2sqrt (example 3.1)

Percentage Accurate: 6.6% → 99.6%

Time: 6.9s

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: 6.6% 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, 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 6.9%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-div N/A

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

   4. pow2 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. unpow1 N/A

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

   9. lower-/.f64 6.7

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 6.7

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 6.7

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

4. Applied rewrites 6.7%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. lift-sqrt.f64 N/A

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. pow-plus N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. pow1/2 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. flip-- N/A

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

   \[\leadsto \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
8. Add Preprocessing

Alternative 2: 98.0% 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 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. *-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 97.9

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

5. Applied rewrites 97.9%

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

6. Final simplification 97.9%

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

Alternative 3: 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 6.9%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-div N/A

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

   4. pow2 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. unpow1 N/A

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

   9. lower-/.f64 6.7

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 6.7

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 6.7

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

4. Applied rewrites 6.7%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. lift-sqrt.f64 N/A

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. pow-plus N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. pow1/2 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. flip-- N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
8. 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 97.9

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

9. Applied rewrites 97.9%

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

11. Applied rewrites 97.6%

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

Alternative 4: 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 6.9%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-div N/A

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

   4. pow2 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. unpow1 N/A

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

   9. lower-/.f64 6.7

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 6.7

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 6.7

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

4. Applied rewrites 6.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)} - \sqrt{x} \]
6. 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 4.6

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

7. Applied rewrites 4.6%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 4.6%

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

Alternative 5: 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

4. Applied rewrites 9.5%

   \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt{x}, x, {\left(1 + x\right)}^{1.5}\right)} \cdot \left(\left(\left(1 + x\right) + x\right) - \mathsf{hypot}\left(\sqrt{x}, x\right)\right)} \]

5. Taylor expanded in x around -inf

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

   1. associate-/r* N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{x}}{\color{blue}{0} \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

   4. mul0-lft N/A

      \[\leadsto \frac{\frac{3}{x}}{\color{blue}{0}} \]

   5. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{3}{0 \cdot x}} \]

   6. mul0-lft N/A

      \[\leadsto \frac{3}{\color{blue}{0}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{3}{\color{blue}{\mathsf{neg}\left(0\right)}} \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{3}{0}\right)} \]

   9. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(3\right)}{0}} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(3\right)}{0}} \]

   11. metadata-eval 0.8

       \[\leadsto \frac{\color{blue}{-3}}{0} \]

7. Applied rewrites 0.8%

   \[\leadsto \color{blue}{\frac{-3}{0}} \]
8. Step-by-step derivation

9. Applied rewrites 3.8%

   \[\leadsto \color{blue}{0} \]
10. 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 2024242 
(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)))