2isqrt (example 3.6)

Percentage Accurate: 38.7% → 98.8%

Time: 8.7s

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: 38.7% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\frac{0.25}{x}} + -0.375 \cdot {x}^{-1.5}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (sqrt (/ 0.25 x)) (* -0.375 (pow x -1.5))) x))

C

double code(double x) {
	return (sqrt((0.25 / x)) + (-0.375 * pow(x, -1.5))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sqrt((0.25d0 / x)) + ((-0.375d0) * (x ** (-1.5d0)))) / x
end function

Java

public static double code(double x) {
	return (Math.sqrt((0.25 / x)) + (-0.375 * Math.pow(x, -1.5))) / x;
}

Python

def code(x):
	return (math.sqrt((0.25 / x)) + (-0.375 * math.pow(x, -1.5))) / x

Julia

function code(x)
	return Float64(Float64(sqrt(Float64(0.25 / x)) + Float64(-0.375 * (x ^ -1.5))) / x)
end

MATLAB

function tmp = code(x)
	tmp = (sqrt((0.25 / x)) + (-0.375 * (x ^ -1.5))) / x;
end

Wolfram

code[x_] := N[(N[(N[Sqrt[N[(0.25 / x), $MachinePrecision]], $MachinePrecision] + N[(-0.375 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 34.5%

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

3. Taylor expanded in x around inf 77.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
5. Step-by-step derivation

   1. fma-define 98.8%

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

   2. distribute-rgt-out 98.8%

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

   3. metadata-eval 98.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

6. Simplified 98.8%

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

   1. *-un-lft-identity 98.8%

      \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]

   2. div-inv 98.8%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot 0.375\right) \cdot \frac{1}{x}}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   3. *-commutative 98.8%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \color{blue}{\left(0.375 \cdot \sqrt{\frac{1}{x}}\right)} \cdot \frac{1}{x}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   4. associate-*l* 98.8%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, \color{blue}{0.375 \cdot \left(\sqrt{\frac{1}{x}} \cdot \frac{1}{x}\right)}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   5. *-commutative 98.8%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, 0.375 \cdot \color{blue}{\left(\frac{1}{x} \cdot \sqrt{\frac{1}{x}}\right)}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   6. pow1 98.8%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, 0.375 \cdot \left(\color{blue}{{\left(\frac{1}{x}\right)}^{1}} \cdot \sqrt{\frac{1}{x}}\right), 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   7. pow1/2 98.8%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, 0.375 \cdot \left({\left(\frac{1}{x}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{0.5}}\right), 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   8. pow-prod-up 98.8%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, 0.375 \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{\left(1 + 0.5\right)}}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   9. inv-pow 98.8%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, 0.375 \cdot {\color{blue}{\left({x}^{-1}\right)}}^{\left(1 + 0.5\right)}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   10. metadata-eval 98.8%

       \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, 0.375 \cdot {\left({x}^{-1}\right)}^{\color{blue}{1.5}}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   11. pow-pow 98.8%

       \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, 0.375 \cdot \color{blue}{{x}^{\left(-1 \cdot 1.5\right)}}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   12. metadata-eval 98.8%

       \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, 0.375 \cdot {x}^{\color{blue}{-1.5}}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   13. sqrt-div 98.8%

       \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, 0.375 \cdot {x}^{-1.5}, 0.5 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)}{x} \]

   14. metadata-eval 98.8%

       \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, 0.375 \cdot {x}^{-1.5}, 0.5 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)}{x} \]

   15. un-div-inv 98.8%

       \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, 0.375 \cdot {x}^{-1.5}, \color{blue}{\frac{0.5}{\sqrt{x}}}\right)}{x} \]

8. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{fma}\left(-1, 0.375 \cdot {x}^{-1.5}, \frac{0.5}{\sqrt{x}}\right)}{x}} \]
9. Step-by-step derivation

   1. *-lft-identity 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, 0.375 \cdot {x}^{-1.5}, \frac{0.5}{\sqrt{x}}\right)}{x}} \]

