2isqrt (example 3.6)

Percentage Accurate: 39.0% → 99.6%

Time: 8.0s

Alternatives: 8

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: 39.0% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 43.0%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lower--.f64 99.2

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

6. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. +-inverses N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity 99.2

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

   7. lift-*.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. remove-double-neg N/A

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

   11. distribute-rgt-neg-out N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lift-*.f64 N/A

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 98.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 43.0%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lower--.f64 99.2

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

6. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. +-inverses N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity 99.2

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

   7. lift-*.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. remove-double-neg N/A

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

   11. distribute-rgt-neg-out N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lift-*.f64 N/A

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around inf

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

    1. distribute-lft-in N/A

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

    2. *-commutative N/A

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

    3. metadata-eval N/A

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

    4. rem-square-sqrt N/A

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

    5. unpow2 N/A

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

    6. *-commutative N/A

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

    7. associate-*r* N/A

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

    8. rgt-mult-inverse N/A

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

    9. metadata-eval N/A

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

    10. lower-fma.f64 N/A

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

    11. unpow2 N/A

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

    12. rem-square-sqrt N/A

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

    13. metadata-eval 98.1

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

11. Applied rewrites 98.1%

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

12. Final simplification 98.1%

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

Alternative 3: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\sqrt{x + 1}\right)}^{-1}}{2 \cdot x} \end{array} \]

FPCore

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

C

double code(double x) {
	return pow(sqrt((x + 1.0)), -1.0) / (2.0 * x);
}

Fortran

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

Java

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

Python

def code(x):
	return math.pow(math.sqrt((x + 1.0)), -1.0) / (2.0 * x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 43.0%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lower--.f64 99.2

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

6. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. +-inverses N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity 99.2

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

   7. lift-*.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. remove-double-neg N/A

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

   11. distribute-rgt-neg-out N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lift-*.f64 N/A

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around inf

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

    1. metadata-eval N/A

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

    2. rem-square-sqrt N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. unpow2 N/A

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

    6. rem-square-sqrt N/A

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

    7. metadata-eval 96.7

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

11. Applied rewrites 96.7%

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

12. Final simplification 96.7%

    \[\leadsto \frac{{\left(\sqrt{x + 1}\right)}^{-1}}{2 \cdot x} \]
13. Add Preprocessing

Alternative 4: 5.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sqrt((x ^ -1.0))
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.7

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

5. Applied rewrites 5.7%

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

6. Final simplification 5.7%

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

Alternative 5: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.7%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. flip-+ N/A

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

   5. sqrt-div N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower--.f64 8.6

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

4. Applied rewrites 8.6%

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

6. Applied rewrites 43.0%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-+r- N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval 98.5

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

8. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lift-neg.f64 N/A

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

   5. distribute-rgt-neg-out N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-neg.f64 98.8

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

10. Applied rewrites 98.8%

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

Alternative 6: 97.6% accurate, 1.5× speedup

Math

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

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 41.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 41.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 96.6

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

7. Applied rewrites 96.6%

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

Alternative 7: 81.2% 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 41.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 85.4%

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

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 84.1%

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

Alternative 8: 37.4% 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 41.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.7

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

5. Applied rewrites 5.7%

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

7. Applied rewrites 39.1%

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

Developer Target 1: 39.0% 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 2024313 
(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)))))