2isqrt (example 3.6)

Percentage Accurate: 38.8% → 98.0%

Time: 10.9s

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: 38.8% 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: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in x around inf 79.6%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]

4. Taylor expanded in x around inf 98.5%

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

   1. metadata-eval 98.5%

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

   2. cancel-sign-sub-inv 98.5%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} - -0.5 \cdot \sqrt{\frac{1}{x}}}}{x} \]

   3. distribute-lft-out-- 98.5%

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

   4. associate-/l* 98.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{\frac{1}{{x}^{3}}} - \sqrt{\frac{1}{x}}}{x}} \]

   5. div-sub 98.5%

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

   6. unpow1/2 98.5%

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

   7. rem-exp-log 94.2%

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

   8. exp-neg 94.2%

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

   9. exp-prod 94.2%

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

   10. distribute-lft-neg-out 94.2%

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

   11. distribute-rgt-neg-in 94.2%

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

   12. metadata-eval 94.2%

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

   13. exp-to-pow 98.6%

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

   14. *-lft-identity 98.6%

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

   15. associate-*l/ 98.4%

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

   16. unpow-1 98.4%

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

   17. metadata-eval 98.4%

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

   18. pow-sqr 98.0%

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

6. Simplified 98.8%

   \[\leadsto \color{blue}{-0.5 \cdot \left(\frac{\sqrt{\frac{1}{{x}^{3}}}}{x} - {x}^{-1.5}\right)} \]
7. Step-by-step derivation

   1. *-commutative 98.8%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{1}{{x}^{3}}}}{x} - {x}^{-1.5}\right) \cdot -0.5} \]

   2. pow-flip 98.8%

      \[\leadsto \left(\frac{\sqrt{\color{blue}{{x}^{\left(-3\right)}}}}{x} - {x}^{-1.5}\right) \cdot -0.5 \]

   3. sqrt-pow1 98.8%

      \[\leadsto \left(\frac{\color{blue}{{x}^{\left(\frac{-3}{2}\right)}}}{x} - {x}^{-1.5}\right) \cdot -0.5 \]

   4. metadata-eval 98.8%

      \[\leadsto \left(\frac{{x}^{\left(\frac{\color{blue}{-3}}{2}\right)}}{x} - {x}^{-1.5}\right) \cdot -0.5 \]

   5. metadata-eval 98.8%

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

8. Applied egg-rr 98.8%

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

   1. div-inv 98.8%

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

   2. fma-neg 98.8%

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

10. Applied egg-rr 98.8%

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

    1. fma-undefine 98.8%

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

    2. *-commutative 98.8%

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

    3. neg-mul-1 98.8%

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

    4. distribute-rgt-out 98.8%

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

12. Simplified 98.8%

    \[\leadsto \color{blue}{\left({x}^{-1.5} \cdot \left(\frac{1}{x} + -1\right)\right)} \cdot -0.5 \]
13. Add Preprocessing

Alternative 2: 97.9% 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 34.7%

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

3. Taylor expanded in x around inf 61.6%

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

   1. *-commutative 61.6%

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

   2. pow-flip 61.9%

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

   3. sqrt-pow1 98.7%

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

   4. metadata-eval 98.7%

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

   5. metadata-eval 98.7%

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

5. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} \]
6. 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 34.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 34.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)\right)} - \frac{1}{\sqrt{x + 1}} \]

   2. expm1-undefine 4.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]

   3. inv-pow 4.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{x}\right)}^{-1}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   4. sqrt-pow2 4.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   5. metadata-eval 4.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

4. Applied egg-rr 4.8%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]
5. Step-by-step derivation

   1. log1p-undefine 4.8%

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

   2. rem-exp-log 4.8%

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

   3. +-commutative 4.8%

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

   4. associate--l+ 27.1%

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

   5. metadata-eval 27.1%

      \[\leadsto \left({x}^{-0.5} + \color{blue}{0}\right) - \frac{1}{\sqrt{x + 1}} \]

   6. +-rgt-identity 27.1%

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

6. Simplified 27.1%

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

7. Taylor expanded in x around 0 5.5%

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

   1. unpow1/2 5.5%

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

   2. rem-exp-log 5.5%

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

   3. exp-neg 5.5%

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

   4. exp-prod 5.5%

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

   5. distribute-lft-neg-out 5.5%

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

   6. distribute-rgt-neg-in 5.5%

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

   7. metadata-eval 5.5%

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

   8. exp-to-pow 5.5%

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

9. Simplified 5.5%

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

Developer target: 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 2024096 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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