2sqrt (example 3.1)

Percentage Accurate: 7.0% → 99.8%

Time: 6.3s

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: 7.0% 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

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if ((t_0 - sqrt(x)) <= 0.0) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else {
		tmp = ((1.0 + x) - x) / (t_0 + sqrt(x));
	}
	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 = 0.5d0 * sqrt((1.0d0 / x))
    else
        tmp = ((1.0d0 + x) - x) / (t_0 + sqrt(x))
    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 = 0.5 * Math.sqrt((1.0 / x));
	} else {
		tmp = ((1.0 + x) - x) / (t_0 + Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if (t_0 - math.sqrt(x)) <= 0.0:
		tmp = 0.5 * math.sqrt((1.0 / x))
	else:
		tmp = ((1.0 + x) - x) / (t_0 + math.sqrt(x))
	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(0.5 * sqrt(Float64(1.0 / x)));
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64(t_0 + sqrt(x)));
	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 = 0.5 * sqrt((1.0 / x));
	else
		tmp = ((1.0 + x) - x) / (t_0 + sqrt(x));
	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[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(t$95$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)) < 0.0

   1. Initial program 3.8%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 0.0 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 61.7%

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

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 98.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 99.3%

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

7. Applied rewrites 99.3%

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

9. Applied rewrites 99.3%

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

10. Final simplification 99.3%

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

Alternative 3: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 98.8

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

5. Applied rewrites 98.8%

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

6. Final simplification 98.8%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 97.5

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

5. Applied rewrites 97.5%

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

6. Final simplification 97.5%

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

Alternative 5: 97.5% 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 7.2%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 97.5

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

5. Applied rewrites 97.5%

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

7. Applied rewrites 97.3%

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

Alternative 6: 4.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. lower-fma.f64 4.5

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

5. Applied rewrites 4.5%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 4.5%

   \[\leadsto 0.5 \cdot \color{blue}{x} - \sqrt{x} \]
9. Add Preprocessing

Alternative 7: 1.6% 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 7.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.6%

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

Developer Target 1: 97.9% 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 2024327 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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