2isqrt (example 3.6)

Percentage Accurate: 39.3% → 99.7%

Time: 13.6s

Alternatives: 8

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: 39.3% 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: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[x, -0.5], $MachinePrecision] / N[(N[Sqrt[N[Sqrt[x], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

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

   1. frac-sub 35.4%

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

   2. *-rgt-identity 35.4%

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

   3. *-un-lft-identity 35.4%

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

   4. +-commutative 35.4%

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

   5. sqrt-unprod 35.4%

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

   6. +-commutative 35.4%

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

4. Applied egg-rr 35.4%

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

   1. flip-- 36.5%

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

   2. add-sqr-sqrt 36.6%

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

   3. add-sqr-sqrt 38.1%

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

   4. add-sqr-sqrt 38.1%

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

   5. hypot-1-def 38.1%

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

6. Applied egg-rr 38.1%

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

   1. associate--l+ 85.6%

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

   2. +-inverses 85.6%

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

   3. metadata-eval 85.6%

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

   4. +-commutative 85.6%

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

8. Simplified 85.6%

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

   1. *-un-lft-identity 85.6%

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

   2. sqrt-prod 99.3%

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

   3. +-commutative 99.3%

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

   4. times-frac 99.2%

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

   5. pow1/2 99.2%

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

   6. pow-flip 99.4%

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

   7. metadata-eval 99.4%

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

   8. hypot-undefine 99.4%

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

   9. metadata-eval 99.4%

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

   10. add-sqr-sqrt 99.4%

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

   11. +-commutative 99.4%

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

10. Applied egg-rr 99.4%

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

    1. associate-/l/ 99.3%

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

    2. associate-*r/ 99.4%

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

    3. *-rgt-identity 99.4%

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

12. Simplified 99.4%

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

    1. distribute-rgt-in 99.4%

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

    2. sqrt-prod 85.5%

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

    3. distribute-rgt-in 85.5%

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

    4. *-un-lft-identity 85.5%

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

    5. add-sqr-sqrt 85.5%

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

    6. hypot-define 99.5%

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

    7. add-sqr-sqrt 99.7%

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

14. Applied egg-rr 99.7%

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

15. Final simplification 99.7%

    \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(\sqrt{x}, x\right) + \left(x + 1\right)} \]
16. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.3%

