Main:bigenough3 from C

Percentage Accurate: 54.1% → 99.7%

Time: 9.5s

Alternatives: 8

Speedup: 0.7×

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: 54.1% 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 47.4%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. lower-/.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lift-+.f64 N/A

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

   13. associate--l+ N/A

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

   14. metadata-eval N/A

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

   15. *-rgt-identity N/A

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

   16. lower-+.f64 N/A

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

   17. metadata-eval N/A

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

   18. *-rgt-identity N/A

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

   19. lower--.f64 N/A

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

   20. lower-+.f64 48.1

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

4. Applied rewrites 48.1%

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

   1. associate-+r- N/A

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

   4. +-inverses N/A

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

   5. metadata-eval 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 2e-8) (* 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 <= 2e-8) {
		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 <= 2d-8) 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 <= 2e-8) {
		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 <= 2e-8:
		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 <= 2e-8)
		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 <= 2e-8)
		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, 2e-8], 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)) < 2e-8

   1. Initial program 4.1%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   if 2e-8 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. Initial program 5.2%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 99.0

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

   5. Applied rewrites 99.0%

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

   if 0.0050000000000000001 < (-.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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)\right) - \sqrt{x}} \]
   7. 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. +-commutative N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-neg-out N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.4%

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

Alternative 4: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. Initial program 5.2%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 99.0

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

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
   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 98.8

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

   7. Applied rewrites 98.8%

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

   if 0.0050000000000000001 < (-.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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)\right) - \sqrt{x}} \]
   7. 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. +-commutative N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-neg-out N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.4%

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

Alternative 5: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. Initial program 5.2%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 99.0

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

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
   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 98.8

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

   7. Applied rewrites 98.8%

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

   if 0.0050000000000000001 < (-.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

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}} \]
   7. 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

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-neg-out N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. sub0-neg N/A

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

      12. associate--r- N/A

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

      13. neg-sub0 N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-sqrt.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 99.3%

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

Alternative 6: 52.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ 1.0 x)) (sqrt x)) 0.005) (sqrt x) (- 1.0 (sqrt x))))

C

double code(double x) {
	double tmp;
	if ((sqrt((1.0 + x)) - sqrt(x)) <= 0.005) {
		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)) <= 0.005d0) 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)) <= 0.005) {
		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)) <= 0.005:
		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)) <= 0.005)
		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)) <= 0.005)
		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], 0.005], 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)) < 0.0050000000000000001

   1. Initial program 5.2%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 4.2

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

   5. Applied rewrites 4.2%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-sqrt.f64 4.2

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

   8. Applied rewrites 4.2%

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

      1. lift-sqrt.f64 N/A

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

      2. neg-sub0 N/A

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

      3. flip3-- N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\frac{{0}^{3} - {\left(\sqrt{x}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{\color{blue}{0} - {\left(\sqrt{x}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{\color{blue}{\mathsf{neg}\left({\left(\sqrt{x}\right)}^{3}\right)}}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

      6. distribute-neg-frac N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\sqrt{x}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)}\right)\right)} \]

   10. Applied rewrites 4.2%

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

   11. Taylor expanded in x around 0

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

       1. lower-sqrt.f64 5.1

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

   13. Applied rewrites 5.1%

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

   if 0.0050000000000000001 < (-.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 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.005:\\ \;\;\;\;\sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.4%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 46.8

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

5. Applied rewrites 46.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}} \]
7. 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

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

   4. associate-+l- N/A

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

   5. lower--.f64 N/A

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

   6. sub-neg N/A

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

   7. remove-double-neg N/A

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

   8. distribute-neg-out N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. sub0-neg N/A

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

   12. associate--r- N/A

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

   13. neg-sub0 N/A

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

   14. *-commutative N/A

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

   15. distribute-rgt-neg-in N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-sqrt.f64 46.8

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

8. Applied rewrites 46.8%

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

Alternative 8: 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 47.4%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 46.8

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

5. Applied rewrites 46.8%

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

6. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 3.2

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

8. Applied rewrites 3.2%

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

   1. lift-sqrt.f64 N/A

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

   2. neg-sub0 N/A

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

   3. flip3-- N/A

      \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\frac{{0}^{3} - {\left(\sqrt{x}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)}} \]

   4. metadata-eval N/A

      \[\leadsto x \cdot \frac{1}{2} + \frac{\color{blue}{0} - {\left(\sqrt{x}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   5. neg-sub0 N/A

      \[\leadsto x \cdot \frac{1}{2} + \frac{\color{blue}{\mathsf{neg}\left({\left(\sqrt{x}\right)}^{3}\right)}}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   6. distribute-neg-frac N/A

      \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\sqrt{x}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)}\right)\right)} \]

10. Applied rewrites 5.4%

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

11. Taylor expanded in x around 0

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

    1. lower-sqrt.f64 5.9

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

13. Applied rewrites 5.9%

    \[\leadsto \color{blue}{\sqrt{x}} \]
14. 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 2024216 
(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)))