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

Percentage Accurate: 99.7% → 99.8%

Time: 3.7s

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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000265:\\ \;\;\;\;\frac{x}{\left(4 - 0.25 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot 0.125 + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000265)
   (/ x (* (- 4.0 (* 0.25 (* x x))) (+ (* x 0.125) 0.5)))
   (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.000265) {
		tmp = x / ((4.0 - (0.25 * (x * x))) * ((x * 0.125) + 0.5));
	} 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 <= 0.000265d0) then
        tmp = x / ((4.0d0 - (0.25d0 * (x * x))) * ((x * 0.125d0) + 0.5d0))
    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 <= 0.000265) {
		tmp = x / ((4.0 - (0.25 * (x * x))) * ((x * 0.125) + 0.5));
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000265:
		tmp = x / ((4.0 - (0.25 * (x * x))) * ((x * 0.125) + 0.5))
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000265)
		tmp = Float64(x / Float64(Float64(4.0 - Float64(0.25 * Float64(x * x))) * Float64(Float64(x * 0.125) + 0.5)));
	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 <= 0.000265)
		tmp = x / ((4.0 - (0.25 * (x * x))) * ((x * 0.125) + 0.5));
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000265], N[(x / N[(N[(4.0 - N[(0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * 0.125), $MachinePrecision] + 0.5), $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 < 2.6499999999999999e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-+r+ 99.4%

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

      3. metadata-eval 99.4%

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

      4. flip-+ 99.4%

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

      5. metadata-eval 99.4%

         \[\leadsto \frac{x}{\frac{\color{blue}{4} - \left(0.5 \cdot x\right) \cdot \left(0.5 \cdot x\right)}{2 - 0.5 \cdot x}} \]

      6. swap-sqr 99.4%

         \[\leadsto \frac{x}{\frac{4 - \color{blue}{\left(0.5 \cdot 0.5\right) \cdot \left(x \cdot x\right)}}{2 - 0.5 \cdot x}} \]

      7. metadata-eval 99.4%

         \[\leadsto \frac{x}{\frac{4 - \color{blue}{0.25} \cdot \left(x \cdot x\right)}{2 - 0.5 \cdot x}} \]

      8. *-commutative 99.4%

         \[\leadsto \frac{x}{\frac{4 - 0.25 \cdot \left(x \cdot x\right)}{2 - \color{blue}{x \cdot 0.5}}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \frac{x}{\color{blue}{\frac{4 - 0.25 \cdot \left(x \cdot x\right)}{2 - x \cdot 0.5}}} \]
   5. Step-by-step derivation

      1. div-inv 99.4%

         \[\leadsto \frac{x}{\color{blue}{\left(4 - 0.25 \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{2 - x \cdot 0.5}}} \]

   6. Applied egg-rr 99.4%

      \[\leadsto \frac{x}{\color{blue}{\left(4 - 0.25 \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{2 - x \cdot 0.5}}} \]

   7. Taylor expanded in x around 0 99.8%

      \[\leadsto \frac{x}{\left(4 - 0.25 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(0.125 \cdot x + 0.5\right)}} \]

   if 2.6499999999999999e-4 < x

   1. Initial program 99.2%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. 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 99.9%

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

      4. +-commutative 99.9%

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

      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%

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

   3. Applied egg-rr 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. remove-double-neg 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. *-inverses 99.8%

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

      6. metadata-eval 99.8%

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

      7. distribute-lft-in 99.8%

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

      8. neg-mul-1 99.8%

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

      9. remove-double-neg 99.8%

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

      10. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000265:\\ \;\;\;\;\frac{x}{\left(4 - 0.25 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot 0.125 + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

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: 68.1% 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 69.7%

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

3. Final simplification 69.7%

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

Alternative 4: 67.4% 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 68.7%

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

3. Final simplification 68.7%

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

Alternative 5: 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. Taylor expanded in x around 0 69.7%

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

3. Taylor expanded in x around inf 4.8%

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

4. Final simplification 4.8%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023207 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))