2isqrt (example 3.6)

Percentage Accurate: 38.6% → 99.6%

Time: 8.4s

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: 38.6% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.2%

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

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 42.8%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.4

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

7. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 99.4

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.4

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

9. Applied rewrites 99.4%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lower-fma.f64 99.4

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

    5. lift-+.f64 N/A

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

    6. +-commutative N/A

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

    7. lower-+.f64 99.4

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

    8. lift-*.f64 N/A

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

    9. lift-neg.f64 N/A

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

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

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

    11. pow2 N/A

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

    12. lift-sqrt.f64 N/A

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

    13. pow1/2 N/A

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

    14. lift-+.f64 N/A

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

    15. +-commutative N/A

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

    16. pow-pow N/A

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

    17. metadata-eval N/A

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

    18. unpow1 N/A

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

    19. lower-neg.f64 N/A

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

11. Applied rewrites 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 98.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.2%

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

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 42.8%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.4

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

7. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

9. Applied rewrites 99.2%

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

10. Taylor expanded in x around inf

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

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{8} \cdot 1}{x}} - \frac{1}{2}}{x}}{-\sqrt{1 + x}} \]

    2. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{1}{8}}}{x} - \frac{1}{2}}{x}}{-\sqrt{1 + x}} \]

    3. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1}{8}}{x} - \frac{1}{2}}{x}}}{-\sqrt{1 + x}} \]

    4. lower--.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{8}}{x} - \frac{1}{2}}}{x}}{-\sqrt{1 + x}} \]

    5. lower-/.f64 98.7

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

12. Applied rewrites 98.7%

    \[\leadsto \frac{\color{blue}{\frac{\frac{0.125}{x} - 0.5}{x}}}{-\sqrt{1 + x}} \]
13. Add Preprocessing

Alternative 3: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, -1.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (sqrt (/ 1.0 x))) (fma -2.0 x -1.5)))

C

double code(double x) {
	return -sqrt((1.0 / x)) / fma(-2.0, x, -1.5);
}

Julia

function code(x)
	return Float64(Float64(-sqrt(Float64(1.0 / x))) / fma(-2.0, x, -1.5))
end

Wolfram

code[x_] := N[((-N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]) / N[(-2.0 * x + -1.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.2%

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

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 42.8%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.4

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

7. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 99.4

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.4

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in x around inf

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

    1. distribute-lft-in N/A

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

    2. distribute-rgt-in N/A

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

    3. metadata-eval N/A

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

    4. metadata-eval N/A

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

    5. rem-square-sqrt N/A

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

    6. unpow2 N/A

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

    7. associate-*l* N/A

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

    8. unpow2 N/A

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

    9. rem-square-sqrt N/A

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

    10. metadata-eval N/A

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

    11. metadata-eval N/A

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

    12. metadata-eval N/A

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

    13. *-commutative N/A

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

    14. *-commutative N/A

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

    15. associate-*r* N/A

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

    16. rgt-mult-inverse N/A

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

    17. metadata-eval N/A

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

    18. metadata-eval N/A

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

12. Applied rewrites 98.7%

    \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\mathsf{fma}\left(-2, x, -1.5\right)}} \]
13. Add Preprocessing

Alternative 4: 97.6% accurate, 1.3× 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) / Float64(-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 40.2%

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

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 42.8%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.4

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

7. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

9. Applied rewrites 99.2%

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

10. Taylor expanded in x around inf

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

    1. lower-/.f64 97.5

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

12. Applied rewrites 97.5%

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

Alternative 5: 97.5% 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 40.2%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

5. Applied rewrites 84.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, \mathsf{fma}\left(0.5, x, 1\right), -\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right)\right), -0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x \cdot x}} \]

6. Taylor expanded in x around -inf

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

8. Applied rewrites 97.2%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt{\frac{1}{x}}}{x}, 0.625, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{\color{blue}{x}} \]

9. Taylor expanded in x around inf

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

11. Applied rewrites 97.3%

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

12. Final simplification 97.3%

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

Alternative 6: 37.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} + x\\ \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (sqrt x) x)))
   (if (<= x 4.6e+153) (/ 1.0 t_0) (/ (- (+ 1.0 x) x) t_0))))

C

double code(double x) {
	double t_0 = sqrt(x) + x;
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / t_0;
	} else {
		tmp = ((1.0 + x) - x) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) + x
    if (x <= 4.6d+153) then
        tmp = 1.0d0 / t_0
    else
        tmp = ((1.0d0 + x) - x) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt(x) + x;
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / t_0;
	} else {
		tmp = ((1.0 + x) - x) / t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt(x) + x
	tmp = 0
	if x <= 4.6e+153:
		tmp = 1.0 / t_0
	else:
		tmp = ((1.0 + x) - x) / t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(x) + x)
	tmp = 0.0
	if (x <= 4.6e+153)
		tmp = Float64(1.0 / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt(x) + x;
	tmp = 0.0;
	if (x <= 4.6e+153)
		tmp = 1.0 / t_0;
	else
		tmp = ((1.0 + x) - x) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, 4.6e+153], N[(1.0 / t$95$0), $MachinePrecision], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.6000000000000003e153

   1. Initial program 10.9%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. clear-num N/A

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

      5. frac-sub N/A

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

      6. /-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. flip-- N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. /-rgt-identity N/A

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

      15. associate-/l/ N/A

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

   4. Applied rewrites 16.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.4%

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

   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. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 8.6

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

   10. Applied rewrites 8.6%

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

   if 4.6000000000000003e153 < x

   1. Initial program 70.4%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. clear-num N/A

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

      5. frac-sub N/A

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

      6. /-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. flip-- N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. /-rgt-identity N/A

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

      15. associate-/l/ N/A

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 70.4

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

   7. Applied rewrites 70.4%

      \[\leadsto \frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x} + x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\sqrt{x} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\sqrt{x} + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% 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 40.2%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

5. Applied rewrites 84.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, \mathsf{fma}\left(0.5, x, 1\right), -\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right)\right), -0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x \cdot x}} \]

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 82.8%

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

Alternative 8: 7.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.2%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. clear-num N/A

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

   5. frac-sub N/A

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

   6. /-rgt-identity N/A

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

   7. *-lft-identity N/A

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

   8. metadata-eval N/A

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

   9. div-inv N/A

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

   10. flip-- N/A

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

   11. *-rgt-identity N/A

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

   14. /-rgt-identity N/A

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

   15. associate-/l/ N/A

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

4. Applied rewrites 42.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.6%

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

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. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. rem-square-sqrt N/A

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

   4. lower-+.f64 N/A

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

   5. lower-sqrt.f64 8.0

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

10. Applied rewrites 8.0%

    \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + x}} \]
11. 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 40.2%

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

Developer Target 1: 38.6% 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 2024254 
(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)))))