2isqrt (example 3.6)

Percentage Accurate: 38.0% → 98.4%

Time: 11.1s

Alternatives: 7

Speedup: 1.8×

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: 38.0% 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

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(1 + x, \sqrt{x}, x \cdot \sqrt{1 + x}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (fma (+ 1.0 x) (sqrt x) (* x (sqrt (+ 1.0 x))))))

C

double code(double x) {
	return 1.0 / fma((1.0 + x), sqrt(x), (x * sqrt((1.0 + x))));
}

Julia

function code(x)
	return Float64(1.0 / fma(Float64(1.0 + x), sqrt(x), Float64(x * sqrt(Float64(1.0 + x)))))
end

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. clear-num N/A

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

   2. frac-sub N/A

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

   3. /-rgt-identity N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. flip-- N/A

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

   8. *-rgt-identity N/A

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

   9. metadata-eval N/A

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

   10. div-inv N/A

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

   11. /-rgt-identity N/A

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

   12. associate-/l/ N/A

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

4. Applied egg-rr 37.6%

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

   1. associate--l+ N/A

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

   2. +-inverses N/A

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

   3. metadata-eval 82.5

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

6. Applied egg-rr 82.5%

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

   1. distribute-rgt-in N/A

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

   2. distribute-lft1-in N/A

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

   3. +-commutative N/A

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

   4. sqrt-prod N/A

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

   5. associate-*r* N/A

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

   6. rem-square-sqrt N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

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

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

   9. sqrt-lowering-sqrt.f64 N/A

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

   10. distribute-lft1-in N/A

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

   11. +-commutative N/A

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

   12. sqrt-prod N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. rem-square-sqrt N/A

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

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

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

   17. sqrt-lowering-sqrt.f64 N/A

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

   18. +-lowering-+.f64 98.7

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

8. Applied egg-rr 98.7%

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

Alternative 2: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. clear-num N/A

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

   2. frac-sub N/A

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

   3. /-rgt-identity N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. flip-- N/A

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

   8. *-rgt-identity N/A

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

   9. metadata-eval N/A

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

   10. div-inv N/A

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

   11. /-rgt-identity N/A

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

   12. associate-/l/ N/A

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

4. Applied egg-rr 37.6%

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

   1. associate--l+ N/A

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

   2. +-inverses N/A

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

   3. metadata-eval 82.5

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

6. Applied egg-rr 82.5%

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

7. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-*l* N/A

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

   5. lft-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. +-lowering-+.f64 97.7

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

9. Simplified 97.7%

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

10. Final simplification 97.7%

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

Alternative 3: 97.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{\sqrt{\frac{1}{x}}}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 81.4%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
7. Step-by-step derivation

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

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. cube-mult N/A

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 67.2

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

8. Simplified 67.2%

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

   1. associate-/r* N/A

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

   2. sqrt-div N/A

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

   3. sqrt-unprod N/A

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

   4. rem-square-sqrt N/A

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

   5. /-lowering-/.f64 N/A

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

   6. sqrt-div N/A

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

   7. metadata-eval N/A

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

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

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

   9. sqrt-lowering-sqrt.f64 97.4

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

10. Applied egg-rr 97.4%

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

11. Taylor expanded in x around 0

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

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

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

    2. /-lowering-/.f64 97.5

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

13. Simplified 97.5%

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

Alternative 4: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x}}{\sqrt{x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 81.4%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
7. Step-by-step derivation

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

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. cube-mult N/A

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 67.2

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

8. Simplified 67.2%

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

   1. associate-/r* N/A

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

   2. sqrt-div N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. sqrt-unprod N/A

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

   6. rem-square-sqrt N/A

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

   7. associate-/r* N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

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

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

   11. /-lowering-/.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 97.4

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

10. Applied egg-rr 97.4%

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

    1. associate-*r/ N/A

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

    2. un-div-inv N/A

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

    3. metadata-eval N/A

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

    4. distribute-neg-frac N/A

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

    5. /-lowering-/.f64 N/A

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

    6. distribute-neg-frac N/A

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

    7. metadata-eval N/A

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

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

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

    9. sqrt-lowering-sqrt.f64 97.4

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

12. Applied egg-rr 97.4%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{x}}{\sqrt{x}}} \]
13. Add Preprocessing

Alternative 5: 37.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.6e+153) (/ 1.0 (+ x (sqrt x))) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / (x + sqrt(x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.6d+153) then
        tmp = 1.0d0 / (x + sqrt(x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / (x + Math.sqrt(x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.6e+153:
		tmp = 1.0 / (x + math.sqrt(x))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.6e+153)
		tmp = Float64(1.0 / Float64(x + sqrt(x)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.6e+153)
		tmp = 1.0 / (x + sqrt(x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.6e+153], N[(1.0 / N[(x + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.6000000000000003e153

   1. Initial program 10.2%

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

      1. clear-num N/A

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

      2. frac-sub N/A

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

      3. /-rgt-identity N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. div-inv N/A

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

      7. flip-- N/A

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

      8. *-rgt-identity N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. /-rgt-identity N/A

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

      12. associate-/l/ N/A

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

   4. Applied egg-rr 15.0%

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

      1. associate--l+ N/A

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

      2. +-inverses N/A

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

      3. metadata-eval 99.5

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. rem-square-sqrt N/A

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

      4. *-rgt-identity N/A

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

      5. +-lowering-+.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 8.5

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

   9. Simplified 8.5%

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

   if 4.6000000000000003e153 < x

   1. Initial program 63.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. /-lowering-/.f64 43.0

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

   5. Simplified 43.0%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. +-inverses 63.2

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.5 (* x (sqrt x))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(0.5 / Float64(x * sqrt(x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 81.4%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
7. Step-by-step derivation

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

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. cube-mult N/A

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 67.2

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

8. Simplified 67.2%

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

   1. associate-/r* N/A

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

   2. sqrt-div N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. sqrt-unprod N/A

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

   6. rem-square-sqrt N/A

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

   7. associate-/r* N/A

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

   8. *-commutative N/A

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

   9. un-div-inv N/A

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

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

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

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

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

   12. sqrt-lowering-sqrt.f64 96.6

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

10. Applied egg-rr 96.6%

    \[\leadsto \color{blue}{\frac{0.5}{x \cdot \sqrt{x}}} \]
11. Add Preprocessing

Alternative 7: 35.1% accurate, 49.0× 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 35.1%

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

3. Taylor expanded in x around inf

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

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

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

   2. /-lowering-/.f64 22.6

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

5. Simplified 22.6%

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

   1. metadata-eval N/A

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

   2. sqrt-div N/A

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

   3. +-inverses 31.9

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 31.9%

   \[\leadsto \color{blue}{0} \]
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]

Developer Target 2: 38.0% 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 2024197 
(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)))))