2sqrt (example 3.1)

Percentage Accurate: 6.7% → 99.6%

Time: 8.5s

Alternatives: 4

Speedup: 205.0×

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: 6.7% 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: 99.6% accurate, 1.0× 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 8.4%

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

   1. flip-- 9.5%

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

   2. add-sqr-sqrt 9.7%

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

   3. add-sqr-sqrt 10.6%

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

   4. associate--l+ 10.6%

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

4. Applied egg-rr 10.6%

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

5. Taylor expanded in x around 0 99.6%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 32500000:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 32500000.0) (- (sqrt (+ x 1.0)) (sqrt x)) (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 32500000.0) {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	} else {
		tmp = 0.5 * pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 32500000.0d0) then
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    else
        tmp = 0.5d0 * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 32500000.0) {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 32500000.0:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	else:
		tmp = 0.5 * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 32500000.0)
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	else
		tmp = Float64(0.5 * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 32500000.0)
		tmp = sqrt((x + 1.0)) - sqrt(x);
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 32500000.0], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.25e7

   1. Initial program 90.0%

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

   if 3.25e7 < x

   1. Initial program 5.5%

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

   3. Taylor expanded in x around inf 99.0%

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

      1. inv-pow 99.0%

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

      2. sqrt-pow1 99.2%

         \[\leadsto 0.5 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \]

      3. metadata-eval 99.2%

         \[\leadsto 0.5 \cdot {x}^{\color{blue}{-0.5}} \]

   5. Applied egg-rr 99.2%

      \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.1% accurate, 2.0× 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]

Derivation

1. Initial program 8.4%

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

3. Taylor expanded in x around inf 96.6%

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

   1. inv-pow 96.6%

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

   2. sqrt-pow1 96.8%

      \[\leadsto 0.5 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \]

   3. metadata-eval 96.8%

      \[\leadsto 0.5 \cdot {x}^{\color{blue}{-0.5}} \]

5. Applied egg-rr 96.8%

   \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]
6. Add Preprocessing

Alternative 4: 7.0% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.375 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.375)

C

double code(double x) {
	return 0.375;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.375d0
end function

Java

public static double code(double x) {
	return 0.375;
}

Python

def code(x):
	return 0.375

Julia

function code(x)
	return 0.375
end

MATLAB

function tmp = code(x)
	tmp = 0.375;
end

Wolfram

code[x_] := 0.375

Derivation

1. Initial program 8.4%

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

3. Taylor expanded in x around inf 97.6%

   \[\leadsto \color{blue}{\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{x}}{x}} \]
4. Step-by-step derivation

   1. flip-+ 97.5%

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

   2. *-commutative 97.5%

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

   3. *-commutative 97.5%

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

   4. swap-sqr 97.5%

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

   5. add-sqr-sqrt 97.5%

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

   6. metadata-eval 97.5%

      \[\leadsto \frac{\frac{\frac{1}{x} \cdot \color{blue}{0.015625} - \left(0.5 \cdot \sqrt{x}\right) \cdot \left(0.5 \cdot \sqrt{x}\right)}{-0.125 \cdot \sqrt{\frac{1}{x}} - 0.5 \cdot \sqrt{x}}}{x} \]

   7. *-commutative 97.5%

      \[\leadsto \frac{\frac{\frac{1}{x} \cdot 0.015625 - \color{blue}{\left(\sqrt{x} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{x}\right)}{-0.125 \cdot \sqrt{\frac{1}{x}} - 0.5 \cdot \sqrt{x}}}{x} \]

   8. *-commutative 97.5%

      \[\leadsto \frac{\frac{\frac{1}{x} \cdot 0.015625 - \left(\sqrt{x} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{x} \cdot 0.5\right)}}{-0.125 \cdot \sqrt{\frac{1}{x}} - 0.5 \cdot \sqrt{x}}}{x} \]

   9. swap-sqr 97.5%

      \[\leadsto \frac{\frac{\frac{1}{x} \cdot 0.015625 - \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(0.5 \cdot 0.5\right)}}{-0.125 \cdot \sqrt{\frac{1}{x}} - 0.5 \cdot \sqrt{x}}}{x} \]

   10. add-sqr-sqrt 97.5%

       \[\leadsto \frac{\frac{\frac{1}{x} \cdot 0.015625 - \color{blue}{x} \cdot \left(0.5 \cdot 0.5\right)}{-0.125 \cdot \sqrt{\frac{1}{x}} - 0.5 \cdot \sqrt{x}}}{x} \]

   11. metadata-eval 97.5%

       \[\leadsto \frac{\frac{\frac{1}{x} \cdot 0.015625 - x \cdot \color{blue}{0.25}}{-0.125 \cdot \sqrt{\frac{1}{x}} - 0.5 \cdot \sqrt{x}}}{x} \]

   12. cancel-sign-sub-inv 97.5%

       \[\leadsto \frac{\frac{\frac{1}{x} \cdot 0.015625 - x \cdot 0.25}{\color{blue}{-0.125 \cdot \sqrt{\frac{1}{x}} + \left(-0.5\right) \cdot \sqrt{x}}}}{x} \]

   13. sqrt-div 97.5%

       \[\leadsto \frac{\frac{\frac{1}{x} \cdot 0.015625 - x \cdot 0.25}{-0.125 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} + \left(-0.5\right) \cdot \sqrt{x}}}{x} \]

   14. metadata-eval 97.5%

       \[\leadsto \frac{\frac{\frac{1}{x} \cdot 0.015625 - x \cdot 0.25}{-0.125 \cdot \frac{\color{blue}{1}}{\sqrt{x}} + \left(-0.5\right) \cdot \sqrt{x}}}{x} \]

   15. un-div-inv 97.5%

       \[\leadsto \frac{\frac{\frac{1}{x} \cdot 0.015625 - x \cdot 0.25}{\color{blue}{\frac{-0.125}{\sqrt{x}}} + \left(-0.5\right) \cdot \sqrt{x}}}{x} \]

5. Applied egg-rr 97.5%

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x} \cdot 0.015625 - x \cdot 0.25}{\frac{-0.125}{\sqrt{x}} + -0.5 \cdot \sqrt{x}}}}{x} \]

7. Applied egg-rr 7.0%

   \[\leadsto \frac{\color{blue}{x \cdot 0.5 + x \cdot -0.125}}{x} \]

8. Taylor expanded in x around 0 7.0%

   \[\leadsto \color{blue}{0.375} \]
9. Add Preprocessing

Developer target: 99.6% accurate, 1.0× 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 2024077 -o generate:simplify
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

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