2sqrt (example 3.1)

Percentage Accurate: 6.9% → 99.8%

Time: 8.9s

Alternatives: 7

Speedup: 1.2×

Specification

\[x > 1 \land x < 10^{+308}\]

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: 6.9% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;t\_0 - \sqrt{x} \leq 0:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt{x} + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (- t_0 (sqrt x)) 0.0)
     (* (sqrt (/ 1.0 x)) 0.5)
     (/ (+ x (- 1.0 x)) (+ (sqrt x) t_0)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if ((t_0 - sqrt(x)) <= 0.0) {
		tmp = sqrt((1.0 / x)) * 0.5;
	} else {
		tmp = (x + (1.0 - x)) / (sqrt(x) + 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((1.0d0 + x))
    if ((t_0 - sqrt(x)) <= 0.0d0) then
        tmp = sqrt((1.0d0 / x)) * 0.5d0
    else
        tmp = (x + (1.0d0 - x)) / (sqrt(x) + t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if ((t_0 - Math.sqrt(x)) <= 0.0) {
		tmp = Math.sqrt((1.0 / x)) * 0.5;
	} else {
		tmp = (x + (1.0 - x)) / (Math.sqrt(x) + t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if (t_0 - math.sqrt(x)) <= 0.0:
		tmp = math.sqrt((1.0 / x)) * 0.5
	else:
		tmp = (x + (1.0 - x)) / (math.sqrt(x) + t_0)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0

   1. Initial program 3.9%

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

      3. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

   5. Simplified 99.8%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 59.0%

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

      1. flip-- N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right), \color{blue}{\left(\sqrt{x + 1} + \sqrt{x}\right)}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) - \sqrt{x} \cdot \sqrt{x}\right), \left(\sqrt{\color{blue}{x + 1}} + \sqrt{x}\right)\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) - x\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + \left(1 - x\right)\right), \left(\color{blue}{\sqrt{x + 1}} + \sqrt{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + \left(1 \cdot 1 - x\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + \left(1 \cdot 1 - x \cdot 1\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 \cdot 1 - x \cdot 1\right)\right), \left(\color{blue}{\sqrt{x + 1}} + \sqrt{x}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 - x \cdot 1\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 - x\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\left(\sqrt{x + 1}\right), \color{blue}{\left(\sqrt{x}\right)}\right)\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(\left(x + 1\right)\right), \left(\sqrt{\color{blue}{x}}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left(\sqrt{x}\right)\right)\right) \]

      15. sqrt-lowering-sqrt.f64 99.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   4. Applied egg-rr 99.2%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \sqrt{x} \cdot 0.5\right)\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma
   -0.125
   (sqrt (/ 1.0 x))
   (fma 0.0625 (sqrt (/ 1.0 (* x (* x x)))) (* (sqrt x) 0.5)))
  x))

C

double code(double x) {
	return fma(-0.125, sqrt((1.0 / x)), fma(0.0625, sqrt((1.0 / (x * (x * x)))), (sqrt(x) * 0.5))) / x;
}

Julia

function code(x)
	return Float64(fma(-0.125, sqrt(Float64(1.0 / x)), fma(0.0625, sqrt(Float64(1.0 / Float64(x * Float64(x * x)))), Float64(sqrt(x) * 0.5))) / x)
end

Wolfram

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

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right), \color{blue}{x}\right) \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{1}{x}}\right), \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right), x\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right), x\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right), x\right) \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{fma.f64}\left(\frac{1}{16}, \left(\sqrt{\frac{1}{{x}^{3}}}\right), \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{sqrt.f64}\left(\left(\frac{1}{{x}^{3}}\right)\right), \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{3}\right)\right)\right), \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot \left(x \cdot x\right)\right)\right)\right), \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right), \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\sqrt{x} \cdot \frac{1}{2}\right)\right)\right), x\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\sqrt{x}\right), \frac{1}{2}\right)\right)\right), x\right) \]

   13. sqrt-lowering-sqrt.f64 99.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{fma.f64}\left(\frac{1}{16}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{1}{2}\right)\right)\right), x\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \sqrt{x} \cdot 0.5\right)\right)}{x}} \]
6. Add Preprocessing

Alternative 3: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{x}, 0.5, \frac{-0.125}{\sqrt{x}}\right)}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fma (sqrt x) 0.5 (/ -0.125 (sqrt x))) x))

C

double code(double x) {
	return fma(sqrt(x), 0.5, (-0.125 / sqrt(x))) / x;
}

Julia

function code(x)
	return Float64(fma(sqrt(x), 0.5, Float64(-0.125 / sqrt(x))) / x)
end

