Main:bigenough3 from C

Percentage Accurate: 53.3% → 97.4%

Time: 7.9s

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: 53.3% 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: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 10^{-7}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x, -0.125\right), x, 0.5\right), x, 1 - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 1e-7)
   (* 0.5 (sqrt (/ 1.0 x)))
   (fma (fma (fma 0.0625 x -0.125) x 0.5) x (- 1.0 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 1e-7) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else {
		tmp = fma(fma(fma(0.0625, x, -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)) <= 1e-7)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	else
		tmp = fma(fma(fma(0.0625, x, -0.125), x, 0.5), x, Float64(1.0 - sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 1e-7], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0625 * x + -0.125), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 9.9999999999999995e-8

   1. Initial program 4.1%

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

   3. Taylor expanded in x around inf

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

      1. *-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 99.7

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

   5. Applied rewrites 99.7%

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

   if 9.9999999999999995e-8 < (-.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

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{16} \cdot x - \frac{1}{8}, x, \frac{1}{2}\right)}, x, 1 - \sqrt{x}\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{16} \cdot x + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) \]

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{16}, x, \frac{-1}{8}\right)}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, x, \frac{-1}{8}\right), x, \frac{1}{2}\right), x, \color{blue}{1 - \sqrt{x}}\right) \]

      12. lower-sqrt.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x, -0.125\right), x, 0.5\right), x, 1 - \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x, -0.125\right), x, 0.5\right), x, 1 - \sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 10^{-7}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x, -0.125\right), x, 0.5\right), x, 1 - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 10^{-7}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 1e-7)
   (* 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)) <= 1e-7) {
		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)) <= 1e-7)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	else
		tmp = fma(fma(-0.125, x, 0.5), x, Float64(1.0 - sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 1e-7], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 9.9999999999999995e-8

   1. Initial program 4.1%

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

   3. Taylor expanded in x around inf

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

      1. *-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 99.7

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

   5. Applied rewrites 99.7%

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

   if 9.9999999999999995e-8 < (-.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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 10^{-7}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 10^{-7}:\\ \;\;\;\;\frac{0.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 1e-7)
   (/ 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)) <= 1e-7) {
		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)) <= 1e-7)
		tmp = Float64(0.5 / sqrt(x));
	else
		tmp = fma(fma(-0.125, x, 0.5), x, Float64(1.0 - sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 1e-7], N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 9.9999999999999995e-8

   1. Initial program 4.1%

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

   3. Taylor expanded in x around inf

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

      1. *-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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.5%

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

   if 9.9999999999999995e-8 < (-.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 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 10^{-7}:\\ \;\;\;\;\frac{0.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 1e-7)
   (/ 0.5 (sqrt x))
   (fma 0.5 x (- 1.0 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 1e-7) {
		tmp = 0.5 / sqrt(x);
	} else {
		tmp = fma(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)) <= 1e-7)
		tmp = Float64(0.5 / sqrt(x));
	else
		tmp = fma(0.5, x, Float64(1.0 - sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 1e-7], N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * x + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 9.9999999999999995e-8

   1. Initial program 4.1%

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

   3. Taylor expanded in x around inf

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

      1. *-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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.5%

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

   if 9.9999999999999995e-8 < (-.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 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 5: 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 51.6%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 51.9

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

   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 51.9

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

4. Applied rewrites 51.9%

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

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 6: 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 51.6%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 51.5

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

5. Applied rewrites 51.5%

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

Alternative 7: 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 51.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 49.4%

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

Alternative 8: 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 51.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{16} \cdot x - \frac{1}{8}, x, \frac{1}{2}\right)}, x, 1 - \sqrt{x}\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{16} \cdot x + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) \]

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{16}, x, \frac{-1}{8}\right)}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) \]

   11. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, x, \frac{-1}{8}\right), x, \frac{1}{2}\right), x, \color{blue}{1 - \sqrt{x}}\right) \]

   12. lower-sqrt.f64 51.1

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x, -0.125\right), x, 0.5\right), x, 1 - \color{blue}{\sqrt{x}}\right) \]

5. Applied rewrites 51.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x, -0.125\right), x, 0.5\right), x, 1 - \sqrt{x}\right)} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 2.9%

   \[\leadsto \mathsf{fma}\left(0.0625, x, -0.125\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{8} \cdot \left(x \cdot x\right) \]
10. Step-by-step derivation

11. Applied rewrites 1.9%

    \[\leadsto -0.125 \cdot \left(x \cdot x\right) \]

12. Final simplification 1.9%

    \[\leadsto \left(x \cdot x\right) \cdot -0.125 \]
13. 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 2024268 
(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)))