2isqrt (example 3.6)

Percentage Accurate: 38.3% → 98.8%

Time: 10.5s

Alternatives: 7

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: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\sqrt{1 + x}}}{x + \left(x + 0.5\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

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

4. Applied rewrites 41.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. +-inverses N/A

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

   8. metadata-eval N/A

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

   9. associate-/r* N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lower-/.f64 84.8

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

6. Applied rewrites 84.8%

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

7. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. lower-+.f64 98.9

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

9. Applied rewrites 98.9%

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

10. Final simplification 98.9%

    \[\leadsto \frac{\frac{1}{\sqrt{1 + x}}}{x + \left(x + 0.5\right)} \]
11. Add Preprocessing

Alternative 2: 97.7% accurate, 1.4× 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) / 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 38.3%

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

4. Applied rewrites 40.5%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.7

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

7. Applied rewrites 97.7%

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

Developer Target 1: 98.3% 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]

Developer Target 2: 38.4% 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 2024222 
(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))))))

  :alt
  (! :herbie-platform default (- (pow x -1/2) (pow (+ x 1) -1/2)))

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