2isqrt (example 3.6)

Percentage Accurate: 38.6% → 99.6%

Time: 10.4s

Alternatives: 9

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.6% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

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

   1. frac-sub 39.4%

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

   2. *-un-lft-identity 39.4%

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

   3. +-commutative 39.4%

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

   4. *-rgt-identity 39.4%

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

   5. sqrt-unprod 39.4%

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

   6. +-commutative 39.4%

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

4. Applied egg-rr 39.4%

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

   1. flip-- 40.2%

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

   2. add-sqr-sqrt 39.7%

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

   3. add-sqr-sqrt 41.5%

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

6. Applied egg-rr 41.5%

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

   1. associate--l+ 85.7%

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

   2. +-inverses 85.7%

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

   3. metadata-eval 85.7%

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

   4. +-commutative 85.7%

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

8. Simplified 85.7%

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

   1. *-un-lft-identity 85.7%

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

   2. associate-/l/ 85.7%

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

   3. *-commutative 85.7%

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

   4. distribute-rgt-in 85.7%

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

   5. *-un-lft-identity 85.7%

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

   6. add-sqr-sqrt 85.7%

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

   7. hypot-define 98.8%

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

10. Applied egg-rr 98.8%

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

    1. *-lft-identity 98.8%

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

    2. *-commutative 98.8%

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

    3. associate-/r* 99.7%

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

12. Simplified 99.7%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

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

   1. frac-sub 39.4%

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

   2. *-un-lft-identity 39.4%

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

   3. +-commutative 39.4%

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

   4. *-rgt-identity 39.4%

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

   5. sqrt-unprod 39.4%

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

   6. +-commutative 39.4%

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

4. Applied egg-rr 39.4%

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

   1. flip-- 40.2%

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

   2. add-sqr-sqrt 39.7%

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

   3. add-sqr-sqrt 41.5%

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

6. Applied egg-rr 41.5%

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

   1. associate--l+ 85.7%

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

   2. +-inverses 85.7%

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

   3. metadata-eval 85.7%

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

   4. +-commutative 85.7%

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

8. Simplified 85.7%

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

9. Taylor expanded in x around -inf 0.0%

   \[\leadsto \frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)}} \]
10. Step-by-step derivation

    1. mul-1-neg 0.0%

       \[\leadsto \frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\color{blue}{-x \cdot \left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)}} \]

    2. *-commutative 0.0%

       \[\leadsto \frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{-\color{blue}{\left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right) \cdot x}} \]

    3. distribute-rgt-neg-in 0.0%

       \[\leadsto \frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\color{blue}{\left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right) \cdot \left(-x\right)}} \]

11. Simplified 99.5%

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

12. Taylor expanded in x around inf 99.5%

    \[\leadsto \frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\left(-1 - \color{blue}{\frac{0.5 - 0.125 \cdot \frac{1}{x}}{x}}\right) \cdot \left(-x\right)} \]
13. Step-by-step derivation

