Main:bigenough3 from C

Percentage Accurate: 53.9% → 99.8%

Time: 7.8s

Alternatives: 11

Speedup: 0.5×

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: 53.9% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {\left({\left(\sqrt{{\left(1 + x\right)}^{-1}}\right)}^{-1} + \sqrt{x}\right)}^{-1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

   6. inv-pow N/A

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

   7. metadata-eval N/A

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

   8. lower-pow.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. metadata-eval N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. rem-square-sqrt N/A

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

4. Applied rewrites 54.2%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow-1 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate--l+ N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 99.7

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 99.7

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

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

   4. clear-num N/A

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

   14. lift-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 73.3

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

8. Applied rewrites 73.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-fma.f64 N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lift--.f64 N/A

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

   8. flip-+ N/A

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

   9. +-commutative N/A

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

   10. lower-/.f64 N/A

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

   11. lower-+.f64 99.7

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

    \[\leadsto {\left({\left(\sqrt{{\left(1 + x\right)}^{-1}}\right)}^{-1} + \sqrt{x}\right)}^{-1} \]
12. Add Preprocessing

Alternative 2: 99.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 4e-5) {
		tmp = sqrt(pow(x, -1.0)) * 0.5;
	} 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((x + 1.0d0)) - sqrt(x)
    if (t_0 <= 4d-5) then
        tmp = sqrt((x ** (-1.0d0))) * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-5], N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.5), $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 4.6%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

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

   1. Initial program 99.7%

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

4. Final simplification 99.6%

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

Alternative 3: 98.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 6.5%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 98.0

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

   5. Applied rewrites 98.0%

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

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

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 98.5%

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

Alternative 4: 59.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.2:\\ \;\;\;\;\sqrt{{x}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{fma}\left(-0.5, x, \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 0.2)
   (sqrt (pow x -1.0))
   (- 1.0 (fma -0.5 x (sqrt x)))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 0.2) {
		tmp = sqrt(pow(x, -1.0));
	} else {
		tmp = 1.0 - fma(-0.5, x, sqrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 0.2)
		tmp = sqrt((x ^ -1.0));
	else
		tmp = Float64(1.0 - fma(-0.5, x, sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.2], N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision], N[(1.0 - N[(-0.5 * x + 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)) < 0.20000000000000001

   1. Initial program 5.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. inv-pow N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

   4. Applied rewrites 7.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 18.8

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

   7. Applied rewrites 18.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \sqrt{\frac{1}{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.7%

       \[\leadsto \sqrt{\frac{1}{x}} \]

   if 0.20000000000000001 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 98.4

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

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.1%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.2:\\ \;\;\;\;\sqrt{{x}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{fma}\left(-0.5, x, \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{-1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

   6. inv-pow N/A

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

   7. metadata-eval N/A

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

   8. lower-pow.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. metadata-eval N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. rem-square-sqrt N/A

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

4. Applied rewrites 54.2%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow-1 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate--l+ N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 99.7

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 99.7

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

   \[\leadsto {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{-1} \]
8. Add Preprocessing

Alternative 6: 58.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{x} + 1\right)}^{-1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

   6. inv-pow N/A

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

   7. metadata-eval N/A

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

   8. lower-pow.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. metadata-eval N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. rem-square-sqrt N/A

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

4. Applied rewrites 54.2%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-sqrt.f64 58.5

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

7. Applied rewrites 58.5%

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

8. Final simplification 58.5%

   \[\leadsto {\left(\sqrt{x} + 1\right)}^{-1} \]
9. Add Preprocessing

Alternative 7: 59.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.2:\\ \;\;\;\;\frac{\sqrt{x} - 1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{fma}\left(-0.5, x, \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 0.2)
   (/ (- (sqrt x) 1.0) x)
   (- 1.0 (fma -0.5 x (sqrt x)))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 0.2) {
		tmp = (sqrt(x) - 1.0) / x;
	} else {
		tmp = 1.0 - fma(-0.5, x, sqrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 0.2)
		tmp = Float64(Float64(sqrt(x) - 1.0) / x);
	else
		tmp = Float64(1.0 - fma(-0.5, x, sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.2], N[(N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(-0.5 * x + 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)) < 0.20000000000000001

   1. Initial program 5.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. inv-pow N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

   4. Applied rewrites 7.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 18.8

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

   7. Applied rewrites 18.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 18.8%

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

   if 0.20000000000000001 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 98.4

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

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.1%

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

Alternative 8: 52.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.3%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-sqrt.f64 50.1

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

5. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 50.1%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 51.6%

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

Alternative 9: 50.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 49.9%

   \[\leadsto \color{blue}{1} - \sqrt{x} \]
6. Add Preprocessing

Alternative 10: 48.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 1 - \left(0.125 \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* (* 0.125 x) x)))

C

double code(double x) {
	return 1.0 - ((0.125 * x) * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - ((0.125d0 * x) * x)
end function

Java

public static double code(double x) {
	return 1.0 - ((0.125 * x) * x);
}

Python

def code(x):
	return 1.0 - ((0.125 * x) * x)

Julia

function code(x)
	return Float64(1.0 - Float64(Float64(0.125 * x) * x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - ((0.125 * x) * x);
end

Wolfram

code[x_] := N[(1.0 - N[(N[(0.125 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-sqrt.f64 50.1

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

5. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 50.1%

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

8. Taylor expanded in x around inf

   \[\leadsto 1 - \frac{1}{8} \cdot \color{blue}{{x}^{2}} \]
9. Step-by-step derivation

10. Applied rewrites 48.7%

    \[\leadsto 1 - \left(0.125 \cdot x\right) \cdot \color{blue}{x} \]
11. Add Preprocessing

Alternative 11: 1.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot -0.125 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x x) -0.125))

C

double code(double x) {
	return (x * x) * -0.125;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (-0.125d0)
end function

Java

public static double code(double x) {
	return (x * x) * -0.125;
}

Python

def code(x):
	return (x * x) * -0.125

Julia

function code(x)
	return Float64(Float64(x * x) * -0.125)
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * -0.125;
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * -0.125), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-sqrt.f64 50.1

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

5. Applied rewrites 50.1%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{-1}{8} \cdot \color{blue}{{x}^{2}} \]
7. Step-by-step derivation

8. Applied rewrites 1.9%

   \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{-0.125} \]
9. Add Preprocessing

Developer Target 1: 99.8% 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 2024324 
(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)))