2isqrt (example 3.6)

Percentage Accurate: 38.3% → 99.6%

Time: 8.4s

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: 38.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 40.7%

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

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 99.3

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

7. Applied rewrites 99.3%

   \[\leadsto \frac{\color{blue}{-\sqrt{\frac{1}{x}}}}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left(-\sqrt{x + 1}\right)} \]
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-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-*.f64 99.3

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.3

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

9. Applied rewrites 99.3%

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

    1. lift-fma.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. *-commutative N/A

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

    5. lift-neg.f64 N/A

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

    6. distribute-lft-neg-in N/A

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

    7. distribute-rgt-neg-in N/A

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

    8. lower-fma.f64 N/A

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

    9. lift-+.f64 N/A

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

    10. +-commutative N/A

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

    11. lower-+.f64 N/A

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

    12. lower-neg.f64 N/A

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

    13. lift-neg.f64 N/A

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

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

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

    15. pow2 N/A

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

    16. lift-sqrt.f64 N/A

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

    17. pow1/2 N/A

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

    18. lift-+.f64 N/A

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

    19. +-commutative N/A

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

    20. pow-pow N/A

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 2: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{0.125}{x \cdot x}, x, \mathsf{fma}\left(-2, x, -1.5\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- (sqrt (/ 1.0 x))) (fma (/ 0.125 (* x x)) x (fma -2.0 x -1.5))))

C

double code(double x) {
	return -sqrt((1.0 / x)) / fma((0.125 / (x * x)), x, fma(-2.0, x, -1.5));
}

Julia

function code(x)
	return Float64(Float64(-sqrt(Float64(1.0 / x))) / fma(Float64(0.125 / Float64(x * x)), x, fma(-2.0, x, -1.5)))
end

Wolfram

code[x_] := N[((-N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]) / N[(N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision] * x + N[(-2.0 * x + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 40.7%

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

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 99.3

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

7. Applied rewrites 99.3%

   \[\leadsto \frac{\color{blue}{-\sqrt{\frac{1}{x}}}}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left(-\sqrt{x + 1}\right)} \]
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-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-*.f64 99.3

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.3

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

9. Applied rewrites 99.3%

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

10. Taylor expanded in x around inf

    \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{x \cdot \left(\frac{\frac{1}{8}}{{x}^{2}} - \left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)}} \]
11. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{x \cdot \color{blue}{\left(\frac{\frac{1}{8}}{{x}^{2}} + \left(\mathsf{neg}\left(\left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)\right)\right)}} \]

    2. distribute-rgt-in N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\frac{\frac{1}{8}}{{x}^{2}} \cdot x + \left(\mathsf{neg}\left(\left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)\right) \cdot x}} \]

    3. distribute-lft-neg-in N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\frac{\frac{1}{8}}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\left(2 + \frac{3}{2} \cdot \frac{1}{x}\right) \cdot x\right)\right)}} \]

    4. *-commutative N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\frac{\frac{1}{8}}{{x}^{2}} \cdot x + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)}\right)\right)} \]

    5. mul-1-neg N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\frac{\frac{1}{8}}{{x}^{2}} \cdot x + \color{blue}{-1 \cdot \left(x \cdot \left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)}} \]

    6. lower-fma.f64 N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{8}}{{x}^{2}}, x, -1 \cdot \left(x \cdot \left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)\right)}} \]

    7. lower-/.f64 N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{8}}{{x}^{2}}}, x, -1 \cdot \left(x \cdot \left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)\right)} \]

    8. unpow2 N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{\color{blue}{x \cdot x}}, x, -1 \cdot \left(x \cdot \left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)\right)} \]

    9. lower-*.f64 N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{\color{blue}{x \cdot x}}, x, -1 \cdot \left(x \cdot \left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)\right)} \]

    10. distribute-rgt-in N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, -1 \cdot \color{blue}{\left(2 \cdot x + \left(\frac{3}{2} \cdot \frac{1}{x}\right) \cdot x\right)}\right)} \]

