2isqrt (example 3.6)

Percentage Accurate: 38.6% → 99.6%

Time: 5.8s

Alternatives: 12

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: 38.6% 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: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[(N[Sqrt[N[Sqrt[x], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lift-/.f64 N/A

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

   13. frac-2neg N/A

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

   14. metadata-eval N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 41.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

9. Applied rewrites 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \frac{\frac{1}{\frac{x}{\sqrt{x}}}}{\left(t\_0 + \sqrt{x}\right) \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (/ (/ 1.0 (/ x (sqrt x))) (* (+ t_0 (sqrt x)) t_0))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return (1.0 / (x / sqrt(x))) / ((t_0 + sqrt(x)) * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((1.0d0 + x))
    code = (1.0d0 / (x / sqrt(x))) / ((t_0 + sqrt(x)) * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	return (1.0 / (x / Math.sqrt(x))) / ((t_0 + Math.sqrt(x)) * t_0);
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	return (1.0 / (x / math.sqrt(x))) / ((t_0 + math.sqrt(x)) * t_0)

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	return Float64(Float64(1.0 / Float64(x / sqrt(x))) / Float64(Float64(t_0 + sqrt(x)) * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = (1.0 / (x / sqrt(x))) / ((t_0 + sqrt(x)) * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 / N[(x / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lift-/.f64 N/A

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

   13. frac-2neg N/A

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

   14. metadata-eval N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 41.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-neg.f64 99.3

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.3

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.3

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.3

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 99.3

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

9. Applied rewrites 99.3%

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

    1. unpow1 N/A

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

    2. metadata-eval N/A

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

    3. pow-prod-up N/A

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

    4. pow2 N/A

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

    5. lift-sqrt.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. rem-square-sqrt N/A

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

    8. inv-pow N/A

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

    9. div-inv N/A

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

    10. lower-/.f64 99.6

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 3: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \frac{\frac{\frac{1}{\sqrt{x}}}{t\_0}}{t\_0 + \sqrt{x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x)))) (/ (/ (/ 1.0 (sqrt x)) t_0) (+ t_0 (sqrt x)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return ((1.0 / sqrt(x)) / t_0) / (t_0 + sqrt(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((1.0d0 + x))
    code = ((1.0d0 / sqrt(x)) / t_0) / (t_0 + sqrt(x))
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	return ((1.0 / Math.sqrt(x)) / t_0) / (t_0 + Math.sqrt(x));
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	return ((1.0 / math.sqrt(x)) / t_0) / (t_0 + math.sqrt(x))

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	return Float64(Float64(Float64(1.0 / sqrt(x)) / t_0) / Float64(t_0 + sqrt(x)))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = ((1.0 / sqrt(x)) / t_0) / (t_0 + sqrt(x));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lift-/.f64 N/A

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

   13. frac-2neg N/A

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

   14. metadata-eval N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 41.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

9. Applied rewrites 99.3%

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

10. Final simplification 99.3%

    \[\leadsto \frac{\frac{\frac{1}{\sqrt{x}}}{\sqrt{1 + x}}}{\sqrt{1 + x} + \sqrt{x}} \]
11. Add Preprocessing

Alternative 4: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \frac{\frac{1}{\sqrt{x}}}{\left(t\_0 + \sqrt{x}\right) \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x)))) (/ (/ 1.0 (sqrt x)) (* (+ t_0 (sqrt x)) t_0))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return (1.0 / sqrt(x)) / ((t_0 + sqrt(x)) * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((1.0d0 + x))
    code = (1.0d0 / sqrt(x)) / ((t_0 + sqrt(x)) * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	return (1.0 / Math.sqrt(x)) / ((t_0 + Math.sqrt(x)) * t_0);
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	return (1.0 / math.sqrt(x)) / ((t_0 + math.sqrt(x)) * t_0)

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	return Float64(Float64(1.0 / sqrt(x)) / Float64(Float64(t_0 + sqrt(x)) * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = (1.0 / sqrt(x)) / ((t_0 + sqrt(x)) * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lift-/.f64 N/A

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

   13. frac-2neg N/A

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

   14. metadata-eval N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 41.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-neg.f64 99.3

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.3

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.3

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.3

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 99.3

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

9. Applied rewrites 99.3%

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

10. Final simplification 99.3%

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

Alternative 5: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(\frac{0.125}{x} - 0.5\right) - \frac{0.0625}{x \cdot x}}{x}}{-\sqrt{1 + x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lift-/.f64 N/A

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

   13. frac-2neg N/A

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

   14. metadata-eval N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 41.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in x around inf