Wolfram

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

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}\right), \color{blue}{x}\right) \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{1}{x}}\right), \left(\frac{1}{2} \cdot \sqrt{x}\right)\right), x\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\frac{1}{2} \cdot \sqrt{x}\right)\right), x\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\frac{1}{2} \cdot \sqrt{x}\right)\right), x\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\sqrt{x} \cdot \frac{1}{2}\right)\right), x\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\left(\sqrt{x}\right), \frac{1}{2}\right)\right), x\right) \]

   7. sqrt-lowering-sqrt.f64 98.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{1}{2}\right)\right), x\right) \]

5. Simplified 98.7%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \sqrt{x} \cdot \frac{1}{2}\right), \color{blue}{x}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{x} \cdot \frac{1}{2} + \frac{-1}{8} \cdot \sqrt{\frac{1}{x}}\right), x\right) \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(\sqrt{x}\right), \frac{1}{2}, \left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}}\right)\right), x\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{1}{2}, \left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}}\right)\right), x\right) \]

   5. sqrt-div N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{1}{2}, \left(\frac{-1}{8} \cdot \frac{\sqrt{1}}{\sqrt{x}}\right)\right), x\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{1}{2}, \left(\frac{-1}{8} \cdot \frac{1}{\sqrt{x}}\right)\right), x\right) \]

   7. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{1}{2}, \left(\frac{\frac{-1}{8}}{\sqrt{x}}\right)\right), x\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{8}, \left(\sqrt{x}\right)\right)\right), x\right) \]

   9. sqrt-lowering-sqrt.f64 98.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(x\right)\right)\right), x\right) \]

7. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{x}, 0.5, \frac{-0.125}{\sqrt{x}}\right)}{x}} \]
8. Add Preprocessing

Alternative 4: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{x}} \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

   3. /-lowering-/.f64 97.8%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

5. Simplified 97.8%

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

6. Final simplification 97.8%

   \[\leadsto \sqrt{\frac{1}{x}} \cdot 0.5 \]
7. Add Preprocessing

Alternative 5: 97.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.5 (sqrt x)))

C

double code(double x) {
	return 0.5 / sqrt(x);
}

Fortran

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

Java

public static double code(double x) {
	return 0.5 / Math.sqrt(x);
}

Python

def code(x):
	return 0.5 / math.sqrt(x)

Julia

function code(x)
	return Float64(0.5 / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / sqrt(x);
end

Wolfram

code[x_] := N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

   3. /-lowering-/.f64 97.8%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

5. Simplified 97.8%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{x}\right)}\right) \]

   5. sqrt-lowering-sqrt.f64 97.6%

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right) \]

7. Applied egg-rr 97.6%

   \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x}}} \]
8. Add Preprocessing

Alternative 6: 4.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0.5, 0 - \sqrt{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 0.5 (- 0.0 (sqrt x))))

C

double code(double x) {
	return fma(x, 0.5, (0.0 - sqrt(x)));
}

Julia

function code(x)
	return fma(x, 0.5, Float64(0.0 - sqrt(x)))
end

Wolfram

code[x_] := N[(x * 0.5 + N[(0.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

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

   2. associate--l+ N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{2}}, \left(1 - \sqrt{x}\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{2}, \mathsf{\_.f64}\left(1, \left(\sqrt{x}\right)\right)\right) \]

   6. sqrt-lowering-sqrt.f64 4.5%

      \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{2}, \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

5. Simplified 4.5%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{2}, \color{blue}{\left(-1 \cdot \sqrt{x}\right)}\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{2}, \left(\mathsf{neg}\left(\sqrt{x}\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{2}, \left(0 - \sqrt{x}\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{2}, \mathsf{\_.f64}\left(0, \left(\sqrt{x}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 4.5%

      \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{2}, \mathsf{\_.f64}\left(0, \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

8. Simplified 4.5%

   \[\leadsto \mathsf{fma}\left(x, 0.5, \color{blue}{0 - \sqrt{x}}\right) \]
9. Add Preprocessing

Alternative 7: 3.8% accurate, 27.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\sqrt{x}\right)}, \mathsf{sqrt.f64}\left(x\right)\right) \]
4. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 3.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right) \]

5. Simplified 3.8%

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

   1. +-inverses 3.8%

      \[\leadsto 0 \]

7. Applied egg-rr 3.8%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.6% 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]

Developer Target 2: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -0.5)))

C

double code(double x) {
	return 0.5 * pow(x, -0.5);
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5 * math.pow(x, -0.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -0.5))
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024193 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (sqrt (+ x 1)) (sqrt x))))

  :alt
  (! :herbie-platform default (* 1/2 (pow x -1/2)))

  (- (sqrt (+ x 1.0)) (sqrt x)))