2isqrt (example 3.6)

Percentage Accurate: 38.7% → 99.3%

Time: 7.7s

Alternatives: 10

Speedup: 1.5×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.7% 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: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 37.4%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lower--.f64 99.2

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

6. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. +-inverses N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity 99.2

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 99.2

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

8. Applied rewrites 99.2%

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

9. Final simplification 99.2%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\frac{0.0625}{x} - 0.125}{x} + 0.5}{x}}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (+ (/ (- (/ 0.0625 x) 0.125) x) 0.5) x) (sqrt (+ x 1.0))))

C

double code(double x) {
	return (((((0.0625 / x) - 0.125) / x) + 0.5) / x) / sqrt((x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((((0.0625d0 / x) - 0.125d0) / x) + 0.5d0) / x) / sqrt((x + 1.0d0))
end function

Java

public static double code(double x) {
	return (((((0.0625 / x) - 0.125) / x) + 0.5) / x) / Math.sqrt((x + 1.0));
}

Python

def code(x):
	return (((((0.0625 / x) - 0.125) / x) + 0.5) / x) / math.sqrt((x + 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(0.0625 / x) - 0.125) / x) + 0.5) / x) / sqrt(Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (((((0.0625 / x) - 0.125) / x) + 0.5) / x) / sqrt((x + 1.0));
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(0.0625 / x), $MachinePrecision] - 0.125), $MachinePrecision] / x), $MachinePrecision] + 0.5), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-/.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lower--.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. sqrt-unprod N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lower-*.f64 8.7

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

6. Applied rewrites 8.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 8.6

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

8. Applied rewrites 8.6%

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

9. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{2}}\right) - \frac{1}{8} \cdot \frac{1}{x}}}{x}}{\sqrt{x + 1}} \]
10. Step-by-step derivation

    1. associate--l+ N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} + \left(\frac{\frac{1}{16}}{{x}^{2}} - \frac{1}{8} \cdot \frac{1}{x}\right)}}{x}}{\sqrt{x + 1}} \]

    2. +-commutative N/A

       \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\frac{1}{16}}{{x}^{2}} - \frac{1}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}}{x}}{\sqrt{x + 1}} \]

    3. unpow2 N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{1}{16}}{\color{blue}{x \cdot x}} - \frac{1}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x + 1}} \]

    4. associate-/r* N/A

       \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{\frac{1}{16}}{x}}{x}} - \frac{1}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x + 1}} \]

    5. metadata-eval N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{\color{blue}{\frac{1}{16} \cdot 1}}{x}}{x} - \frac{1}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x + 1}} \]

    6. associate-*r/ N/A

       \[\leadsto \frac{\frac{\left(\frac{\color{blue}{\frac{1}{16} \cdot \frac{1}{x}}}{x} - \frac{1}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x + 1}} \]

    7. associate-*r/ N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{1}{16} \cdot \frac{1}{x}}{x} - \color{blue}{\frac{\frac{1}{8} \cdot 1}{x}}\right) + \frac{1}{2}}{x}}{\sqrt{x + 1}} \]

    8. metadata-eval N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{1}{16} \cdot \frac{1}{x}}{x} - \frac{\color{blue}{\frac{1}{8}}}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x + 1}} \]

    9. div-sub N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{16} \cdot \frac{1}{x} - \frac{1}{8}}{x}} + \frac{1}{2}}{x}}{\sqrt{x + 1}} \]

    10. lower-+.f64 N/A

        \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{16} \cdot \frac{1}{x} - \frac{1}{8}}{x} + \frac{1}{2}}}{x}}{\sqrt{x + 1}} \]

    11. lower-/.f64 N/A

        \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{16} \cdot \frac{1}{x} - \frac{1}{8}}{x}} + \frac{1}{2}}{x}}{\sqrt{x + 1}} \]

    12. lower--.f64 N/A

        \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{1}{16} \cdot \frac{1}{x} - \frac{1}{8}}}{x} + \frac{1}{2}}{x}}{\sqrt{x + 1}} \]

    13. associate-*r/ N/A

        \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{\frac{1}{16} \cdot 1}{x}} - \frac{1}{8}}{x} + \frac{1}{2}}{x}}{\sqrt{x + 1}} \]

    14. metadata-eval N/A

        \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{\frac{1}{16}}}{x} - \frac{1}{8}}{x} + \frac{1}{2}}{x}}{\sqrt{x + 1}} \]

    15. lower-/.f64 98.5

        \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{0.0625}{x}} - 0.125}{x} + 0.5}{x}}{\sqrt{x + 1}} \]

11. Applied rewrites 98.5%

    \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{0.0625}{x} - 0.125}{x} + 0.5}}{x}}{\sqrt{x + 1}} \]
12. Add Preprocessing

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 - \frac{0.125}{x}}{x}}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (- 0.5 (/ 0.125 x)) x) (sqrt (+ x 1.0))))

C

double code(double x) {
	return ((0.5 - (0.125 / x)) / x) / sqrt((x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 - (0.125d0 / x)) / x) / sqrt((x + 1.0d0))
end function

Java

public static double code(double x) {
	return ((0.5 - (0.125 / x)) / x) / Math.sqrt((x + 1.0));
}

Python

def code(x):
	return ((0.5 - (0.125 / x)) / x) / math.sqrt((x + 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(0.5 - Float64(0.125 / x)) / x) / sqrt(Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 - (0.125 / x)) / x) / sqrt((x + 1.0));
end

Wolfram

code[x_] := N[(N[(N[(0.5 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 34.5%