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

   1. frac-sub 35.4%

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

   2. *-rgt-identity 35.4%

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

   3. *-un-lft-identity 35.4%

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

   4. +-commutative 35.4%

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

   5. sqrt-unprod 35.4%

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

   6. +-commutative 35.4%

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

4. Applied egg-rr 35.4%

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

   1. flip-- 36.5%

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

   2. add-sqr-sqrt 36.6%

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

   3. add-sqr-sqrt 38.1%

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

   4. add-sqr-sqrt 38.1%

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

   5. hypot-1-def 38.1%

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

6. Applied egg-rr 38.1%

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

   1. associate--l+ 85.6%

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

   2. +-inverses 85.6%

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

   3. metadata-eval 85.6%

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

   4. +-commutative 85.6%

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

8. Simplified 85.6%

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

   1. *-un-lft-identity 85.6%

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

   2. hypot-undefine 85.6%

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

   3. metadata-eval 85.6%

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

   4. add-sqr-sqrt 85.6%

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

10. Applied egg-rr 85.6%

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

    1. *-lft-identity 85.6%

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

12. Simplified 85.6%

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

13. Taylor expanded in x around inf 99.0%

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

    1. associate-*r/ 97.6%

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

    2. metadata-eval 97.6%

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

15. Simplified 99.0%

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

16. Final simplification 99.0%

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

Alternative 3: 97.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.3%

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

   1. frac-sub 35.4%

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

   2. *-rgt-identity 35.4%

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

   3. *-un-lft-identity 35.4%

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

   4. +-commutative 35.4%

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

   5. sqrt-unprod 35.4%

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

   6. +-commutative 35.4%

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

4. Applied egg-rr 35.4%

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

5. Taylor expanded in x around inf 83.5%

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

   1. *-commutative 83.5%

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

7. Simplified 83.5%

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

8. Taylor expanded in x around inf 97.6%

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

   1. associate-*r/ 97.6%

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

   2. metadata-eval 97.6%

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

10. Simplified 97.6%

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

11. Final simplification 97.6%

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

Alternative 4: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.3%

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

   1. frac-sub 35.4%

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

   2. *-rgt-identity 35.4%

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

   3. *-un-lft-identity 35.4%

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

   4. +-commutative 35.4%

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

   5. sqrt-unprod 35.4%

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

   6. +-commutative 35.4%

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

4. Applied egg-rr 35.4%

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

5. Taylor expanded in x around inf 83.5%

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

   1. *-commutative 83.5%

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

7. Simplified 83.5%

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

   1. associate-/l* 83.4%

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

   2. inv-pow 83.4%

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

   3. sqrt-pow1 83.5%

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

   4. metadata-eval 83.5%

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

   5. *-un-lft-identity 83.5%

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

   6. associate-*r/ 83.6%

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

   7. sqrt-prod 97.4%

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

   8. times-frac 97.4%

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

   9. metadata-eval 97.4%

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

   10. pow-flip 97.2%

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

   11. pow1/2 97.2%

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

   12. associate-/r* 97.2%

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

   13. add-sqr-sqrt 97.5%

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

9. Applied egg-rr 97.5%

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

    1. *-lft-identity 97.5%

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

    2. associate-*r/ 97.5%

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

    3. associate-*l/ 97.5%

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

    4. metadata-eval 97.5%

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

11. Simplified 97.5%

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

12. Final simplification 97.5%

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

Alternative 5: 97.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-0.5}}{x \cdot 2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (pow x -0.5) (* x 2.0)))

C

double code(double x) {
	return pow(x, -0.5) / (x * 2.0);
}

Fortran

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

Java

public static double code(double x) {
	return Math.pow(x, -0.5) / (x * 2.0);
}

Python

def code(x):
	return math.pow(x, -0.5) / (x * 2.0)

Julia

function code(x)
	return Float64((x ^ -0.5) / Float64(x * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.5) / (x * 2.0);
end

Wolfram

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

Derivation

1. Initial program 35.3%

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

   1. frac-sub 35.4%

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

   2. *-rgt-identity 35.4%

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

   3. *-un-lft-identity 35.4%

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

   4. +-commutative 35.4%

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

   5. sqrt-unprod 35.4%

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

   6. +-commutative 35.4%

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

4. Applied egg-rr 35.4%

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

   1. flip-- 36.5%

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

   2. add-sqr-sqrt 36.6%

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

   3. add-sqr-sqrt 38.1%

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

   4. add-sqr-sqrt 38.1%

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

   5. hypot-1-def 38.1%

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

6. Applied egg-rr 38.1%

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

   1. associate--l+ 85.6%

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

   2. +-inverses 85.6%

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

   3. metadata-eval 85.6%

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

   4. +-commutative 85.6%

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

8. Simplified 85.6%

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

   1. *-un-lft-identity 85.6%

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

   2. sqrt-prod 99.3%

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

   3. +-commutative 99.3%

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

   4. times-frac 99.2%

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

   5. pow1/2 99.2%

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

   6. pow-flip 99.4%

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

   7. metadata-eval 99.4%

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

   8. hypot-undefine 99.4%

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

   9. metadata-eval 99.4%

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

   10. add-sqr-sqrt 99.4%

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

   11. +-commutative 99.4%

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

10. Applied egg-rr 99.4%

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

    1. associate-/l/ 99.3%

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

    2. associate-*r/ 99.4%

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

    3. *-rgt-identity 99.4%

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

12. Simplified 99.4%

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

13. Taylor expanded in x around inf 97.5%

    \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{2 \cdot x}} \]
14. Step-by-step derivation

    1. *-commutative 97.5%

       \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{x \cdot 2}} \]

15. Simplified 97.5%

    \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{x \cdot 2}} \]

16. Final simplification 97.5%

    \[\leadsto \frac{{x}^{-0.5}}{x \cdot 2} \]
17. Add Preprocessing

Alternative 6: 96.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{{x}^{1.5}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.5 (pow x 1.5)))