    1. associate-*r/ 99.5%

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

    2. metadata-eval 99.5%

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

14. Simplified 99.5%

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

15. Final simplification 99.5%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

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

   1. frac-sub 39.4%

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

   2. *-un-lft-identity 39.4%

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

   3. +-commutative 39.4%

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

   4. *-rgt-identity 39.4%

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

   5. sqrt-unprod 39.4%

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

   6. +-commutative 39.4%

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

4. Applied egg-rr 39.4%

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

   1. flip-- 40.2%

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

   2. add-sqr-sqrt 39.7%

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

   3. add-sqr-sqrt 41.5%

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

6. Applied egg-rr 41.5%

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

   1. associate--l+ 85.7%

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

   2. +-inverses 85.7%

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

   3. metadata-eval 85.7%

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

   4. +-commutative 85.7%

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

8. Simplified 85.7%

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

9. Taylor expanded in x around inf 99.1%

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

    1. associate-*r/ 99.1%

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

    2. metadata-eval 99.1%

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

11. Simplified 99.1%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

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

   1. frac-sub 39.4%

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

   2. *-un-lft-identity 39.4%

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

   3. +-commutative 39.4%

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

   4. *-rgt-identity 39.4%

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

   5. sqrt-unprod 39.4%

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

   6. +-commutative 39.4%

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

4. Applied egg-rr 39.4%

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

   1. flip-- 40.2%

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

   2. add-sqr-sqrt 39.7%

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

   3. add-sqr-sqrt 41.5%

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

6. Applied egg-rr 41.5%

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

   1. associate--l+ 85.7%

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

   2. +-inverses 85.7%

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

   3. metadata-eval 85.7%

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

   4. +-commutative 85.7%

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

8. Simplified 85.7%

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

   1. *-un-lft-identity 85.7%

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

   2. associate-/l/ 85.7%

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

   3. *-commutative 85.7%

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

   4. distribute-rgt-in 85.7%

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

   5. *-un-lft-identity 85.7%

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

   6. add-sqr-sqrt 85.7%

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

   7. hypot-define 98.8%

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

10. Applied egg-rr 98.8%

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

    1. *-lft-identity 98.8%

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

    2. *-commutative 98.8%

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

    3. associate-/r* 99.7%

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

12. Simplified 99.7%

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

13. Taylor expanded in x around inf 99.0%

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

    1. associate-*r/ 99.0%

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

    2. metadata-eval 99.0%

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

15. Simplified 99.0%

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

Alternative 5: 97.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

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

   1. frac-sub 39.4%

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

   2. *-un-lft-identity 39.4%

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

   3. +-commutative 39.4%

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

   4. *-rgt-identity 39.4%

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

   5. sqrt-unprod 39.4%

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

   6. +-commutative 39.4%

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

4. Applied egg-rr 39.4%

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

5. Taylor expanded in x around inf 84.3%

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

   1. *-commutative 84.3%

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

   2. unpow1/2 84.3%

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

   3. rem-exp-log 81.4%

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

   4. exp-neg 81.4%

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

   5. exp-prod 81.4%

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

   6. distribute-lft-neg-out 81.4%

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

   7. distribute-rgt-neg-in 81.4%

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

   8. metadata-eval 81.4%

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

   9. exp-to-pow 84.3%

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

7. Simplified 84.3%

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

8. Taylor expanded in x around inf 98.3%

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

   1. associate-*r/ 99.1%

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

   2. metadata-eval 99.1%

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

10. Simplified 98.3%

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

11. Final simplification 98.3%

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

Alternative 6: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

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

   1. frac-sub 39.4%

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

   2. *-un-lft-identity 39.4%

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

   3. +-commutative 39.4%

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

   4. *-rgt-identity 39.4%

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

   5. sqrt-unprod 39.4%

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

   6. +-commutative 39.4%

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

4. Applied egg-rr 39.4%

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

5. Taylor expanded in x around inf 84.3%

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

   1. *-commutative 84.3%

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

   2. unpow1/2 84.3%

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

   3. rem-exp-log 81.4%

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

   4. exp-neg 81.4%

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

   5. exp-prod 81.4%

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

   6. distribute-lft-neg-out 81.4%

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

   7. distribute-rgt-neg-in 81.4%

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

   8. metadata-eval 81.4%

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

   9. exp-to-pow 84.3%

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

7. Simplified 84.3%

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

8. Taylor expanded in x around inf 98.2%

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

9. Final simplification 98.2%

   \[\leadsto \frac{0.5 \cdot {x}^{-0.5}}{x} \]
10. Add Preprocessing

Alternative 7: 37.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\left(x \cdot x\right)}^{-0.25} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (* x x) -0.25))

C

double code(double x) {
	return pow((x * x), -0.25);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) ** (-0.25d0)
end function

Java

public static double code(double x) {
	return Math.pow((x * x), -0.25);
}

Python

def code(x):
	return math.pow((x * x), -0.25)

Julia

function code(x)
	return Float64(x * x) ^ -0.25
end

MATLAB

function tmp = code(x)
	tmp = (x * x) ^ -0.25;
end

Wolfram

code[x_] := N[Power[N[(x * x), $MachinePrecision], -0.25], $MachinePrecision]

Derivation

1. Initial program 39.3%

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

3. Taylor expanded in x around 0 5.6%

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

   1. pow1/2 5.6%

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

   2. inv-pow 5.6%

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

   3. pow-pow 5.6%

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

   4. metadata-eval 5.6%

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

   5. sqr-pow 5.6%

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

   6. pow-prod-down 38.2%

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

   7. pow2 38.2%

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

   8. metadata-eval 38.2%

      \[\leadsto {\left({x}^{2}\right)}^{\color{blue}{-0.25}} \]

5. Applied egg-rr 38.2%

   \[\leadsto \color{blue}{{\left({x}^{2}\right)}^{-0.25}} \]
6. Step-by-step derivation

