Main:bigenough3 from C

Percentage Accurate: 52.9% → 99.7%

Time: 6.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1020:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, {x}^{-1.5}, \frac{-0.125}{\sqrt{x}}\right) + \mathsf{fma}\left(\sqrt{x}, 0.5, -0.0390625 \cdot {x}^{-2.5}\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1020.0)
   (- (sqrt (+ x 1.0)) (sqrt x))
   (/
    (+
     (fma 0.0625 (pow x -1.5) (/ -0.125 (sqrt x)))
     (fma (sqrt x) 0.5 (* -0.0390625 (pow x -2.5))))
    x)))

C

double code(double x) {
	double tmp;
	if (x <= 1020.0) {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	} else {
		tmp = (fma(0.0625, pow(x, -1.5), (-0.125 / sqrt(x))) + fma(sqrt(x), 0.5, (-0.0390625 * pow(x, -2.5)))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1020.0)
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	else
		tmp = Float64(Float64(fma(0.0625, (x ^ -1.5), Float64(-0.125 / sqrt(x))) + fma(sqrt(x), 0.5, Float64(-0.0390625 * (x ^ -2.5)))) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1020.0], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0625 * N[Power[x, -1.5], $MachinePrecision] + N[(-0.125 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 0.5 + N[(-0.0390625 * N[Power[x, -2.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1020

   1. Initial program 100.0%

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

   if 1020 < x

   1. Initial program 7.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{-5}{128} \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{-5}{128} \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x}} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.0390625, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, 0.0625, \mathsf{fma}\left(\sqrt{\frac{1}{x}}, -0.125, 0.5 \cdot \sqrt{x}\right)\right)\right)}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, {x}^{-1.5}, \frac{-0.125}{\sqrt{x}}\right) + \mathsf{fma}\left(\sqrt{x}, 0.5, -0.0390625 \cdot {x}^{-2.5}\right)}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1020:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{-2.5}, -0.0390625, \mathsf{fma}\left(0.0625, {x}^{-1.5}, \mathsf{fma}\left(\sqrt{x}, 0.5, \frac{-0.125}{\sqrt{x}}\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1020.0)
   (- (sqrt (+ x 1.0)) (sqrt x))
   (/
    (fma
     (pow x -2.5)
     -0.0390625
     (fma 0.0625 (pow x -1.5) (fma (sqrt x) 0.5 (/ -0.125 (sqrt x)))))
    x)))

C

double code(double x) {
	double tmp;
	if (x <= 1020.0) {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	} else {
		tmp = fma(pow(x, -2.5), -0.0390625, fma(0.0625, pow(x, -1.5), fma(sqrt(x), 0.5, (-0.125 / sqrt(x))))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1020.0)
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	else
		tmp = Float64(fma((x ^ -2.5), -0.0390625, fma(0.0625, (x ^ -1.5), fma(sqrt(x), 0.5, Float64(-0.125 / sqrt(x))))) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1020.0], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, -2.5], $MachinePrecision] * -0.0390625 + N[(0.0625 * N[Power[x, -1.5], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 0.5 + N[(-0.125 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1020

   1. Initial program 100.0%

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

   if 1020 < x

   1. Initial program 7.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{-5}{128} \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{-5}{128} \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x}} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.0390625, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, 0.0625, \mathsf{fma}\left(\sqrt{\frac{1}{x}}, -0.125, 0.5 \cdot \sqrt{x}\right)\right)\right)}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{-2.5}, -0.0390625, \mathsf{fma}\left(0.0625, {x}^{-1.5}, \mathsf{fma}\left(\sqrt{x}, 0.5, \frac{-0.125}{\sqrt{x}}\right)\right)\right)}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9000:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x}, 0.5, \mathsf{fma}\left(0.0625, {x}^{-1.5}, \frac{-0.125}{\sqrt{x}}\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 9000.0)
   (- (sqrt (+ x 1.0)) (sqrt x))
   (/ (fma (sqrt x) 0.5 (fma 0.0625 (pow x -1.5) (/ -0.125 (sqrt x)))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 9000.0) {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	} else {
		tmp = fma(sqrt(x), 0.5, fma(0.0625, pow(x, -1.5), (-0.125 / sqrt(x)))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 9000.0)
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	else
		tmp = Float64(fma(sqrt(x), 0.5, fma(0.0625, (x ^ -1.5), Float64(-0.125 / sqrt(x)))) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 9000.0], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 0.5 + N[(0.0625 * N[Power[x, -1.5], $MachinePrecision] + N[(-0.125 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9e3

