2isqrt (example 3.6)

Percentage Accurate: 38.5% → 98.3%

Time: 10.8s

Alternatives: 8

Speedup: 1.8×

Specification

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

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

Fortran

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

Java

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

Python

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end

Wolfram

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

Initial Program: 38.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

Fortran

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

Java

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

Python

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end

Wolfram

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (fma (sqrt (+ 1.0 x)) x (* (+ 1.0 x) (sqrt x)))))

C

double code(double x) {
	return 1.0 / fma(sqrt((1.0 + x)), x, ((1.0 + x) * sqrt(x)));
}

Julia

function code(x)
	return Float64(1.0 / fma(sqrt(Float64(1.0 + x)), x, Float64(Float64(1.0 + x) * sqrt(x))))
end

Wolfram

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

Derivation

1. Initial program 37.1%

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

4. Applied rewrites 39.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 84.5%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. sqrt-unprod N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. sqrt-unprod N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lower-*.f64 65.1

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

9. Applied rewrites 65.1%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-sqrt.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-fma.f64 N/A

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

    6. distribute-lft-in N/A

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

    7. distribute-lft1-in N/A

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

    8. +-commutative N/A

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

    9. lift-+.f64 N/A

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

    10. sqrt-prod N/A

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

    11. lift-sqrt.f64 N/A

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

    12. sqrt-unprod N/A

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

    13. rem-square-sqrt N/A

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

    14. lower-fma.f64 65.1

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

    15. lift-sqrt.f64 N/A

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

    16. lift-*.f64 N/A

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

    17. lift-fma.f64 N/A

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

    18. distribute-lft1-in N/A

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

    19. +-commutative N/A

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

    20. lift-+.f64 N/A

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

    21. associate-*r* N/A

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

11. Applied rewrites 99.4%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt{1 + x}, x, \left(1 + x\right) \cdot \sqrt{x}\right)}} \]
12. Add Preprocessing

Alternative 2: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(x + 0.5\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

4. Applied rewrites 39.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 84.5%

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

8. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 98.7

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

10. Applied rewrites 98.7%

    \[\leadsto \frac{1}{\color{blue}{\left(x + 0.5\right)} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} \]
11. Add Preprocessing

Alternative 3: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x + 0.5}}{\sqrt{x} \cdot 2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 1.0 (+ x 0.5)) (* (sqrt x) 2.0)))

C

double code(double x) {
	return (1.0 / (x + 0.5)) / (sqrt(x) * 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x + 0.5d0)) / (sqrt(x) * 2.0d0)
end function

Java

public static double code(double x) {
	return (1.0 / (x + 0.5)) / (Math.sqrt(x) * 2.0);
}

Python

def code(x):
	return (1.0 / (x + 0.5)) / (math.sqrt(x) * 2.0)

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 0.5)) / Float64(sqrt(x) * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x + 0.5)) / (sqrt(x) * 2.0);
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x + 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.1%

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

4. Applied rewrites 39.3%

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

6. Applied rewrites 84.5%

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

7. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 98.8

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

9. Applied rewrites 98.8%

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

10. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-sqrt.f64 98.0

       \[\leadsto \frac{\frac{1}{x + 0.5}}{\color{blue}{\sqrt{x}} \cdot 2} \]

12. Applied rewrites 98.0%

    \[\leadsto \frac{\frac{1}{x + 0.5}}{\color{blue}{\sqrt{x} \cdot 2}} \]
13. Add Preprocessing

Alternative 4: 97.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-out-- N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 82.8

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

5. Applied rewrites 82.8%

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

7. Applied rewrites 98.0%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 97.9%

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

Alternative 5: 97.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

4. Applied rewrites 39.3%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{{x}^{3}}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{1}{{x}^{3}}}} \]

   4. cube-mult N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 63.4

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

7. Applied rewrites 63.4%

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

9. Applied rewrites 97.8%

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

11. Applied rewrites 97.8%

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

Alternative 6: 96.6% accurate, 1.8× speedup

Math

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

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(0.5 / Float64(x * sqrt(x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

4. Applied rewrites 39.3%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{{x}^{3}}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{1}{{x}^{3}}}} \]

   4. cube-mult N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 63.4

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

7. Applied rewrites 63.4%

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

9. Applied rewrites 97.7%

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

Alternative 7: 7.8% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ x (sqrt x))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(1.0 / Float64(x + sqrt(x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

4. Applied rewrites 39.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 84.5%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. rem-square-sqrt N/A

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

   4. *-rgt-identity N/A

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

   5. lower-+.f64 N/A

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

   6. lower-sqrt.f64 7.7

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

10. Applied rewrites 7.7%

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

Alternative 8: 5.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.6

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

5. Applied rewrites 5.6%

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

Developer Target 1: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 38.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} - {\left(x + 1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (pow x -0.5) (pow (+ x 1.0) -0.5)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024227 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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

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