2sqrt (example 3.1)

Percentage Accurate: 6.5% → 98.9%

Time: 6.3s

Alternatives: 5

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: 6.5% 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: 98.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.6%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{-5}{128} \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{-5}{128} \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x}} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, -0.0390625, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, 0.0625, \mathsf{fma}\left(\sqrt{\frac{1}{x}}, -0.125, 0.5 \cdot \sqrt{x}\right)\right)\right)}{x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
7. Step-by-step derivation

8. Applied rewrites 99.5%

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

10. Applied rewrites 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 98.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.6%

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

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

5. Applied rewrites 99.3%

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

6. Final simplification 99.3%

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

Alternative 3: 98.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.6%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Final simplification 98.9%

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

Alternative 4: 97.8% 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 5.6%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 99.5%

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

7. Applied rewrites 98.7%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\sqrt{x}, 0.5, \mathsf{fma}\left(0.0625, {x}^{-1.5}, \frac{0.125}{\sqrt{x}}\right)\right)}{\sqrt{x}}}{\color{blue}{\sqrt{x}}} \]

8. Taylor expanded in x around inf

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

10. Applied rewrites 98.8%

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

Alternative 5: 5.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sqrt(x)
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[x], $MachinePrecision]

Derivation

1. Initial program 5.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.6%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-prod N/A

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

   3. sqr-neg-rev N/A

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

   4. sqrt-prod N/A

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

   5. rem-square-sqrt N/A

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

   6. lower-neg.f64 5.3

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

7. Applied rewrites 5.3%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 5.3

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

10. Applied rewrites 5.3%

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

11. Taylor expanded in x around -inf

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

13. Applied rewrites 5.3%

    \[\leadsto \sqrt{x} \]
14. Add Preprocessing

Developer Target 1: 98.2% 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 2024339 
(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)))