    11. distribute-rgt-in N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \color{blue}{\left(2 \cdot x\right) \cdot -1 + \left(\left(\frac{3}{2} \cdot \frac{1}{x}\right) \cdot x\right) \cdot -1}\right)} \]

    12. metadata-eval N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \left(\color{blue}{\left(1 + 1\right)} \cdot x\right) \cdot -1 + \left(\left(\frac{3}{2} \cdot \frac{1}{x}\right) \cdot x\right) \cdot -1\right)} \]

    13. metadata-eval N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \left(\left(1 + \color{blue}{-1 \cdot -1}\right) \cdot x\right) \cdot -1 + \left(\left(\frac{3}{2} \cdot \frac{1}{x}\right) \cdot x\right) \cdot -1\right)} \]

    14. rem-square-sqrt N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \left(\left(1 + -1 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot x\right) \cdot -1 + \left(\left(\frac{3}{2} \cdot \frac{1}{x}\right) \cdot x\right) \cdot -1\right)} \]

    15. unpow2 N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \left(\left(1 + -1 \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}\right) \cdot x\right) \cdot -1 + \left(\left(\frac{3}{2} \cdot \frac{1}{x}\right) \cdot x\right) \cdot -1\right)} \]

    16. *-commutative N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \color{blue}{\left(x \cdot \left(1 + -1 \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot -1 + \left(\left(\frac{3}{2} \cdot \frac{1}{x}\right) \cdot x\right) \cdot -1\right)} \]

12. Applied rewrites 99.1%

    \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\mathsf{fma}\left(\frac{0.125}{x \cdot x}, x, \mathsf{fma}\left(-2, x, -1.5\right)\right)}} \]
13. Add Preprocessing

Alternative 3: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 + \frac{\frac{0.3125}{x} - 0.375}{x}}{x}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (+ 0.5 (/ (- (/ 0.3125 x) 0.375) x)) x) (sqrt x)))

C

double code(double x) {
	return ((0.5 + (((0.3125 / x) - 0.375) / x)) / x) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 + (((0.3125d0 / x) - 0.375d0) / x)) / x) / sqrt(x)
end function

Java

public static double code(double x) {
	return ((0.5 + (((0.3125 / x) - 0.375) / x)) / x) / Math.sqrt(x);
}

Python

def code(x):
	return ((0.5 + (((0.3125 / x) - 0.375) / x)) / x) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(0.5 + Float64(Float64(Float64(0.3125 / x) - 0.375) / x)) / x) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 + (((0.3125 / x) - 0.375) / x)) / x) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(0.5 + N[(N[(N[(0.3125 / x), $MachinePrecision] - 0.375), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 39.6%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 6.1%

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

7. Taylor expanded in x around -inf

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

   1. unpow2 N/A

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

   2. rem-square-sqrt N/A

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

   3. metadata-eval 36.9

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

9. Applied rewrites 36.9%

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

10. Taylor expanded in x around inf

    \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{\frac{5}{16}}{{x}^{2}}\right) - \frac{3}{8} \cdot \frac{1}{x}}{x}}}{\sqrt{x}} \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{\frac{5}{16}}{{x}^{2}}\right) - \frac{3}{8} \cdot \frac{1}{x}}{x}}}{\sqrt{x}} \]

    2. associate--l+ N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} + \left(\frac{\frac{5}{16}}{{x}^{2}} - \frac{3}{8} \cdot \frac{1}{x}\right)}}{x}}{\sqrt{x}} \]

    3. +-commutative N/A

       \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\frac{5}{16}}{{x}^{2}} - \frac{3}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}}{x}}{\sqrt{x}} \]

    4. unpow2 N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16}}{\color{blue}{x \cdot x}} - \frac{3}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x}} \]

    5. associate-/r* N/A

       \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{\frac{5}{16}}{x}}{x}} - \frac{3}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x}} \]

    6. metadata-eval N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{\color{blue}{\frac{5}{16} \cdot 1}}{x}}{x} - \frac{3}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x}} \]

    7. associate-*r/ N/A

       \[\leadsto \frac{\frac{\left(\frac{\color{blue}{\frac{5}{16} \cdot \frac{1}{x}}}{x} - \frac{3}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x}} \]