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

    1. lower-/.f64 N/A

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

    2. associate--r+ N/A

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

    3. lower--.f64 N/A

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

    4. lower--.f64 N/A

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

    5. associate-*r/ N/A

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

    6. metadata-eval N/A

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

    7. lower-/.f64 N/A

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

    8. lower-/.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 99.3

        \[\leadsto \frac{\frac{\left(\frac{0.125}{x} - 0.5\right) - \frac{0.0625}{\color{blue}{x \cdot x}}}{x}}{-\sqrt{1 + x}} \]

12. Applied rewrites 99.3%

    \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.125}{x} - 0.5\right) - \frac{0.0625}{x \cdot x}}{x}}}{-\sqrt{1 + x}} \]
13. Add Preprocessing

Alternative 6: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (1.0 / (((0.5 / x) + 2.0) * x)) / sqrt((1.0 + x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (((0.5d0 / x) + 2.0d0) * x)) / sqrt((1.0d0 + x))
end function

Java

public static double code(double x) {
	return (1.0 / (((0.5 / x) + 2.0) * x)) / Math.sqrt((1.0 + x));
}

Python

def code(x):
	return (1.0 / (((0.5 / x) + 2.0) * x)) / math.sqrt((1.0 + x))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(Float64(Float64(0.5 / x) + 2.0) * x)) / sqrt(Float64(1.0 + x)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (((0.5 / x) + 2.0) * x)) / sqrt((1.0 + x));
end

Wolfram

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

Derivation

1. Initial program 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lift-/.f64 N/A

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

   13. frac-2neg N/A

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

   14. metadata-eval N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 41.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-+.f64 N/A

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

    4. associate-*r/ N/A

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

    5. metadata-eval N/A

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

    6. lower-/.f64 99.0

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

12. Applied rewrites 99.0%

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

13. Final simplification 99.0%

    \[\leadsto \frac{\frac{1}{\left(\frac{0.5}{x} + 2\right) \cdot x}}{\sqrt{1 + x}} \]
14. Add Preprocessing

Alternative 7: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lift-/.f64 N/A

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

   13. frac-2neg N/A

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

   14. metadata-eval N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 41.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in x around inf

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

    1. lower-/.f64 N/A

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

    2. lower--.f64 N/A

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

    5. lower-/.f64 98.9

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

12. Applied rewrites 98.9%

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

Alternative 8: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 - \frac{0.375}{x}}{x}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (- 0.5 (/ 0.375 x)) x) (sqrt x)))

C

double code(double x) {
	return ((0.5 - (0.375 / x)) / x) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 - (0.375d0 / x)) / x) / sqrt(x)
end function

Java

public static double code(double x) {
	return ((0.5 - (0.375 / x)) / x) / Math.sqrt(x);
}

Python

def code(x):
	return ((0.5 - (0.375 / x)) / x) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(0.5 - Float64(0.375 / x)) / x) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 - (0.375 / x)) / x) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(0.5 - N[(0.375 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 39.2%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}}{\sqrt{x}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}}{\sqrt{x}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}}{x}}{\sqrt{x}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{4} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}\right)} - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{4} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}\right)} - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{{x}^{2}}} + \frac{1}{2}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{\left(\frac{\color{blue}{\frac{1}{4}}}{{x}^{2}} + \frac{1}{2}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{4}}{{x}^{2}}} + \frac{1}{2}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{4}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{4}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   10. div-add-rev N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{1}{4}}{x \cdot x} + \frac{1}{2}\right) - \color{blue}{\frac{\frac{1}{8} + \frac{1}{4}}{x}}}{x}}{\sqrt{x}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{1}{4}}{x \cdot x} + \frac{1}{2}\right) - \frac{\color{blue}{\frac{3}{8}}}{x}}{x}}{\sqrt{x}} \]

   12. lower-/.f64 98.9

       \[\leadsto \frac{\frac{\left(\frac{0.25}{x \cdot x} + 0.5\right) - \color{blue}{\frac{0.375}{x}}}{x}}{\sqrt{x}} \]

7. Applied rewrites 98.9%

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.25}{x \cdot x} + 0.5\right) - \frac{0.375}{x}}{x}}}{\sqrt{x}} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{\frac{1}{2} - \frac{3}{8} \cdot \frac{1}{x}}{x}}{\sqrt{x}} \]
9. Step-by-step derivation

10. Applied rewrites 98.8%

    \[\leadsto \frac{\frac{0.5 - \frac{0.375}{x}}{x}}{\sqrt{x}} \]
11. Add Preprocessing

Alternative 9: 97.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lift-/.f64 N/A

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

   13. frac-2neg N/A

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

   14. metadata-eval N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 41.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in x around inf

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

    1. lower-/.f64 97.5

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

12. Applied rewrites 97.5%

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

Alternative 10: 97.8% 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 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 39.2%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.4

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

7. Applied rewrites 97.4%

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

Alternative 11: 37.2% 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 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 37.4%

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

Alternative 12: 5.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

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

Developer Target 1: 38.6% 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 2024312 
(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)))))