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

Percentage Accurate: 99.7% → 100.0%

Time: 6.0s

Alternatives: 10

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: 100.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ x (+ 1.0 (/ (+ 1.0 x) (sqrt (+ 1.0 x))))))

C

double code(double x) {
	return x / (1.0 + ((1.0 + x) / sqrt((1.0 + x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (1.0d0 + ((1.0d0 + x) / sqrt((1.0d0 + x))))
end function

Java

public static double code(double x) {
	return x / (1.0 + ((1.0 + x) / Math.sqrt((1.0 + x))));
}

Python

def code(x):
	return x / (1.0 + ((1.0 + x) / math.sqrt((1.0 + x))))

Julia

function code(x)
	return Float64(x / Float64(1.0 + Float64(Float64(1.0 + x) / sqrt(Float64(1.0 + x)))))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 + ((1.0 + x) / sqrt((1.0 + x))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-div N/A

      \[\leadsto \frac{x}{1 + \color{blue}{\frac{{\left(\sqrt{x + 1}\right)}^{2}}{{\left(\sqrt{x + 1}\right)}^{1}}}} \]

   4. pow2 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. unpow1 N/A

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

   9. lower-/.f64 100.0

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 100.0

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \frac{x}{1 + \color{blue}{\frac{1 + x}{\sqrt{1 + x}}}} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0390625, x, 0.0625\right) \cdot x - 0.125, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 0.005)
   (* (fma (- (* (fma -0.0390625 x 0.0625) x) 0.125) x 0.5) x)
   (- (sqrt x) 1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 0.005) {
		tmp = fma(((fma(-0.0390625, x, 0.0625) * x) - 0.125), x, 0.5) * x;
	} 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)))) <= 0.005)
		tmp = Float64(fma(Float64(Float64(fma(-0.0390625, x, 0.0625) * x) - 0.125), x, 0.5) * x);
	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], 0.005], N[(N[(N[(N[(N[(-0.0390625 * x + 0.0625), $MachinePrecision] * x), $MachinePrecision] - 0.125), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $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))))) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}\right) + \frac{1}{2}\right)} \cdot x \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}\right) \cdot x} + \frac{1}{2}\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}, x, \frac{1}{2}\right)} \cdot x \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}}, x, \frac{1}{2}\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) \cdot x} - \frac{1}{8}, x, \frac{1}{2}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) \cdot x} - \frac{1}{8}, x, \frac{1}{2}\right) \cdot x \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-5}{128} \cdot x + \frac{1}{16}\right)} \cdot x - \frac{1}{8}, x, \frac{1}{2}\right) \cdot x \]

      10. lower-fma.f64 99.8

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.0390625, x, 0.0625\right)} \cdot x - 0.125, x, 0.5\right) \cdot x \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0390625, x, 0.0625\right) \cdot x - 0.125, x, 0.5\right) \cdot x} \]

   if 0.0050000000000000001 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x, -0.125\right) \cdot x, x, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 0.005)
   (fma (* (fma 0.0625 x -0.125) x) x (* x 0.5))
   (- (sqrt x) 1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 0.005) {
		tmp = fma((fma(0.0625, x, -0.125) * x), x, (x * 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)))) <= 0.005)
		tmp = fma(Float64(fma(0.0625, x, -0.125) * x), x, Float64(x * 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], 0.005], N[(N[(N[(0.0625 * x + -0.125), $MachinePrecision] * x), $MachinePrecision] * x + N[(x * 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))))) < 0.0050000000000000001

   1. Initial program 100.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-div N/A

         \[\leadsto \frac{x}{1 + \color{blue}{\frac{{\left(\sqrt{x + 1}\right)}^{2}}{{\left(\sqrt{x + 1}\right)}^{1}}}} \]

      4. pow2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. unpow1 N/A

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

      9. lower-/.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{16} - \frac{1}{8} \cdot \frac{1}{x}\right), x, \frac{1}{2}\right) \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 99.7%

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

   12. Applied rewrites 99.7%

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

   if 0.0050000000000000001 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x, -0.125\right), x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 0.005)
   (* (fma (fma 0.0625 x -0.125) x 0.5) x)
   (- (sqrt x) 1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 0.005) {
		tmp = fma(fma(0.0625, x, -0.125), x, 0.5) * x;
	} 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)))) <= 0.005)
		tmp = Float64(fma(fma(0.0625, x, -0.125), x, 0.5) * x);
	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], 0.005], N[(N[(N[(0.0625 * x + -0.125), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $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))))) < 0.0050000000000000001

   1. Initial program 100.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-div N/A

         \[\leadsto \frac{x}{1 + \color{blue}{\frac{{\left(\sqrt{x + 1}\right)}^{2}}{{\left(\sqrt{x + 1}\right)}^{1}}}} \]

      4. pow2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. unpow1 N/A

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

      9. lower-/.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{16} - \frac{1}{8} \cdot \frac{1}{x}\right), x, \frac{1}{2}\right) \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 99.7%

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

   if 0.0050000000000000001 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.125, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 0.005)
   (fma (* x x) -0.125 (* 0.5 x))
   (- (sqrt x) 1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 0.005) {
		tmp = fma((x * x), -0.125, (0.5 * x));
	} 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)))) <= 0.005)
		tmp = fma(Float64(x * x), -0.125, Float64(0.5 * x));
	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], 0.005], N[(N[(x * x), $MachinePrecision] * -0.125 + N[(0.5 * x), $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))))) < 0.0050000000000000001

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

   if 0.0050000000000000001 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(-0.125, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 0.005)
   (* (fma -0.125 x 0.5) x)
   (- (sqrt x) 1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 0.005) {
		tmp = fma(-0.125, x, 0.5) * x;
	} 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)))) <= 0.005)
		tmp = Float64(fma(-0.125, x, 0.5) * x);
	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], 0.005], N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x), $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))))) < 0.0050000000000000001

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.0050000000000000001 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 7: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(-0.125, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 0.005)
   (* (fma -0.125 x 0.5) x)
   (sqrt x)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 0.005) {
		tmp = fma(-0.125, x, 0.5) * x;
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0)))) <= 0.005)
		tmp = Float64(fma(-0.125, x, 0.5) * x);
	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], 0.005], N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x), $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))))) < 0.0050000000000000001

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.0050000000000000001 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 99.1

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

   5. Applied rewrites 99.1%

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

Alternative 8: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.005:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 0.005) (* 0.5 x) (sqrt x)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 0.005) {
		tmp = 0.5 * x;
	} 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)))) <= 0.005d0) then
        tmp = 0.5d0 * x
    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)))) <= 0.005) {
		tmp = 0.5 * x;
	} else {
		tmp = Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x / (1.0 + math.sqrt((x + 1.0)))) <= 0.005:
		tmp = 0.5 * x
	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)))) <= 0.005)
		tmp = Float64(0.5 * x);
	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)))) <= 0.005)
		tmp = 0.5 * x;
	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], 0.005], N[(0.5 * x), $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))))) < 0.0050000000000000001

   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 98.1

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

   5. Applied rewrites 98.1%

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

   if 0.0050000000000000001 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 99.1

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

   5. Applied rewrites 99.1%

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

Alternative 9: 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 99.8%

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

Alternative 10: 68.0% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 0.5 * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
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 69.8

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

5. Applied rewrites 69.8%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))