    8. associate-*r/ N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16} \cdot \frac{1}{x}}{x} - \color{blue}{\frac{\frac{3}{8} \cdot 1}{x}}\right) + \frac{1}{2}}{x}}{\sqrt{x}} \]

    9. metadata-eval N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16} \cdot \frac{1}{x}}{x} - \frac{\color{blue}{\frac{3}{8}}}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x}} \]

    10. div-sub N/A

        \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{5}{16} \cdot \frac{1}{x} - \frac{3}{8}}{x}} + \frac{1}{2}}{x}}{\sqrt{x}} \]

    11. lower-+.f64 N/A

        \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{5}{16} \cdot \frac{1}{x} - \frac{3}{8}}{x} + \frac{1}{2}}}{x}}{\sqrt{x}} \]

    12. lower-/.f64 N/A

        \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{5}{16} \cdot \frac{1}{x} - \frac{3}{8}}{x}} + \frac{1}{2}}{x}}{\sqrt{x}} \]

    13. lower--.f64 N/A

        \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{5}{16} \cdot \frac{1}{x} - \frac{3}{8}}}{x} + \frac{1}{2}}{x}}{\sqrt{x}} \]

    14. associate-*r/ N/A

        \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{\frac{5}{16} \cdot 1}{x}} - \frac{3}{8}}{x} + \frac{1}{2}}{x}}{\sqrt{x}} \]

    15. metadata-eval N/A

        \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{\frac{5}{16}}}{x} - \frac{3}{8}}{x} + \frac{1}{2}}{x}}{\sqrt{x}} \]

    16. lower-/.f64 98.9

        \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{0.3125}{x}} - 0.375}{x} + 0.5}{x}}{\sqrt{x}} \]

12. Applied rewrites 98.9%

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{0.3125}{x} - 0.375}{x} + 0.5}{x}}}{\sqrt{x}} \]

13. Final simplification 98.9%

    \[\leadsto \frac{\frac{0.5 + \frac{\frac{0.3125}{x} - 0.375}{x}}{x}}{\sqrt{x}} \]
14. Add Preprocessing

Alternative 4: 98.8% 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 39.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 40.7%

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

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 99.3

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

7. Applied rewrites 99.3%

   \[\leadsto \frac{\color{blue}{-\sqrt{\frac{1}{x}}}}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left(-\sqrt{x + 1}\right)} \]
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-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-*.f64 99.3

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.3

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

9. Applied rewrites 99.3%

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

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-rgt-in N/A

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

    2. distribute-rgt-in N/A

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

    3. metadata-eval N/A

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

    4. metadata-eval N/A

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

    5. rem-square-sqrt N/A

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

    6. unpow2 N/A

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

    7. *-commutative N/A

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

    8. associate-*l* N/A

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

    9. lft-mult-inverse N/A

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

    10. metadata-eval N/A

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

    11. metadata-eval N/A

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

12. Applied rewrites 98.6%

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

Alternative 5: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 39.6%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 6.1%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 97.8%

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

10. Final simplification 97.8%

    \[\leadsto \frac{\frac{0.5}{1 + x}}{\sqrt{x}} \]
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 39.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 39.6%

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

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 97.7

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

7. Applied rewrites 97.7%

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

Alternative 7: 81.0% 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 39.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 83.7%

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

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 82.9%

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

Alternative 8: 35.3% accurate, 49.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 39.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 39.6%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 6.1%

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

7. Taylor expanded in x around -inf

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

   1. unpow2 N/A

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

   2. rem-square-sqrt N/A

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

   3. metadata-eval 36.9

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

9. Applied rewrites 36.9%

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

10. Taylor expanded in x around -inf

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

    1. distribute-rgt-in N/A

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

    2. unpow2 N/A

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

    3. rem-square-sqrt N/A

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

    4. distribute-rgt-in N/A

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

    5. metadata-eval N/A

       \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{0} \]

    6. mul0-rgt 36.9

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

12. Applied rewrites 36.9%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Developer Target 1: 38.3% 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 2024267 
(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)))))