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

Percentage Accurate: 99.7% → 99.8%

Time: 5.5s

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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0011:\\ \;\;\;\;\frac{x}{2 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0011)
   (/ x (+ 2.0 (* x (+ 0.5 (* x (- (* x 0.0625) 0.125))))))
   (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0011) {
		tmp = x / (2.0 + (x * (0.5 + (x * ((x * 0.0625) - 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 <= 0.0011d0) then
        tmp = x / (2.0d0 + (x * (0.5d0 + (x * ((x * 0.0625d0) - 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 <= 0.0011) {
		tmp = x / (2.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125)))));
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0011:
		tmp = x / (2.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125)))))
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0011)
		tmp = Float64(x / Float64(2.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.0625) - 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 <= 0.0011)
		tmp = x / (2.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125)))));
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0011], N[(x / N[(2.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $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 < 0.00110000000000000007

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{x}{\color{blue}{2 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)}} \]

   if 0.00110000000000000007 < x

   1. Initial program 99.2%

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

      1. add-log-exp 6.6%

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

      2. *-un-lft-identity 6.6%

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

      3. log-prod 6.6%

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

      4. metadata-eval 6.6%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{x}{1 + \sqrt{x + 1}}}\right) \]

      5. add-log-exp 99.2%

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

      6. frac-2neg 99.2%

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

      7. distribute-frac-neg2 99.2%

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

      8. neg-sub0 99.2%

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

      9. metadata-eval 99.2%

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

      10. associate--r+ 99.1%

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

      11. metadata-eval 99.1%

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

      12. +-commutative 99.1%

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

      13. add-sqr-sqrt 99.8%

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

      14. flip-- 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0 + \left(-\left(1 - \sqrt{x + 1}\right)\right)} \]
   5. Step-by-step derivation

      1. unsub-neg 99.9%

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

      2. associate--r- 99.9%

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

      3. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0011:\\ \;\;\;\;\frac{x}{2 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3:\\ \;\;\;\;\frac{x}{2 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 + \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.0)
   (/ x (+ 2.0 (* x (+ 0.5 (* x (- (* x 0.0625) 0.125))))))
   (+ -1.0 (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.0) {
		tmp = x / (2.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125)))));
	} else {
		tmp = -1.0 + sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.0d0) then
        tmp = x / (2.0d0 + (x * (0.5d0 + (x * ((x * 0.0625d0) - 0.125d0)))))
    else
        tmp = (-1.0d0) + sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.0) {
		tmp = x / (2.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125)))));
	} else {
		tmp = -1.0 + Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.0:
		tmp = x / (2.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125)))))
	else:
		tmp = -1.0 + math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.0)
		tmp = Float64(x / Float64(2.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.0625) - 0.125))))));
	else
		tmp = Float64(-1.0 + sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.0)
		tmp = x / (2.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125)))));
	else
		tmp = -1.0 + sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.0], N[(x / N[(2.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.8%

      \[\leadsto \frac{x}{\color{blue}{2 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)}} \]

   if 3 < x

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3:\\ \;\;\;\;\frac{x}{2 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 + \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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. Add Preprocessing

Alternative 4: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;\frac{x}{2 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.7)
   (/ x (+ 2.0 (* x (+ 0.5 (* x (- (* x 0.0625) 0.125))))))
   (sqrt x)))

C

double code(double x) {
	double tmp;
	if (x <= 3.7) {
		tmp = x / (2.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125)))));
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.7d0) then
        tmp = x / (2.0d0 + (x * (0.5d0 + (x * ((x * 0.0625d0) - 0.125d0)))))
    else
        tmp = sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.7) {
		tmp = x / (2.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125)))));
	} else {
		tmp = Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.7:
		tmp = x / (2.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125)))))
	else:
		tmp = math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.7)
		tmp = Float64(x / Float64(2.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.0625) - 0.125))))));
	else
		tmp = sqrt(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.7)
		tmp = x / (2.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125)))));
	else
		tmp = sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.7], N[(x / N[(2.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.7000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.8%

      \[\leadsto \frac{x}{\color{blue}{2 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)}} \]

   if 3.7000000000000002 < x

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf 96.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;\frac{x}{2 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% 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. Add Preprocessing

3. Taylor expanded in x around 0 63.5%

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

   1. +-commutative 63.5%

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

5. Simplified 63.5%

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

6. Final simplification 63.5%

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

Alternative 6: 67.6% 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 62.9%

   \[\leadsto \frac{x}{\color{blue}{2}} \]
4. Add Preprocessing

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 99.7%

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

3. Taylor expanded in x around 0 61.9%

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

   1. +-commutative 61.9%

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

   2. *-commutative 61.9%

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

5. Simplified 61.9%

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

6. Taylor expanded in x around 0 63.5%

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

7. Taylor expanded in x around inf 5.0%

   \[\leadsto \color{blue}{2} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024180 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))