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

Percentage Accurate: 99.7% → 99.7%

Time: 2.7s

Alternatives: 4

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000136:\\ \;\;\;\;x \cdot \left(0.5 + \left(x \cdot \left(x \cdot 0.0625\right) + x \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000136)
   (* x (+ 0.5 (+ (* x (* x 0.0625)) (* x -0.125))))
   (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.000136) {
		tmp = x * (0.5 + ((x * (x * 0.0625)) + (x * -0.125)));
	} 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 <= 0.000136d0) then
        tmp = x * (0.5d0 + ((x * (x * 0.0625d0)) + (x * (-0.125d0))))
    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 <= 0.000136) {
		tmp = x * (0.5 + ((x * (x * 0.0625)) + (x * -0.125)));
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000136:
		tmp = x * (0.5 + ((x * (x * 0.0625)) + (x * -0.125)))
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000136)
		tmp = Float64(x * Float64(0.5 + Float64(Float64(x * Float64(x * 0.0625)) + Float64(x * -0.125))));
	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 <= 0.000136)
		tmp = x * (0.5 + ((x * (x * 0.0625)) + (x * -0.125)));
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000136], N[(x * N[(0.5 + N[(N[(x * N[(x * 0.0625), $MachinePrecision]), $MachinePrecision] + N[(x * -0.125), $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 < 1.36e-4

   1. Initial program 100.0%

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

      1. flip-+ 3.2%

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

      2. metadata-eval 3.2%

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

      3. add-sqr-sqrt 3.2%

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

      4. +-commutative 3.2%

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

      5. associate--r+ 8.2%

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

      6. metadata-eval 8.2%

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

      7. neg-sub0 8.2%

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

      8. associate-/r/ 8.2%

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

   3. Applied egg-rr 8.2%

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

      1. remove-double-neg 8.2%

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

      2. distribute-frac-neg 8.2%

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

      3. *-inverses 8.2%

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

      4. metadata-eval 8.2%

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

      5. neg-mul-1 8.2%

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

      6. sub-neg 8.2%

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

      7. +-commutative 8.2%

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

      8. distribute-neg-in 8.2%

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

      9. remove-double-neg 8.2%

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

      10. metadata-eval 8.2%

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

   5. Simplified 8.2%

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

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{-0.125 \cdot {x}^{2} + \left(0.5 \cdot x + 0.0625 \cdot {x}^{3}\right)} \]
   7. Step-by-step derivation

      1. unpow2 99.6%

         \[\leadsto -0.125 \cdot \color{blue}{\left(x \cdot x\right)} + \left(0.5 \cdot x + 0.0625 \cdot {x}^{3}\right) \]

      2. *-commutative 99.6%

         \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot -0.125} + \left(0.5 \cdot x + 0.0625 \cdot {x}^{3}\right) \]

      3. associate-*r* 99.6%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.125\right)} + \left(0.5 \cdot x + 0.0625 \cdot {x}^{3}\right) \]

      4. +-commutative 99.6%

         \[\leadsto \color{blue}{\left(0.5 \cdot x + 0.0625 \cdot {x}^{3}\right) + x \cdot \left(x \cdot -0.125\right)} \]

      5. associate-+l+ 99.6%

         \[\leadsto \color{blue}{0.5 \cdot x + \left(0.0625 \cdot {x}^{3} + x \cdot \left(x \cdot -0.125\right)\right)} \]

      6. *-commutative 99.6%

         \[\leadsto \color{blue}{x \cdot 0.5} + \left(0.0625 \cdot {x}^{3} + x \cdot \left(x \cdot -0.125\right)\right) \]

      7. *-commutative 99.6%

         \[\leadsto x \cdot 0.5 + \left(\color{blue}{{x}^{3} \cdot 0.0625} + x \cdot \left(x \cdot -0.125\right)\right) \]

      8. unpow3 99.6%

         \[\leadsto x \cdot 0.5 + \left(\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot 0.0625 + x \cdot \left(x \cdot -0.125\right)\right) \]

      9. associate-*r* 99.6%

         \[\leadsto x \cdot 0.5 + \left(\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot 0.0625\right)} + x \cdot \left(x \cdot -0.125\right)\right) \]

      10. associate-*r* 99.6%

          \[\leadsto x \cdot 0.5 + \left(\left(x \cdot x\right) \cdot \left(x \cdot 0.0625\right) + \color{blue}{\left(x \cdot x\right) \cdot -0.125}\right) \]

      11. distribute-lft-in 99.6%

          \[\leadsto x \cdot 0.5 + \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot 0.0625 + -0.125\right)} \]

      12. fma-udef 99.6%

          \[\leadsto x \cdot 0.5 + \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, 0.0625, -0.125\right)} \]

      13. associate-*l* 99.6%

          \[\leadsto x \cdot 0.5 + \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.0625, -0.125\right)\right)} \]

      14. distribute-lft-out 99.6%

          \[\leadsto \color{blue}{x \cdot \left(0.5 + x \cdot \mathsf{fma}\left(x, 0.0625, -0.125\right)\right)} \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + x \cdot \mathsf{fma}\left(x, 0.0625, -0.125\right)\right)} \]
   9. Step-by-step derivation

      1. fma-udef 99.6%

         \[\leadsto x \cdot \left(0.5 + x \cdot \color{blue}{\left(x \cdot 0.0625 + -0.125\right)}\right) \]

      2. distribute-rgt-in 99.6%

         \[\leadsto x \cdot \left(0.5 + \color{blue}{\left(\left(x \cdot 0.0625\right) \cdot x + -0.125 \cdot x\right)}\right) \]

   10. Applied egg-rr 99.6%

       \[\leadsto x \cdot \left(0.5 + \color{blue}{\left(\left(x \cdot 0.0625\right) \cdot x + -0.125 \cdot x\right)}\right) \]

   if 1.36e-4 < x

   1. Initial program 99.2%

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

      1. flip-+ 99.2%

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

      2. metadata-eval 99.2%

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

      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/ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. *-inverses 100.0%

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

      4. metadata-eval 100.0%

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

      5. neg-mul-1 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. remove-double-neg 100.0%

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

      10. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000136:\\ \;\;\;\;x \cdot \left(0.5 + \left(x \cdot \left(x \cdot 0.0625\right) + x \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

Alternative 2: 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. Final simplification 99.7%

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

Alternative 3: 68.2% 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. Taylor expanded in x around 0 65.6%

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

3. Final simplification 65.6%

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

Alternative 4: 67.5% 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. Taylor expanded in x around 0 64.7%

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

3. Final simplification 64.7%

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

Reproduce

herbie shell --seed 2023217 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))