Main:bigenough3 from C

Percentage Accurate: 52.9% → 99.8%

Time: 6.7s

Alternatives: 8

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.8% 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]

Derivation

1. Initial program 53.8%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. 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 54.1

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

   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 54.1

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

4. Applied rewrites 54.1%

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

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 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 2e-6) (* 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-6) {
		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-6) 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-6) {
		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-6:
		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-6)
		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-6)
		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-6], 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)) < 1.99999999999999991e-6

   1. Initial program 4.2%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

   if 1.99999999999999991e-6 < (-.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.0001:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 0.0001)
   (* 0.5 (sqrt (/ 1.0 x)))
   (- (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)) <= 0.0001) {
		tmp = 0.5 * sqrt((1.0 / x));
	} 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)) <= 0.0001)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	else
		tmp = Float64(fma(fma(-0.125, x, 0.5), x, 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], 0.0001], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 1.00000000000000005e-4

   1. Initial program 4.7%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 1.00000000000000005e-4 < (-.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. *-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 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.0001:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.0001:\\ \;\;\;\;\frac{0.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 0.0001)
   (/ 0.5 (sqrt x))
   (- (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)) <= 0.0001) {
		tmp = 0.5 / sqrt(x);
	} 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)) <= 0.0001)
		tmp = Float64(0.5 / sqrt(x));
	else
		tmp = Float64(fma(fma(-0.125, x, 0.5), x, 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], 0.0001], N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 1.00000000000000005e-4

   1. Initial program 4.7%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 5.4

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

      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 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   6. 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 99.3

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.2%

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

   if 1.00000000000000005e-4 < (-.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. *-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 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 5: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.0001:\\ \;\;\;\;\frac{0.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\sqrt{x} - 0.5 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 0.0001)
   (/ 0.5 (sqrt x))
   (- 1.0 (- (sqrt x) (* 0.5 x)))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 0.0001) {
		tmp = 0.5 / sqrt(x);
	} else {
		tmp = 1.0 - (sqrt(x) - (0.5 * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((sqrt((x + 1.0d0)) - sqrt(x)) <= 0.0001d0) then
        tmp = 0.5d0 / sqrt(x)
    else
        tmp = 1.0d0 - (sqrt(x) - (0.5d0 * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) <= 0.0001) {
		tmp = 0.5 / Math.sqrt(x);
	} else {
		tmp = 1.0 - (Math.sqrt(x) - (0.5 * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.sqrt((x + 1.0)) - math.sqrt(x)) <= 0.0001:
		tmp = 0.5 / math.sqrt(x)
	else:
		tmp = 1.0 - (math.sqrt(x) - (0.5 * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 0.0001)
		tmp = Float64(0.5 / sqrt(x));
	else
		tmp = Float64(1.0 - Float64(sqrt(x) - Float64(0.5 * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 0.0001)
		tmp = 0.5 / sqrt(x);
	else
		tmp = 1.0 - (sqrt(x) - (0.5 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.0001], N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] - N[(0.5 * 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)) < 1.00000000000000005e-4

   1. Initial program 4.7%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 5.4

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

      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 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   6. 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 99.3

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.2%

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

   if 1.00000000000000005e-4 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. 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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.4%

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

Alternative 6: 51.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (sqrt(x) - (0.5d0 * x))
end function

Java

public static double code(double x) {
	return 1.0 - (Math.sqrt(x) - (0.5 * x));
}

Python

def code(x):
	return 1.0 - (math.sqrt(x) - (0.5 * x))

Julia

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

MATLAB

function tmp = code(x)
	tmp = 1.0 - (sqrt(x) - (0.5 * x));
end

Wolfram

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

Derivation

1. Initial program 53.8%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 53.3

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

5. Applied rewrites 53.3%

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

7. Applied rewrites 53.3%

   \[\leadsto 1 - \color{blue}{\left(\sqrt{x} - 0.5 \cdot x\right)} \]
8. Add Preprocessing

Alternative 7: 51.3% 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 53.8%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 53.3

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

5. Applied rewrites 53.3%

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

Alternative 8: 49.3% 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 53.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.4%

   \[\leadsto \color{blue}{1} - \sqrt{x} \]
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 2024244 
(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)))