C

double code(double x) {
	return 0.5 / pow(x, 1.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 / (x ** 1.5d0)
end function

Java

public static double code(double x) {
	return 0.5 / Math.pow(x, 1.5);
}

Python

def code(x):
	return 0.5 / math.pow(x, 1.5)

Julia

function code(x)
	return Float64(0.5 / (x ^ 1.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / (x ^ 1.5);
end

Wolfram

code[x_] := N[(0.5 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

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

   1. flip-- 35.3%

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

   2. clear-num 35.3%

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

   3. inv-pow 35.3%

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

   4. sqrt-pow2 35.3%

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

   5. metadata-eval 35.3%

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

   6. inv-pow 35.3%

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

   7. sqrt-pow2 35.3%

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

   8. +-commutative 35.3%

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

   9. metadata-eval 35.3%

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

   10. frac-times 23.4%

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

   11. metadata-eval 23.4%

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

   12. add-sqr-sqrt 23.6%

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

   13. frac-times 26.4%

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

   14. metadata-eval 26.4%

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

   15. add-sqr-sqrt 35.4%

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

   16. +-commutative 35.4%

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

4. Applied egg-rr 35.4%

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

5. Taylor expanded in x around inf 67.6%

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

   1. *-un-lft-identity 67.6%

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

   2. associate-/r* 67.6%

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

   3. metadata-eval 67.6%

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

   4. sqrt-pow1 96.2%

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

   5. metadata-eval 96.2%

      \[\leadsto 1 \cdot \frac{0.5}{{x}^{\color{blue}{1.5}}} \]

7. Applied egg-rr 96.2%

   \[\leadsto \color{blue}{1 \cdot \frac{0.5}{{x}^{1.5}}} \]
8. Step-by-step derivation

   1. *-lft-identity 96.2%

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

9. Simplified 96.2%

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

10. Final simplification 96.2%

    \[\leadsto \frac{0.5}{{x}^{1.5}} \]
11. Add Preprocessing

Alternative 7: 38.5% accurate, 26.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 6.4e+153) (/ 0.5 x) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 6.4e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0;
	}
	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 = 0.0d0
    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 = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.4e+153:
		tmp = 0.5 / x
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 6.4e+153], N[(0.5 / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.4000000000000003e153

   1. Initial program 10.1%

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

      1. frac-sub 10.2%

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

      2. *-rgt-identity 10.2%

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

      3. *-un-lft-identity 10.2%

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

      4. +-commutative 10.2%

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

      5. sqrt-unprod 10.2%

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

      6. +-commutative 10.2%

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

   4. Applied egg-rr 10.2%

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

   5. Taylor expanded in x around inf 95.9%

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

      1. *-commutative 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in x around 0 8.4%

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

   if 6.4000000000000003e153 < x

   1. Initial program 67.7%

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

      1. sub-neg 67.7%

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

      2. +-commutative 67.7%

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

      3. add-cube-cbrt 11.4%

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

      4. distribute-lft-neg-in 11.4%

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

      5. fma-define 4.5%

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

   4. Applied egg-rr 4.5%

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

   5. Taylor expanded in x around inf 67.7%

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

      1. distribute-rgt1-in 67.7%

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

      2. metadata-eval 67.7%

         \[\leadsto \color{blue}{0} \cdot \sqrt{\frac{1}{x}} \]

      3. mul0-lft 67.7%

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

   7. Simplified 67.7%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.5% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 35.3%

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

   1. sub-neg 35.3%

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

   2. +-commutative 35.3%

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

   3. add-cube-cbrt 10.9%

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

   4. distribute-lft-neg-in 10.9%

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

   5. fma-define 7.9%

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

4. Applied egg-rr 8.1%

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

5. Taylor expanded in x around inf 32.0%

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

   1. distribute-rgt1-in 32.0%

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

   2. metadata-eval 32.0%

      \[\leadsto \color{blue}{0} \cdot \sqrt{\frac{1}{x}} \]

   3. mul0-lft 32.0%

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

7. Simplified 32.0%

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

8. Final simplification 32.0%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer target: 98.3% 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 2024071 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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