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

Percentage Accurate: 99.7% → 99.7%

Time: 7.4s

Alternatives: 9

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

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

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 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.5:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.0234375, 0.0625\right), -0.125\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 0.5)
   (* x (fma x (fma x (fma x -0.0234375 0.0625) -0.125) 0.5))
   (+ (sqrt x) -1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 0.5) {
		tmp = x * fma(x, fma(x, fma(x, -0.0234375, 0.0625), -0.125), 0.5);
	} else {
		tmp = sqrt(x) + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0)))) <= 0.5)
		tmp = Float64(x * fma(x, fma(x, fma(x, -0.0234375, 0.0625), -0.125), 0.5));
	else
		tmp = Float64(sqrt(x) + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], N[(x * N[(x * N[(x * N[(x * -0.0234375 + 0.0625), $MachinePrecision] + -0.125), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{2 + x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{-1}{8} \cdot x, 2\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, \color{blue}{\frac{-1}{8} \cdot x + \frac{1}{2}}, 2\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{8}} + \frac{1}{2}, 2\right)} \]

      5. lower-fma.f64 99.6

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.125, 0.5\right)}, 2\right)} \]

   5. Simplified 99.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 2\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{16} + \frac{-3}{128} \cdot x\right) - \frac{1}{8}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{16} + \frac{-3}{128} \cdot x\right) - \frac{1}{8}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{16} + \frac{-3}{128} \cdot x\right) - \frac{1}{8}\right) + \frac{1}{2}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{16} + \frac{-3}{128} \cdot x\right) - \frac{1}{8}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{16} + \frac{-3}{128} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, \frac{1}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, x \cdot \left(\frac{1}{16} + \frac{-3}{128} \cdot x\right) + \color{blue}{\frac{-1}{8}}, \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{16} + \frac{-3}{128} \cdot x, \frac{-1}{8}\right)}, \frac{1}{2}\right) \]

      7. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-3}{128} \cdot x + \frac{1}{16}}, \frac{-1}{8}\right), \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-3}{128}} + \frac{1}{16}, \frac{-1}{8}\right), \frac{1}{2}\right) \]

      9. lower-fma.f64 99.6

         \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.0234375, 0.0625\right)}, -0.125\right), 0.5\right) \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.0234375, 0.0625\right), -0.125\right), 0.5\right)} \]

   if 0.5 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 99.1

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

   5. Simplified 99.1%

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

Alternative 3: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right) + \frac{1}{2}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{16} \cdot x - \frac{1}{8}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{16} \cdot x + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{16}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), \frac{1}{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{16} + \color{blue}{\frac{-1}{8}}, \frac{1}{2}\right) \]

      7. lower-fma.f64 99.5

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

   5. Simplified 99.5%

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

   if 0.5 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 99.1

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

   5. Simplified 99.1%

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

Alternative 4: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.5:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.03125, -0.125\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 0.5)
   (* x (fma x (fma x 0.03125 -0.125) 0.5))
   (+ (sqrt x) -1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 0.5) {
		tmp = x * fma(x, fma(x, 0.03125, -0.125), 0.5);
	} else {
		tmp = sqrt(x) + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0)))) <= 0.5)
		tmp = Float64(x * fma(x, fma(x, 0.03125, -0.125), 0.5));
	else
		tmp = Float64(sqrt(x) + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], N[(x * N[(x * N[(x * 0.03125 + -0.125), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.5

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0.5, x, 2\right)}} \]

   5. Simplified 99.5%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0.5, x, 2\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{32} \cdot x - \frac{1}{8}\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{32} \cdot x - \frac{1}{8}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{32} \cdot x - \frac{1}{8}\right) + \frac{1}{2}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{32} \cdot x - \frac{1}{8}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{32} \cdot x + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{32}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), \frac{1}{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{32} + \color{blue}{\frac{-1}{8}}, \frac{1}{2}\right) \]

      7. lower-fma.f64 99.5

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.03125, -0.125\right)}, 0.5\right) \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.03125, -0.125\right), 0.5\right)} \]

   if 0.5 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 99.1

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

   5. Simplified 99.1%

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

Alternative 5: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{8} \cdot x + \frac{1}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{x \cdot \frac{-1}{8}} + \frac{1}{2}\right) \]

      4. lower-fma.f64 99.5

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

   5. Simplified 99.5%

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

   if 0.5 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 99.1

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

   5. Simplified 99.1%

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

Alternative 6: 98.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{8} \cdot x + \frac{1}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{x \cdot \frac{-1}{8}} + \frac{1}{2}\right) \]

      4. lower-fma.f64 99.5

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

   5. Simplified 99.5%

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

   if 0.5 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 96.2

         \[\leadsto \color{blue}{\sqrt{x}} \]

   5. Simplified 96.2%

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

Alternative 7: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.5:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 98.8

         \[\leadsto \color{blue}{0.5 \cdot x} \]

   5. Simplified 98.8%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if 0.5 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 96.2

         \[\leadsto \color{blue}{\sqrt{x}} \]

   5. Simplified 96.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.5:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.2% accurate, 4.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x 0.5))

C

double code(double x) {
	return x * 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return x * 0.5

Julia

function code(x)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.5;
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 69.7

      \[\leadsto \color{blue}{0.5 \cdot x} \]

5. Simplified 69.7%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

6. Final simplification 69.7%

   \[\leadsto x \cdot 0.5 \]
7. Add Preprocessing

Alternative 9: 4.9% accurate, 28.0× 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

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

   1. +-commutative N/A

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

   2. lower-fma.f64 70.2

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0.5, x, 2\right)}} \]

5. Simplified 70.2%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0.5, x, 2\right)}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{2} \]
7. Step-by-step derivation

8. Simplified 4.8%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))