Main:bigenough3 from C

Percentage Accurate: 53.8% → 99.7%

Time: 8.5s

Alternatives: 10

Speedup: 1.1×

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: 53.8% 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

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

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]

Derivation

1. Initial program 57.4%

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

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 57.9

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 57.9

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

4. Applied rewrites 57.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 2e-8) (* 0.5 (sqrt (/ 1.0 x))) t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - 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((x + 1.0d0)) - 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((x + 1.0)) - 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((x + 1.0)) - 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(x + 1.0)) - 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((x + 1.0)) - 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[(x + 1.0), $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 4.0%

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

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

   5. Applied rewrites 99.8%

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

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

   1. Initial program 99.6%

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

4. Final simplification 99.7%

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

Alternative 3: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.4:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{fma}\left(\mathsf{fma}\left(0.125, x, -0.5\right), x, \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 0.4)
   (* 0.5 (sqrt (/ 1.0 x)))
   (- 1.0 (fma (fma 0.125 x -0.5) x (sqrt x)))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 0.4) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else {
		tmp = 1.0 - fma(fma(0.125, x, -0.5), x, sqrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 0.4)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	else
		tmp = Float64(1.0 - fma(fma(0.125, x, -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], 0.4], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.125 * x + -0.5), $MachinePrecision] * 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)) < 0.40000000000000002

   1. Initial program 6.1%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-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 98.2

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

   5. Applied rewrites 98.2%

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

   if 0.40000000000000002 < (-.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. *-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 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.4:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{fma}\left(\mathsf{fma}\left(0.125, x, -0.5\right), x, \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;1 - \mathsf{fma}\left(-0.5, x, \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} - 1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85) (- 1.0 (fma -0.5 x (sqrt x))) (/ (- (sqrt x) 1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = 1.0 - fma(-0.5, x, sqrt(x));
	} else {
		tmp = (sqrt(x) - 1.0) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(1.0 - fma(-0.5, x, sqrt(x)));
	else
		tmp = Float64(Float64(sqrt(x) - 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(1.0 - N[(-0.5 * x + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   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. *-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 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

   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 98.3%

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

   if 1.8500000000000001 < x

   1. Initial program 6.1%

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

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 7.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 7.2

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

   4. Applied rewrites 7.2%

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

   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%

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

Alternative 5: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;1 - \mathsf{fma}\left(-0.5, x, \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4) (- 1.0 (fma -0.5 x (sqrt x))) (sqrt (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0 - fma(-0.5, x, sqrt(x));
	} else {
		tmp = sqrt((1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = Float64(1.0 - fma(-0.5, x, sqrt(x)));
	else
		tmp = sqrt(Float64(1.0 / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], N[(1.0 - N[(-0.5 * x + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   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. *-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 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

   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 98.3%

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

   if 2.39999999999999991 < x

   1. Initial program 6.1%

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

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 7.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 7.2

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

   4. Applied rewrites 7.2%

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

   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%

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

Alternative 6: 58.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.4%

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

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 57.9

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 57.9

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

4. Applied rewrites 57.9%

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

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 61.5

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

7. Applied rewrites 61.5%

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

Alternative 7: 52.3% 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 57.4%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 54.5

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

5. Applied rewrites 54.5%

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

7. Applied rewrites 54.5%

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

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 55.8%

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

Alternative 8: 50.3% 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 57.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 53.7%

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

Alternative 9: 48.9% 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 57.4%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 54.5

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

5. Applied rewrites 54.5%

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

7. Applied rewrites 54.5%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 51.9%

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

Alternative 10: 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 57.4%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 54.5

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

5. Applied rewrites 54.5%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 2.0%

   \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{-0.125} \]
9. 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 2024243 
(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)))