2isqrt (example 3.6)

Percentage Accurate: 39.4% → 98.6%

Time: 8.5s

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: 39.4% 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: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.5 + x \cdot 0.125}{{x}^{1.5}} + \frac{-0.5 \cdot \left(\frac{1}{x} - x\right)}{\frac{x + 1}{\sqrt{x}}}}{x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/
   (+
    (/ (+ 0.5 (* x 0.125)) (pow x 1.5))
    (/ (* -0.5 (- (/ 1.0 x) x)) (/ (+ x 1.0) (sqrt x))))
   x)
  x))

C

double code(double x) {
	return ((((0.5 + (x * 0.125)) / pow(x, 1.5)) + ((-0.5 * ((1.0 / x) - x)) / ((x + 1.0) / sqrt(x)))) / x) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((0.5d0 + (x * 0.125d0)) / (x ** 1.5d0)) + (((-0.5d0) * ((1.0d0 / x) - x)) / ((x + 1.0d0) / sqrt(x)))) / x) / x
end function

Java

public static double code(double x) {
	return ((((0.5 + (x * 0.125)) / Math.pow(x, 1.5)) + ((-0.5 * ((1.0 / x) - x)) / ((x + 1.0) / Math.sqrt(x)))) / x) / x;
}

Python

def code(x):
	return ((((0.5 + (x * 0.125)) / math.pow(x, 1.5)) + ((-0.5 * ((1.0 / x) - x)) / ((x + 1.0) / math.sqrt(x)))) / x) / x

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(0.5 + Float64(x * 0.125)) / (x ^ 1.5)) + Float64(Float64(-0.5 * Float64(Float64(1.0 / x) - x)) / Float64(Float64(x + 1.0) / sqrt(x)))) / x) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((((0.5 + (x * 0.125)) / (x ^ 1.5)) + ((-0.5 * ((1.0 / x) - x)) / ((x + 1.0) / sqrt(x)))) / x) / x;
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.5 + N[(x * 0.125), $MachinePrecision]), $MachinePrecision] / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(N[(1.0 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 41.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. Simplified 84.5%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

   \[\leadsto \frac{\frac{\frac{0.5 + x \cdot 0.125}{{x}^{1.5}} + \frac{-0.5 \cdot \left(\frac{1}{x} - x\right)}{\frac{x + 1}{\sqrt{x}}}}{x}}{x} \]
9. Add Preprocessing

Alternative 2: 97.8% 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 41.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. Simplified 84.5%

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

6. Taylor expanded in x around inf

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

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

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

   2. sqrt-lowering-sqrt.f64 83.7%

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

8. Simplified 83.7%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. pow2 N/A

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

   4. pow1/2 N/A

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

   5. pow-div N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

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

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

   9. pow-flip N/A

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

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

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

   11. metadata-eval 98.5%

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

10. Applied egg-rr 98.5%

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

11. Final simplification 98.5%

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

Alternative 3: 36.3% 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 41.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 28.8%

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

5. Simplified 28.8%

   \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\sqrt{\frac{1}{x}}} \]
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 39.4%

      \[\leadsto 0 \]

7. Applied egg-rr 39.4%

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

Developer Target 1: 98.4% 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 2024149 
(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)))))