   1. unpow2 38.2%

      \[\leadsto {\color{blue}{\left(x \cdot x\right)}}^{-0.25} \]

7. Applied egg-rr 38.2%

   \[\leadsto {\color{blue}{\left(x \cdot x\right)}}^{-0.25} \]
8. Add Preprocessing

Alternative 8: 37.8% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \frac{0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.4e+153) (/ 0.5 x) (+ 1.0 (+ -1.0 (/ 0.5 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 6.4e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 1.0 + (-1.0 + (0.5 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.4d+153) then
        tmp = 0.5d0 / x
    else
        tmp = 1.0d0 + ((-1.0d0) + (0.5d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.4e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 1.0 + (-1.0 + (0.5 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.4e+153:
		tmp = 0.5 / x
	else:
		tmp = 1.0 + (-1.0 + (0.5 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.4e+153)
		tmp = Float64(0.5 / x);
	else
		tmp = Float64(1.0 + Float64(-1.0 + Float64(0.5 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.4e+153)
		tmp = 0.5 / x;
	else
		tmp = 1.0 + (-1.0 + (0.5 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.4e+153], N[(0.5 / x), $MachinePrecision], N[(1.0 + N[(-1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.4000000000000003e153

   1. Initial program 8.8%

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

      1. frac-sub 9.0%

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

      2. *-un-lft-identity 9.0%

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

      3. +-commutative 9.0%

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

      4. *-rgt-identity 9.0%

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

      5. sqrt-unprod 9.0%

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

      6. +-commutative 9.0%

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

   4. Applied egg-rr 9.0%

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

   5. Taylor expanded in x around inf 96.7%

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

      1. *-commutative 96.7%

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

      2. unpow1/2 96.7%

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

      3. rem-exp-log 91.2%

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

      4. exp-neg 91.2%

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

      5. exp-prod 91.2%

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

      6. distribute-lft-neg-out 91.2%

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

      7. distribute-rgt-neg-in 91.2%

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

      8. metadata-eval 91.2%

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

      9. exp-to-pow 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in x around 0 8.4%

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

   if 6.4000000000000003e153 < x

   1. Initial program 70.2%

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

      1. frac-sub 70.2%

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

      2. *-un-lft-identity 70.2%

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

      3. +-commutative 70.2%

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

      4. *-rgt-identity 70.2%

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

      5. sqrt-unprod 70.2%

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

      6. +-commutative 70.2%

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

   4. Applied egg-rr 70.2%

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

   5. Taylor expanded in x around inf 71.7%

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

      1. *-commutative 71.7%

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

      2. unpow1/2 71.7%

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

      3. rem-exp-log 71.5%

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

      4. exp-neg 71.5%

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

      5. exp-prod 71.5%

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

      6. distribute-lft-neg-out 71.5%

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

      7. distribute-rgt-neg-in 71.5%

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

      8. metadata-eval 71.5%

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

      9. exp-to-pow 71.7%

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

   7. Simplified 71.7%

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

   8. Taylor expanded in x around 0 7.1%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 7.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{x}\right)\right)} \]

      2. log1p-define 70.2%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{0.5}{x}\right)}\right) \]

      3. expm1-undefine 70.2%

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

      4. add-exp-log 70.2%

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

   10. Applied egg-rr 70.2%

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

       1. associate--l+ 70.2%

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

   12. Simplified 70.2%

       \[\leadsto \color{blue}{1 + \left(\frac{0.5}{x} - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \frac{0.5}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 7.9% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 0.5 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

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

   1. frac-sub 39.4%

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

   2. *-un-lft-identity 39.4%

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

   3. +-commutative 39.4%

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

   4. *-rgt-identity 39.4%

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

   5. sqrt-unprod 39.4%

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

   6. +-commutative 39.4%

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

4. Applied egg-rr 39.4%

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

5. Taylor expanded in x around inf 84.3%

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

   1. *-commutative 84.3%

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

   2. unpow1/2 84.3%

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

   3. rem-exp-log 81.4%

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

   4. exp-neg 81.4%

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

   5. exp-prod 81.4%

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

   6. distribute-lft-neg-out 81.4%

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

   7. distribute-rgt-neg-in 81.4%

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

   8. metadata-eval 81.4%

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

   9. exp-to-pow 84.3%

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

7. Simplified 84.3%

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

8. Taylor expanded in x around 0 7.7%

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

Developer Target 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024113 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

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