2isqrt (example 3.6)

Percentage Accurate: 38.5% → 98.8%

Time: 9.9s

Alternatives: 3

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(0.5 + \frac{-0.375}{x}\right) \cdot {x}^{-1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (+ 0.5 (/ -0.375 x)) (pow x -1.5)))

C

double code(double x) {
	return (0.5 + (-0.375 / x)) * pow(x, -1.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 + ((-0.375d0) / x)) * (x ** (-1.5d0))
end function

Java

public static double code(double x) {
	return (0.5 + (-0.375 / x)) * Math.pow(x, -1.5);
}

Python

def code(x):
	return (0.5 + (-0.375 / x)) * math.pow(x, -1.5)

Julia

function code(x)
	return Float64(Float64(0.5 + Float64(-0.375 / x)) * (x ^ -1.5))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 + (-0.375 / x)) * (x ^ -1.5);
end

Wolfram

code[x_] := N[(N[(0.5 + N[(-0.375 / x), $MachinePrecision]), $MachinePrecision] * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.2%

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

3. Taylor expanded in x around inf

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

5. Simplified 81.2%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{1}{x \cdot x}}{x}} \cdot \left(0.5 \cdot \left(1 + x \cdot 0.25\right)\right) + -0.5 \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)}{x \cdot x}} \]

7. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\left(0.5 + x \cdot 0.125\right) \cdot {x}^{-1.5} + \frac{-0.5 \cdot \left(\frac{1}{x} - x\right)}{\frac{1 + x}{\sqrt{x}}}}{x}}{x}} \]

8. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{\frac{-1}{8} \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right), \color{blue}{\left(-1 \cdot x\right)}\right) \]

10. Simplified 99.0%

    \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{\frac{1}{x}}\right) \cdot \left(0.5 - \frac{0.375}{x}\right)}{0 - x}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\left(\frac{1}{2} - \frac{\frac{3}{8}}{x}\right) \cdot \left(0 - \sqrt{\frac{1}{x}}\right)}{\color{blue}{0} - x} \]

    2. sub0-neg N/A

       \[\leadsto \frac{\left(\frac{1}{2} - \frac{\frac{3}{8}}{x}\right) \cdot \left(0 - \sqrt{\frac{1}{x}}\right)}{\mathsf{neg}\left(x\right)} \]

    3. associate-/l* N/A

       \[\leadsto \left(\frac{1}{2} - \frac{\frac{3}{8}}{x}\right) \cdot \color{blue}{\frac{0 - \sqrt{\frac{1}{x}}}{\mathsf{neg}\left(x\right)}} \]

    4. sub0-neg N/A

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

    5. frac-2neg N/A

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

    6. div-inv N/A

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

    7. inv-pow N/A

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

    8. sqrt-pow1 N/A

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

    9. metadata-eval N/A

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

    10. inv-pow N/A

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

    11. pow-prod-up N/A

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

    12. metadata-eval N/A

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

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} - \frac{\frac{3}{8}}{x}\right), \color{blue}{\left({x}^{\frac{-3}{2}}\right)}\right) \]

    14. sub-neg N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\frac{3}{8}}{x}\right)\right)\right), \left({\color{blue}{x}}^{\frac{-3}{2}}\right)\right) \]

    15. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{3}{8}}{x}\right)\right)\right), \left({\color{blue}{x}}^{\frac{-3}{2}}\right)\right) \]

    16. distribute-neg-frac N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{3}{8}\right)}{x}\right)\right), \left({x}^{\frac{-3}{2}}\right)\right) \]

    17. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right), x\right)\right), \left({x}^{\frac{-3}{2}}\right)\right) \]

    18. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-3}{8}, x\right)\right), \left({x}^{\frac{-3}{2}}\right)\right) \]

    19. pow-lowering-pow.f64 99.2%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-3}{8}, x\right)\right), \mathsf{pow.f64}\left(x, \color{blue}{\frac{-3}{2}}\right)\right) \]

12. Applied egg-rr 99.2%

    \[\leadsto \color{blue}{\left(0.5 + \frac{-0.375}{x}\right) \cdot {x}^{-1.5}} \]
13. Add Preprocessing

Alternative 2: 97.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -1.5)))

C

double code(double x) {
	return 0.5 * pow(x, -1.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x ** (-1.5d0))
end function

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, -1.5);
}

Python

def code(x):
	return 0.5 * math.pow(x, -1.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -1.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ -1.5);
end

Wolfram

code[x_] := N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.2%

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

3. Taylor expanded in x around inf

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

5. Simplified 81.2%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{1}{x \cdot x}}{x}} \cdot \left(0.5 \cdot \left(1 + x \cdot 0.25\right)\right) + -0.5 \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)}{x \cdot x}} \]

7. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\left(0.5 + x \cdot 0.125\right) \cdot {x}^{-1.5} + \frac{-0.5 \cdot \left(\frac{1}{x} - x\right)}{\frac{1 + x}{\sqrt{x}}}}{x}}{x}} \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}, x\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{\frac{1}{x}}\right)\right), x\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right), x\right) \]

   3. /-lowering-/.f64 98.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), x\right) \]

10. Simplified 98.2%

    \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}}}{x} \]
11. Step-by-step derivation

    1. associate-/l* N/A

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

    2. *-commutative N/A

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

    3. div-inv N/A

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

    4. inv-pow N/A

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

    5. sqrt-pow1 N/A

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

    6. metadata-eval N/A

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

    7. inv-pow N/A

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

    8. pow-prod-up N/A

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

    9. metadata-eval N/A

       \[\leadsto {x}^{\frac{-3}{2}} \cdot \frac{1}{2} \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\left({x}^{\frac{-3}{2}}\right), \color{blue}{\frac{1}{2}}\right) \]

    11. pow-lowering-pow.f64 98.4%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-3}{2}\right), \frac{1}{2}\right) \]

12. Applied egg-rr 98.4%

    \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} \]

13. Final simplification 98.4%

    \[\leadsto 0.5 \cdot {x}^{-1.5} \]
14. Add Preprocessing

Alternative 3: 35.3% accurate, 209.0× 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 39.2%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(x\right)\right), \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]
4. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

   2. /-lowering-/.f64 28.7%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

5. Simplified 28.7%

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

   1. metadata-eval N/A

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

   2. sqrt-div N/A

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

   3. +-inverses 37.0%

      \[\leadsto 0 \]

7. Applied egg-rr 37.0%

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

Developer Target 1: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024288 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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