2isqrt (example 3.6)

Percentage Accurate: 38.8% → 98.4%

Time: 9.7s

Alternatives: 4

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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.9%

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

   1. sub-neg 34.9%

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

   2. inv-pow 34.9%

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

   3. sqrt-pow2 27.5%

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

   4. metadata-eval 27.5%

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

   5. distribute-neg-frac 27.5%

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

   6. metadata-eval 27.5%

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

   7. +-commutative 27.5%

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

4. Applied egg-rr 27.5%

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

   1. *-rgt-identity 27.5%

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

   2. cancel-sign-sub 27.5%

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

   3. distribute-lft-neg-in 27.5%

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

   4. *-rgt-identity 27.5%

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

   5. distribute-neg-frac 27.5%

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

   6. metadata-eval 27.5%

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

   7. unpow1/2 27.5%

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

   8. exp-to-pow 7.5%

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

   9. log1p-undefine 7.5%

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

   10. *-commutative 7.5%

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

   11. exp-neg 7.5%

       \[\leadsto {x}^{-0.5} - \color{blue}{e^{-0.5 \cdot \mathsf{log1p}\left(x\right)}} \]

   12. *-commutative 7.5%

       \[\leadsto {x}^{-0.5} - e^{-\color{blue}{\mathsf{log1p}\left(x\right) \cdot 0.5}} \]

   13. distribute-rgt-neg-in 7.5%

       \[\leadsto {x}^{-0.5} - e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \left(-0.5\right)}} \]

   14. log1p-undefine 7.5%

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

   15. metadata-eval 7.5%

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

   16. exp-to-pow 35.0%

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

6. Simplified 35.0%

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

   1. flip-- 35.0%

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

   2. pow-prod-up 22.3%

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

   3. metadata-eval 22.3%

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

   4. inv-pow 21.9%

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

   5. pow-prod-up 34.7%

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

   6. metadata-eval 34.7%

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

   7. inv-pow 35.1%

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

   8. div-inv 35.1%

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

   9. frac-sub 36.9%

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

   10. frac-times 36.9%

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

   11. *-un-lft-identity 36.9%

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

8. Applied egg-rr 36.9%

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

   1. *-rgt-identity 36.9%

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

   2. *-rgt-identity 36.9%

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

   3. associate--l+ 81.7%

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

   4. +-inverses 81.7%

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

   5. metadata-eval 81.7%

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

   6. associate-*l* 98.4%

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

10. Simplified 98.4%

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

    1. distribute-lft-in 98.4%

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

    2. pow1 98.4%

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

    3. pow-prod-up 98.4%

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

    4. metadata-eval 98.4%

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

    5. pow1/2 98.4%

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

    6. distribute-rgt-in 98.4%

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

12. Applied egg-rr 98.4%

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

13. Final simplification 98.4%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.9%

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

   1. sub-neg 34.9%

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

   2. inv-pow 34.9%

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

   3. sqrt-pow2 27.5%

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

   4. metadata-eval 27.5%

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

   5. distribute-neg-frac 27.5%

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

   6. metadata-eval 27.5%

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

   7. +-commutative 27.5%

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

4. Applied egg-rr 27.5%

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

   1. *-rgt-identity 27.5%

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

   2. cancel-sign-sub 27.5%

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

   3. distribute-lft-neg-in 27.5%

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

   4. *-rgt-identity 27.5%

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

   5. distribute-neg-frac 27.5%

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

   6. metadata-eval 27.5%

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

   7. unpow1/2 27.5%

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

   8. exp-to-pow 7.5%

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

   9. log1p-undefine 7.5%

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

   10. *-commutative 7.5%

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

   11. exp-neg 7.5%

       \[\leadsto {x}^{-0.5} - \color{blue}{e^{-0.5 \cdot \mathsf{log1p}\left(x\right)}} \]

   12. *-commutative 7.5%

       \[\leadsto {x}^{-0.5} - e^{-\color{blue}{\mathsf{log1p}\left(x\right) \cdot 0.5}} \]

   13. distribute-rgt-neg-in 7.5%

       \[\leadsto {x}^{-0.5} - e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \left(-0.5\right)}} \]

   14. log1p-undefine 7.5%

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

   15. metadata-eval 7.5%

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

   16. exp-to-pow 35.0%

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

6. Simplified 35.0%

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

   1. flip-- 35.0%

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

   2. pow-prod-up 22.3%

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

   3. metadata-eval 22.3%

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

   4. inv-pow 21.9%

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

   5. pow-prod-up 34.7%

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

   6. metadata-eval 34.7%

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

   7. inv-pow 35.1%

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

   8. div-inv 35.1%

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

   9. frac-sub 36.9%

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

   10. frac-times 36.9%

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

   11. *-un-lft-identity 36.9%

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

8. Applied egg-rr 36.9%

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

   1. *-rgt-identity 36.9%

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

   2. *-rgt-identity 36.9%

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

   3. associate--l+ 81.7%

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

   4. +-inverses 81.7%

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

   5. metadata-eval 81.7%

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

   6. associate-*l* 98.4%

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

10. Simplified 98.4%

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

    1. distribute-lft-in 98.4%

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

    2. pow1 98.4%

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

    3. pow-prod-up 98.4%

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

    4. metadata-eval 98.4%

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

    5. pow1/2 98.4%

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

12. Applied egg-rr 98.4%

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

Alternative 3: 97.7% 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 34.9%

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

   1. add-cube-cbrt 34.9%

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

   2. pow3 34.9%

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

   3. inv-pow 34.9%

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

   4. sqrt-pow2 27.5%

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

   5. metadata-eval 27.5%

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

   6. inv-pow 27.5%

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

   7. sqrt-pow2 35.0%

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

   8. +-commutative 35.0%

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

   9. metadata-eval 35.0%

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

4. Applied egg-rr 35.0%

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

5. Taylor expanded in x around inf 96.6%

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

   1. unpow-prod-down 96.7%

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

   2. pow3 96.7%

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

   3. add-sqr-sqrt 96.7%

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

   4. sqrt-div 96.6%

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

   5. metadata-eval 96.6%

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

   6. clear-num 96.6%

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

   7. frac-times 95.5%

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

   8. /-rgt-identity 95.5%

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

   9. add-sqr-sqrt 95.5%

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

   10. sqrt-unprod 79.1%

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

   11. sqrt-prod 65.3%

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

   12. unpow3 65.3%

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

   13. metadata-eval 65.3%

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

   14. metadata-eval 65.3%

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

   15. sqrt-div 65.3%

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

   16. pow-flip 65.8%

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

   17. sqrt-pow1 96.5%

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

   18. metadata-eval 96.5%

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

   19. metadata-eval 96.5%

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

   20. rem-cube-cbrt 97.8%

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

7. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} \]
8. Add Preprocessing

Alternative 4: 5.7% 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 34.9%

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

3. Taylor expanded in x around 0 5.7%

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

   1. inv-pow 5.7%

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

   2. sqrt-pow1 5.7%

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

   3. metadata-eval 5.7%

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

   4. *-un-lft-identity 5.7%

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

5. Applied egg-rr 5.7%

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

   1. *-lft-identity 5.7%

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

7. Simplified 5.7%

   \[\leadsto \color{blue}{{x}^{-0.5}} \]
8. 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 2024103 
(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)))))