   1. Initial program 100.0%

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

   if 9e3 < x

   1. Initial program 7.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, 0.0625, \mathsf{fma}\left(\sqrt{\frac{1}{x}}, -0.125, 0.5 \cdot \sqrt{x}\right)\right)}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{x}, 0.5, \mathsf{fma}\left(0.0625, {x}^{-1.5}, \frac{-0.125}{\sqrt{x}}\right)\right)}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 33000000.0)
   (- (sqrt (+ x 1.0)) (sqrt x))
   (* (sqrt (pow x -1.0)) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	} else {
		tmp = sqrt(pow(x, -1.0)) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 33000000.0d0) then
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    else
        tmp = sqrt((x ** (-1.0d0))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	} else {
		tmp = Math.sqrt(Math.pow(x, -1.0)) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 33000000.0:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	else:
		tmp = math.sqrt(math.pow(x, -1.0)) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 33000000.0)
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	else
		tmp = Float64(sqrt((x ^ -1.0)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 33000000.0)
		tmp = sqrt((x + 1.0)) - sqrt(x);
	else
		tmp = sqrt((x ^ -1.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 33000000.0], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e7

   1. Initial program 99.4%

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

   if 3.3e7 < x

   1. Initial program 5.6%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 98.8

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

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.22)
   (- (fma (fma -0.125 x 0.5) x 1.0) (sqrt x))
   (* (sqrt (pow x -1.0)) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.22) {
		tmp = fma(fma(-0.125, x, 0.5), x, 1.0) - sqrt(x);
	} else {
		tmp = sqrt(pow(x, -1.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.22)
		tmp = Float64(fma(fma(-0.125, x, 0.5), x, 1.0) - sqrt(x));
	else
		tmp = Float64(sqrt((x ^ -1.0)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.22], N[(N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.21999999999999997

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 1.21999999999999997 < x

   1. Initial program 7.2%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-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 97.7

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

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 98000:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x}, 0.5, \frac{-0.125}{\sqrt{x}}\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 98000.0)
   (- (sqrt (+ x 1.0)) (sqrt x))
   (/ (fma (sqrt x) 0.5 (/ -0.125 (sqrt x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 98000.0) {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	} else {
		tmp = fma(sqrt(x), 0.5, (-0.125 / sqrt(x))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 98000.0)
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	else
		tmp = Float64(fma(sqrt(x), 0.5, Float64(-0.125 / sqrt(x))) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 98000.0], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 0.5 + N[(-0.125 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 98000

   1. Initial program 99.9%

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

   if 98000 < x

   1. Initial program 6.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.4%

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

Alternative 7: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.22)
   (- (fma (fma -0.125 x 0.5) x 1.0) (sqrt x))
   (sqrt (/ 0.25 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.22) {
		tmp = fma(fma(-0.125, x, 0.5), x, 1.0) - sqrt(x);
	} else {
		tmp = sqrt((0.25 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.22)
		tmp = Float64(fma(fma(-0.125, x, 0.5), x, 1.0) - sqrt(x));
	else
		tmp = sqrt(Float64(0.25 / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.22], N[(N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.25 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.21999999999999997

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 1.21999999999999997 < x

   1. Initial program 7.2%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-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 97.7

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

   5. Applied rewrites 97.7%

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

   7. Applied rewrites 97.6%

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

   9. Applied rewrites 97.7%

      \[\leadsto \sqrt{\frac{0.25}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (fma 0.5 x (- 1.0 (sqrt x))) (sqrt (/ 0.25 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = fma(0.5, x, (1.0 - sqrt(x)));
	} else {
		tmp = sqrt((0.25 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = fma(0.5, x, Float64(1.0 - sqrt(x)));
	else
		tmp = sqrt(Float64(0.25 / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(0.5 * x + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.25 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}} \]
   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. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-sqrt.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 1 < x

   1. Initial program 7.2%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-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 97.7

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

   5. Applied rewrites 97.7%

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

   7. Applied rewrites 97.6%

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

   9. Applied rewrites 97.7%

      \[\leadsto \sqrt{\frac{0.25}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.8%

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

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

   4. lower--.f64 N/A

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

   5. lower-sqrt.f64 49.9

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

5. Applied rewrites 49.9%

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

Alternative 10: 49.4% 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 51.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.0%

   \[\leadsto \color{blue}{1} - \sqrt{x} \]
6. 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 2024339 
(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)))