Main:bigenough3 from C

Percentage Accurate: 54.9% → 99.7%

Time: 10.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

   1. lift-+.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. flip-- N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. rem-square-sqrt N/A

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

   11. flip-- N/A

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

   12. associate-/l/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 55.2%

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 4e-5) (* 0.5 (sqrt (/ 1.0 x))) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 4e-5) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 4d-5) then
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 4e-5) {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 4e-5:
		tmp = 0.5 * math.sqrt((1.0 / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 4e-5)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 4e-5)
		tmp = 0.5 * sqrt((1.0 / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-5], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 4.00000000000000033e-5

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 98.7

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]

   if 4.00000000000000033e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.01:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1 - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ 1.0 x)) (sqrt x)) 0.01)
   (* 0.5 (sqrt (/ 1.0 x)))
   (fma x (fma x -0.125 0.5) (- 1.0 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if ((sqrt((1.0 + x)) - sqrt(x)) <= 0.01) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else {
		tmp = fma(x, fma(x, -0.125, 0.5), (1.0 - sqrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) <= 0.01)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	else
		tmp = fma(x, fma(x, -0.125, 0.5), Float64(1.0 - sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.01], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(x * -0.125 + 0.5), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0100000000000000002

   1. Initial program 7.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 97.5

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]

   if 0.0100000000000000002 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 99.1

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1 - \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1 - \sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.01:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1 - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.01:\\ \;\;\;\;\frac{0.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1 - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ 1.0 x)) (sqrt x)) 0.01)
   (/ 0.5 (sqrt x))
   (fma x (fma x -0.125 0.5) (- 1.0 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if ((sqrt((1.0 + x)) - sqrt(x)) <= 0.01) {
		tmp = 0.5 / sqrt(x);
	} else {
		tmp = fma(x, fma(x, -0.125, 0.5), (1.0 - sqrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) <= 0.01)
		tmp = Float64(0.5 / sqrt(x));
	else
		tmp = fma(x, fma(x, -0.125, 0.5), Float64(1.0 - sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.01], N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(x * N[(x * -0.125 + 0.5), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0100000000000000002

   1. Initial program 7.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 97.5

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lift-sqrt.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 97.4

         \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x}}} \]

   7. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x}}} \]

   if 0.0100000000000000002 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 99.1

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1 - \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1 - \sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.01:\\ \;\;\;\;\frac{0.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1 - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.01:\\ \;\;\;\;\frac{0.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ 1.0 x)) (sqrt x)) 0.01)
   (/ 0.5 (sqrt x))
   (- (fma x (fma x -0.125 0.5) 1.0) (sqrt x))))

C

double code(double x) {
	double tmp;
	if ((sqrt((1.0 + x)) - sqrt(x)) <= 0.01) {
		tmp = 0.5 / sqrt(x);
	} else {
		tmp = fma(x, fma(x, -0.125, 0.5), 1.0) - sqrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) <= 0.01)
		tmp = Float64(0.5 / sqrt(x));
	else
		tmp = Float64(fma(x, fma(x, -0.125, 0.5), 1.0) - sqrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.01], N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * -0.125 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0100000000000000002

   1. Initial program 7.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 97.5

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lift-sqrt.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 97.4

         \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x}}} \]

   7. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x}}} \]

   if 0.0100000000000000002 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.125, 0.5\right)}, 1\right) - \sqrt{x} \]

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1\right)} - \sqrt{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.01:\\ \;\;\;\;\frac{0.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.01:\\ \;\;\;\;\sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ 1.0 x)) (sqrt x)) 0.01) (sqrt x) (- 1.0 (sqrt x))))

C

double code(double x) {
	double tmp;
	if ((sqrt((1.0 + x)) - sqrt(x)) <= 0.01) {
		tmp = sqrt(x);
	} else {
		tmp = 1.0 - sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((sqrt((1.0d0 + x)) - sqrt(x)) <= 0.01d0) then
        tmp = sqrt(x)
    else
        tmp = 1.0d0 - sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.sqrt((1.0 + x)) - Math.sqrt(x)) <= 0.01) {
		tmp = Math.sqrt(x);
	} else {
		tmp = 1.0 - Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.sqrt((1.0 + x)) - math.sqrt(x)) <= 0.01:
		tmp = math.sqrt(x)
	else:
		tmp = 1.0 - math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) <= 0.01)
		tmp = sqrt(x);
	else
		tmp = Float64(1.0 - sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((sqrt((1.0 + x)) - sqrt(x)) <= 0.01)
		tmp = sqrt(x);
	else
		tmp = 1.0 - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.01], N[Sqrt[x], $MachinePrecision], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0100000000000000002

   1. Initial program 7.4%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. flip-- N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 9.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.5

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

   7. Applied rewrites 65.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 65.6

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

   10. Applied rewrites 65.6%

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

   11. Taylor expanded in x around 0

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

       1. lower-sqrt.f64 5.5

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

   13. Applied rewrites 5.5%

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

   if 0.0100000000000000002 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.01:\\ \;\;\;\;\sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 98.7

         \[\leadsto \mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} \]

   if 1 < x

   1. Initial program 7.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 97.5

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lift-sqrt.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 97.4

         \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x}}} \]

   7. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 53.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 0.5 (- 1.0 (sqrt x))))

C

double code(double x) {
	return fma(x, 0.5, (1.0 - sqrt(x)));
}

Julia

function code(x)
	return fma(x, 0.5, Float64(1.0 - sqrt(x)))
end

Wolfram

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

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 52.0

      \[\leadsto \mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) \]

5. Applied rewrites 52.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} \]
6. Add Preprocessing

Alternative 9: 6.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt x))

C

double code(double x) {
	return sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(x)
end function

Java

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

Python

def code(x):
	return math.sqrt(x)

Julia

function code(x)
	return sqrt(x)
end

MATLAB

function tmp = code(x)
	tmp = sqrt(x);
end

Wolfram

code[x_] := N[Sqrt[x], $MachinePrecision]

Derivation

1. Initial program 54.0%

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

   1. lift-+.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. flip-- N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. rem-square-sqrt N/A

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

   11. flip-- N/A

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

   12. associate-/l/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 55.2%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 35.3

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

7. Applied rewrites 35.3%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 36.1

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

10. Applied rewrites 36.1%

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

11. Taylor expanded in x around 0

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

    1. lower-sqrt.f64 6.3

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

13. Applied rewrites 6.3%

    \[\leadsto \color{blue}{\sqrt{x}} \]
14. Add Preprocessing

Alternative 10: 3.5% accurate, 27.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 3.5

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

5. Applied rewrites 3.5%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. +-inverses 3.5

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 3.5%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ (sqrt (+ x 1)) (sqrt x))))

  (- (sqrt (+ x 1.0)) (sqrt x)))