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

Percentage Accurate: 99.7% → 99.7%

Time: 3.6s

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

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

FPCore

(FPCore (x) :precision binary64 (* x (/ 1.0 (+ 1.0 (sqrt (+ 1.0 x))))))

C

double code(double x) {
	return x * (1.0 / (1.0 + sqrt((1.0 + x))));
}

Fortran

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

Java

public static double code(double x) {
	return x * (1.0 / (1.0 + Math.sqrt((1.0 + x))));
}

Python

def code(x):
	return x * (1.0 / (1.0 + math.sqrt((1.0 + x))))

Julia

function code(x)
	return Float64(x * Float64(1.0 / Float64(1.0 + sqrt(Float64(1.0 + x)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 / (1.0 + sqrt((1.0 + x))));
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. clear-num 99.2%

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

   2. associate-/r/ 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{x \cdot 0.5 + 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.1e-5) (/ x (+ (* x 0.5) 2.0)) (+ (sqrt (+ 1.0 x)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 1.1e-5) {
		tmp = x / ((x * 0.5) + 2.0);
	} else {
		tmp = sqrt((1.0 + x)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.1d-5) then
        tmp = x / ((x * 0.5d0) + 2.0d0)
    else
        tmp = sqrt((1.0d0 + x)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.1e-5) {
		tmp = x / ((x * 0.5) + 2.0);
	} else {
		tmp = Math.sqrt((1.0 + x)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.1e-5:
		tmp = x / ((x * 0.5) + 2.0)
	else:
		tmp = math.sqrt((1.0 + x)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.1e-5)
		tmp = Float64(x / Float64(Float64(x * 0.5) + 2.0));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.1e-5)
		tmp = x / ((x * 0.5) + 2.0);
	else
		tmp = sqrt((1.0 + x)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.1e-5], N[(x / N[(N[(x * 0.5), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.8%

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

   if 1.1e-5 < x

   1. Initial program 99.0%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. 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 99.6%

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

      4. +-commutative 99.6%

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

      5. associate--r+ 99.6%

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

      6. metadata-eval 99.6%

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

      7. neg-sub0 99.6%

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

      8. associate-/r/ 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. remove-double-neg 99.6%

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

      4. distribute-frac-neg 99.6%

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

      5. *-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. distribute-lft-in 99.6%

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

      8. neg-mul-1 99.6%

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

      9. remove-double-neg 99.6%

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

      10. metadata-eval 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{x \cdot 0.5 + 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + -1\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ x (+ 1.0 (sqrt (+ 1.0 x)))))

C

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

Fortran

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

Java

public static double code(double x) {
	return x / (1.0 + Math.sqrt((1.0 + x)));
}

Python

def code(x):
	return x / (1.0 + math.sqrt((1.0 + x)))

Julia

function code(x)
	return Float64(x / Float64(1.0 + sqrt(Float64(1.0 + x))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Final simplification 99.6%

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

Alternative 4: 68.5% 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 99.6%

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

   1. clear-num 99.2%

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

   2. associate-/r/ 99.6%

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

3. Applied egg-rr 99.6%

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

   1. associate-*l/ 99.6%

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

   2. associate-/l* 99.2%

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

   3. expm1-log1p-u 96.4%

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

   4. expm1-udef 38.8%

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

   5. log1p-udef 38.8%

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

   6. +-commutative 38.8%

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

   7. add-exp-log 41.6%

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

   8. add-sqr-sqrt 41.7%

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

   9. metadata-eval 41.7%

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

   10. +-commutative 41.7%

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

   11. add-sqr-sqrt 41.6%

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

   12. add-exp-log 38.8%

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

   13. +-commutative 38.8%

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

   14. log1p-udef 38.8%

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

   15. metadata-eval 38.8%

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

   16. expm1-udef 96.4%

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

   17. expm1-log1p-u 99.2%

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

5. Applied egg-rr 99.2%

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

6. Taylor expanded in x around 0 65.8%

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

   1. associate-*r/ 65.8%

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

   2. metadata-eval 65.8%

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

8. Simplified 65.8%

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

9. Final simplification 65.8%

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

Alternative 5: 68.7% 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 99.6%

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

2. Taylor expanded in x around 0 66.2%

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

3. Final simplification 66.2%

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

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 99.6%

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

2. Taylor expanded in x around 0 65.3%

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

3. Final simplification 65.3%

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

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 99.6%

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

2. Taylor expanded in x around 0 66.2%

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

3. Taylor expanded in x around inf 5.0%

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

4. Final simplification 5.0%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))