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

Percentage Accurate: 99.7% → 99.7%

Time: 5.4s

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000226:\\ \;\;\;\;\frac{x}{2 + x \cdot \left(0.5 + x \cdot -0.125\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000226)
   (/ x (+ 2.0 (* x (+ 0.5 (* x -0.125)))))
   (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.000226) {
		tmp = x / (2.0 + (x * (0.5 + (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 <= 0.000226d0) then
        tmp = x / (2.0d0 + (x * (0.5d0 + (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 <= 0.000226) {
		tmp = x / (2.0 + (x * (0.5 + (x * -0.125))));
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000226:
		tmp = x / (2.0 + (x * (0.5 + (x * -0.125))))
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000226)
		tmp = Float64(x / Float64(2.0 + Float64(x * Float64(0.5 + 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 <= 0.000226)
		tmp = x / (2.0 + (x * (0.5 + (x * -0.125))));
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000226], N[(x / N[(2.0 + N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $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 < 2.25999999999999991e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

         \[\leadsto \frac{x}{2 + \color{blue}{\left(0.5 \cdot x + -0.125 \cdot {x}^{2}\right)}} \]

      2. unpow2 99.7%

         \[\leadsto \frac{x}{2 + \left(0.5 \cdot x + -0.125 \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

      3. associate-*r* 99.7%

         \[\leadsto \frac{x}{2 + \left(0.5 \cdot x + \color{blue}{\left(-0.125 \cdot x\right) \cdot x}\right)} \]

      4. distribute-rgt-out 99.7%

         \[\leadsto \frac{x}{2 + \color{blue}{x \cdot \left(0.5 + -0.125 \cdot x\right)}} \]

   4. Simplified 99.7%

      \[\leadsto \frac{x}{\color{blue}{2 + x \cdot \left(0.5 + -0.125 \cdot x\right)}} \]

   if 2.25999999999999991e-4 < x

   1. Initial program 99.3%

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

      1. expm1-log1p-u 92.8%

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

      2. expm1-udef 92.8%

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

      3. div-inv 92.8%

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

      4. div-inv 92.8%

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

      5. +-commutative 92.8%

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

      6. add-sqr-sqrt 92.8%

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

      7. hypot-1-def 92.8%

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

   3. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{1 + \mathsf{hypot}\left(1, \sqrt{x}\right)}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 92.8%

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

      2. expm1-log1p 99.3%

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

   5. Simplified 99.3%

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

      1. flip-+ 99.3%

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

      2. clear-num 99.1%

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

      3. associate-/r/ 99.2%

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

      4. hypot-udef 99.2%

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

      5. metadata-eval 99.2%

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

      6. add-sqr-sqrt 99.1%

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

      7. +-commutative 99.1%

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

      8. metadata-eval 99.1%

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

      9. hypot-udef 99.1%

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

      10. hypot-udef 99.1%

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

      11. rem-square-sqrt 99.1%

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

      12. metadata-eval 99.1%

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

      13. add-sqr-sqrt 99.8%

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

      14. associate--r+ 99.8%

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

      15. metadata-eval 99.8%

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

      16. neg-sub0 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. associate-*r/ 68.4%

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

      2. associate-*l/ 100.0%

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

      3. /-rgt-identity 100.0%

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

      4. remove-double-neg 100.0%

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

      5. distribute-frac-neg 100.0%

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

      6. *-inverses 100.0%

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

      7. metadata-eval 100.0%

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

      8. neg-mul-1 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate--r- 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000226:\\ \;\;\;\;\frac{x}{2 + x \cdot \left(0.5 + x \cdot -0.125\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

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: 68.6% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{2 + x \cdot 0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ 2.0 (* x 0.5))))

C

double code(double x) {
	return x / (2.0 + (x * 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (2.0d0 + (x * 0.5d0))
end function

Java

public static double code(double x) {
	return x / (2.0 + (x * 0.5));
}

Python

def code(x):
	return x / (2.0 + (x * 0.5))

Julia

function code(x)
	return Float64(x / Float64(2.0 + Float64(x * 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = x / (2.0 + (x * 0.5));
end

Wolfram

code[x_] := N[(x / N[(2.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 66.4%

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

   1. +-commutative 66.4%

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

4. Simplified 66.4%

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

5. Final simplification 66.4%

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

Alternative 4: 67.9% 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 65.5%

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

3. Final simplification 65.5%

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

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 66.4%

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

   1. +-commutative 66.4%

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

4. Simplified 66.4%

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

   1. frac-2neg 66.4%

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

   2. clear-num 66.3%

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

   3. associate-/r/ 66.4%

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

   4. neg-sub0 66.4%

      \[\leadsto \frac{1}{\color{blue}{0 - \left(0.5 \cdot x + 2\right)}} \cdot \left(-x\right) \]

   5. +-commutative 66.4%

      \[\leadsto \frac{1}{0 - \color{blue}{\left(2 + 0.5 \cdot x\right)}} \cdot \left(-x\right) \]

   6. associate--r+ 66.4%

      \[\leadsto \frac{1}{\color{blue}{\left(0 - 2\right) - 0.5 \cdot x}} \cdot \left(-x\right) \]

   7. metadata-eval 66.4%

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

   8. *-commutative 66.4%

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

6. Applied egg-rr 66.4%

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

7. Taylor expanded in x around inf 5.1%

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

8. Final simplification 5.1%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))