   2. fma-undefine 98.8%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(0.375 \cdot {x}^{-1.5}\right) + \frac{0.5}{\sqrt{x}}}}{x} \]

   3. neg-mul-1 98.8%

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

   4. +-commutative 98.8%

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

   5. unsub-neg 98.8%

      \[\leadsto \frac{\color{blue}{\frac{0.5}{\sqrt{x}} - 0.375 \cdot {x}^{-1.5}}}{x} \]

   6. cancel-sign-sub-inv 98.8%

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

   7. metadata-eval 98.8%

      \[\leadsto \frac{\frac{0.5}{\sqrt{x}} + \color{blue}{-0.375} \cdot {x}^{-1.5}}{x} \]

10. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{\sqrt{x}} + -0.375 \cdot {x}^{-1.5}}{x}} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 98.4%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{0.5}{\sqrt{x}}} \cdot \sqrt{\frac{0.5}{\sqrt{x}}}} + -0.375 \cdot {x}^{-1.5}}{x} \]

    2. sqrt-unprod 98.8%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{0.5}{\sqrt{x}} \cdot \frac{0.5}{\sqrt{x}}}} + -0.375 \cdot {x}^{-1.5}}{x} \]

    3. frac-times 98.7%

       \[\leadsto \frac{\sqrt{\color{blue}{\frac{0.5 \cdot 0.5}{\sqrt{x} \cdot \sqrt{x}}}} + -0.375 \cdot {x}^{-1.5}}{x} \]

    4. metadata-eval 98.7%

       \[\leadsto \frac{\sqrt{\frac{\color{blue}{0.25}}{\sqrt{x} \cdot \sqrt{x}}} + -0.375 \cdot {x}^{-1.5}}{x} \]

    5. add-sqr-sqrt 98.8%

       \[\leadsto \frac{\sqrt{\frac{0.25}{\color{blue}{x}}} + -0.375 \cdot {x}^{-1.5}}{x} \]

12. Applied egg-rr 98.8%

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{0.25}{x}}} + -0.375 \cdot {x}^{-1.5}}{x} \]
13. Add Preprocessing

Alternative 2: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.5%

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

3. Taylor expanded in x around inf 59.4%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 59.4%

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

   2. pow1/2 59.4%

      \[\leadsto 0.5 \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{{x}^{3}}\right)}^{0.5}}\right) \]

   3. pow-flip 60.2%

      \[\leadsto 0.5 \cdot \left(1 \cdot {\color{blue}{\left({x}^{\left(-3\right)}\right)}}^{0.5}\right) \]

   4. pow-pow 98.3%

      \[\leadsto 0.5 \cdot \left(1 \cdot \color{blue}{{x}^{\left(\left(-3\right) \cdot 0.5\right)}}\right) \]

   5. metadata-eval 98.3%

      \[\leadsto 0.5 \cdot \left(1 \cdot {x}^{\left(\color{blue}{-3} \cdot 0.5\right)}\right) \]

   6. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

   \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \]
6. Step-by-step derivation

   1. *-lft-identity 98.3%

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

7. Simplified 98.3%

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

8. Final simplification 98.3%

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

Alternative 3: 5.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 34.5%

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

3. Taylor expanded in x around 0 5.5%

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

   1. pow1/2 5.5%

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

   2. inv-pow 5.5%

      \[\leadsto {\color{blue}{\left({x}^{-1}\right)}}^{0.5} \]

   3. pow-pow 5.5%

      \[\leadsto \color{blue}{{x}^{\left(-1 \cdot 0.5\right)}} \]

   4. metadata-eval 5.5%

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

   5. *-un-lft-identity 5.5%

      \[\leadsto \color{blue}{1 \cdot {x}^{-0.5}} \]

5. Applied egg-rr 5.5%

   \[\leadsto \color{blue}{1 \cdot {x}^{-0.5}} \]
6. Step-by-step derivation

   1. *-lft-identity 5.5%

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

7. Simplified 5.5%

   \[\leadsto \color{blue}{{x}^{-0.5}} \]
8. Add Preprocessing

Developer Target 1: 98.1% 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 2024116 
(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)))))