2isqrt (example 3.6)

Percentage Accurate: 38.2% → 99.6%

Time: 7.6s

Alternatives: 9

Speedup: 1.5×

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: 38.2% 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: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \frac{\frac{1}{\mathsf{fma}\left(t\_0, \sqrt{x}, x\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x)))) (/ (/ 1.0 (fma t_0 (sqrt x) x)) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return (1.0 / fma(t_0, sqrt(x), x)) / t_0;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	return Float64(Float64(1.0 / fma(t_0, sqrt(x), x)) / t_0)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 / N[(t$95$0 * N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 37.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 37.0%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/l/ N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 38.4%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.5%

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

10. Final simplification 99.5%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 + \frac{-0.125 - \frac{-0.0625}{x}}{x}}{x}}{\sqrt{1 + x}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 + (((-0.125d0) - ((-0.0625d0) / x)) / x)) / x) / sqrt((1.0d0 + x))
end function

Java

public static double code(double x) {
	return ((0.5 + ((-0.125 - (-0.0625 / x)) / x)) / x) / Math.sqrt((1.0 + x));
}

Python

def code(x):
	return ((0.5 + ((-0.125 - (-0.0625 / x)) / x)) / x) / math.sqrt((1.0 + x))

Julia

function code(x)
	return Float64(Float64(Float64(0.5 + Float64(Float64(-0.125 - Float64(-0.0625 / x)) / x)) / x) / sqrt(Float64(1.0 + x)))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 + ((-0.125 - (-0.0625 / x)) / x)) / x) / sqrt((1.0 + x));
end

Wolfram

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

Derivation

1. Initial program 37.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 37.0%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/l/ N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 38.4%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 99.3%

   \[\leadsto \frac{\color{blue}{\frac{\frac{-0.125 - \frac{-0.0625}{x}}{x} + 0.5}{x}}}{\sqrt{x + 1}} \]

10. Final simplification 99.3%

    \[\leadsto \frac{\frac{0.5 + \frac{-0.125 - \frac{-0.0625}{x}}{x}}{x}}{\sqrt{1 + x}} \]
11. Add Preprocessing

Alternative 3: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\mathsf{fma}\left(2, x, 0.5\right)}}{\sqrt{1 + x}} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 37.0%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/l/ N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 38.4%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in x around inf

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

    1. distribute-rgt-in N/A

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

    2. associate-*l* N/A

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

    3. lft-mult-inverse N/A

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

    4. metadata-eval N/A

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

    5. lower-fma.f64 99.2

       \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 0.5\right)}}}{\sqrt{x + 1}} \]

12. Applied rewrites 99.2%

    \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 0.5\right)}}}{\sqrt{x + 1}} \]

13. Final simplification 99.2%

    \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(2, x, 0.5\right)}}{\sqrt{1 + x}} \]
14. Add Preprocessing

Alternative 4: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 37.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. sqrt-unprod N/A

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

   9. lower-sqrt.f64 N/A

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

   10. rem-square-sqrt N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. pow2 N/A

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

   14. rem-square-sqrt N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. pow2 N/A

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

   18. pow-prod-down N/A

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

   19. *-commutative N/A

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

   20. pow-prod-down N/A

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

6. Applied rewrites 37.0%

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

7. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. lower-+.f64 36.8

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

9. Applied rewrites 36.8%

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

10. Taylor expanded in x around inf

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

    1. lower-*.f64 N/A

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

    2. lower-sqrt.f64 N/A

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

    3. lower-/.f64 98.8

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

12. Applied rewrites 98.8%

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

13. Final simplification 98.8%

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

Alternative 5: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

5. Applied rewrites 85.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 84.8%

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

10. Applied rewrites 98.7%

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

11. Final simplification 98.7%

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

Alternative 6: 97.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 37.0%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 98.6

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

7. Applied rewrites 98.6%

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

8. Final simplification 98.6%

   \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{1 + x}} \]
9. Add Preprocessing

Alternative 7: 81.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

5. Applied rewrites 85.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 84.8%

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

Alternative 8: 36.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

7. Applied rewrites 36.5%

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

Alternative 9: 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 37.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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: 38.2% 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 2024263 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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