%

          \[\leadsto \frac{x}{\frac{x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot 0.5\right)}}} \]

   4. Applied egg-rr 92.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{x}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot 0.5\right)}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 92.0%

         \[\leadsto \color{blue}{\frac{x}{x} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot 0.5\right)} \]

      2. *-inverses 92.0%

         \[\leadsto \color{blue}{1} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot 0.5\right) \]

      3. *-un-lft-identity 92.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot 0.5\right)} \]

   6. Applied egg-rr 92.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot 0.5\right)} \]

   7. Taylor expanded in x around -inf 0.0%

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

      1. sub-neg 0.0%

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

      2. metadata-eval 0.0%

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

      3. +-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. exp-prod 0.0%

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

      6. unpow1/2 0.0%

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

      7. mul-1-neg 0.0%

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

      8. unsub-neg 0.0%

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

      9. log-div 92.0%

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

      10. remove-double-neg 92.0%

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

      11. neg-mul-1 92.0%

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

      12. associate-/r* 92.0%

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

      13. metadata-eval 92.0%

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

      14. associate-/l* 92.0%

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

      15. neg-mul-1 92.0%

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

      16. remove-double-neg 92.0%

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

      17. /-rgt-identity 92.0%

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

      18. rem-exp-log 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{x}{x \cdot 0.5 + 2}\\ \mathbf{else}:\\ \;\;\;\;-1 + \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.3% 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. Add Preprocessing

3. Taylor expanded in x around 0 70.4%

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

   1. +-commutative 70.4%

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

5. Simplified 70.4%

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

6. Final simplification 70.4%

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

Alternative 5: 68.6% 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 69.4%

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

4. Final simplification 69.4%

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

Reproduce

herbie shell --seed 2024010 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))