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

5. Taylor expanded in x around inf

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

   1. sub-neg N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{x}\right)\right)}}{x}}{\sqrt{x + 1}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{8} \cdot 1}{x}}\right)\right)}{x}}{\sqrt{x + 1}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{8}}}{x}\right)\right)}{x}}{\sqrt{x + 1}} \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \color{blue}{\frac{\frac{1}{8}}{\mathsf{neg}\left(x\right)}}}{x}}{\sqrt{x + 1}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\color{blue}{\frac{1}{8} \cdot 1}}{\mathsf{neg}\left(x\right)}}{x}}{\sqrt{x + 1}} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\frac{1}{8} \cdot 1}{\color{blue}{-1 \cdot x}}}{x}}{\sqrt{x + 1}} \]

   7. rem-square-sqrt N/A

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

   8. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\frac{1}{8} \cdot 1}{\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot x}}{x}}{\sqrt{x + 1}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \frac{\frac{1}{8} \cdot 1}{\color{blue}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}}{x}}{\sqrt{x + 1}} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{\frac{1}{2} + \color{blue}{\frac{1}{8} \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}}{x}}{\sqrt{x + 1}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x}}}{\sqrt{x + 1}} \]

7. Applied rewrites 98.1%

   \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.125}{x}}{x}}}{\sqrt{x + 1}} \]
8. Add Preprocessing

Alternative 4: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{\sqrt{x + 1}}}{\mathsf{fma}\left(-2, x, -0.5\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ -1.0 (sqrt (+ x 1.0))) (fma -2.0 x -0.5)))

C

double code(double x) {
	return (-1.0 / sqrt((x + 1.0))) / fma(-2.0, x, -0.5);
}

Julia

function code(x)
	return Float64(Float64(-1.0 / sqrt(Float64(x + 1.0))) / fma(-2.0, x, -0.5))
end

Wolfram

code[x_] := N[(N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * x + -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 37.4%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lower--.f64 99.2

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

6. Applied rewrites 99.2%

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

7. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. distribute-lft-in N/A

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

   3. neg-mul-1 N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lft-mult-inverse N/A

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

   8. *-lft-identity N/A

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

   9. distribute-lft-neg-in N/A

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

   10. metadata-eval N/A

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

   11. associate-*l* N/A

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

   12. lft-mult-inverse N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 98.2

       \[\leadsto \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{-1}{\sqrt{x + 1}}}{\color{blue}{\mathsf{fma}\left(-2, x, -0.5\right)}} \]

9. Applied rewrites 98.2%

   \[\leadsto \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{-1}{\sqrt{x + 1}}}{\color{blue}{\mathsf{fma}\left(-2, x, -0.5\right)}} \]

10. Final simplification 98.2%

    \[\leadsto \frac{\frac{-1}{\sqrt{x + 1}}}{\mathsf{fma}\left(-2, x, -0.5\right)} \]
11. Add Preprocessing

Alternative 5: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{\sqrt{1 + x}}}{-2 \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ -1.0 (sqrt (+ 1.0 x))) (* -2.0 x)))

C

double code(double x) {
	return (-1.0 / sqrt((1.0 + x))) / (-2.0 * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-1.0d0) / sqrt((1.0d0 + x))) / ((-2.0d0) * x)
end function

Java

public static double code(double x) {
	return (-1.0 / Math.sqrt((1.0 + x))) / (-2.0 * x);
}

Python

def code(x):
	return (-1.0 / math.sqrt((1.0 + x))) / (-2.0 * x)

Julia

function code(x)
	return Float64(Float64(-1.0 / sqrt(Float64(1.0 + x))) / Float64(-2.0 * x))
end

MATLAB

function tmp = code(x)
	tmp = (-1.0 / sqrt((1.0 + x))) / (-2.0 * x);
end

Wolfram

code[x_] := N[(N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 37.4%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lower--.f64 99.2

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

6. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. +-inverses N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity 99.2

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 99.2

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around inf

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

    1. lower-*.f64 97.4

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

11. Applied rewrites 97.4%

    \[\leadsto \frac{\frac{-1}{\sqrt{1 + x}}}{\color{blue}{-2 \cdot x}} \]
12. Add Preprocessing

Alternative 6: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 34.5%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.4

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

7. Applied rewrites 97.4%

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

Alternative 7: 80.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. Applied rewrites 83.5%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 82.7%

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

Alternative 8: 37.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x}}{\mathsf{fma}\left(0.5, x, 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 0.5 x) (fma 0.5 x 1.0)))

C

double code(double x) {
	return (0.5 / x) / fma(0.5, x, 1.0);
}

Julia

function code(x)
	return Float64(Float64(0.5 / x) / fma(0.5, x, 1.0))
end

Wolfram

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

Derivation

1. Initial program 34.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 34.5%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.4

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

7. Applied rewrites 97.4%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 33.3

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

10. Applied rewrites 33.3%

    \[\leadsto \frac{\frac{0.5}{x}}{\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)}} \]
11. Add Preprocessing

Alternative 9: 37.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.8

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

5. Applied rewrites 5.8%

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

7. Applied rewrites 32.3%

   \[\leadsto \sqrt{\frac{x}{x \cdot x}} \]
8. Add Preprocessing

Alternative 10: 35.8% 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 34.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-/.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lower--.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. sqrt-unprod N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lower-*.f64 8.7

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

6. Applied rewrites 8.7%

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

7. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

   2. associate-*r* N/A

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

   3. distribute-lft-neg-in N/A

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

   4. unpow2 N/A

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

   5. rem-square-sqrt N/A

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

   6. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \color{blue}{0} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot 0\right)} \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \left(\mathsf{neg}\left(0\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{0} \]

   10. mul0-rgt 30.9

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

9. Applied rewrites 30.9%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 38.8% 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 2024324 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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