2isqrt (example 3.6)

Percentage Accurate: 38.3% → 99.6%

Time: 23.3s

Alternatives: 10

Speedup: 1.0×

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: 38.3% 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: 99.6% 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 38.4%

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

   1. add-cbrt-cube 7.9%

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

   2. pow1/3 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}}} \]

   3. add-sqr-sqrt 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{\left(x + 1\right)} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

   4. pow1 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{{\left(x + 1\right)}^{1}} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

   5. pow1/2 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\left(x + 1\right)}^{1} \cdot \color{blue}{{\left(x + 1\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

   6. pow-prod-up 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\color{blue}{\left({\left(x + 1\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

   7. +-commutative 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\color{blue}{\left(1 + x\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}} \]

   8. metadata-eval 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\left(1 + x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

4. Applied egg-rr 5.8%

   \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. unpow1/3 7.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]

6. Simplified 7.8%

   \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]
7. Step-by-step derivation

   1. pow1/3 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

   2. pow-pow 38.4%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left(1 + x\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}} \]

   3. metadata-eval 38.4%

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

   4. pow1/2 38.4%

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

   5. frac-sub 38.5%

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

   6. *-un-lft-identity 38.5%

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

   7. *-rgt-identity 38.5%

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

   8. sqrt-prod 38.5%

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

   9. div-inv 38.5%

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

   10. flip-- 38.9%

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

   11. frac-times 38.9%

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

8. Applied egg-rr 99.1%

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

   1. *-rgt-identity 99.1%

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

   2. +-inverses 99.1%

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

   3. metadata-eval 99.1%

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

   4. *-commutative 99.1%

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

10. Simplified 99.1%

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

    1. expm1-log1p-u 99.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)\right)} \]

    2. expm1-udef 37.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)} - 1} \]

    3. associate-/r* 37.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right)}}{\sqrt{1 + x} + \sqrt{x}}}\right)} - 1 \]

    4. +-commutative 37.4%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right)}}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\right)} - 1 \]

12. Applied egg-rr 37.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right)}}{\sqrt{x} + \sqrt{1 + x}}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 99.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right)}}{\sqrt{x} + \sqrt{1 + x}}\right)\right)} \]

    2. expm1-log1p 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

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(1.0 / Float64(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[(1.0 / N[(N[Sqrt[N[Sqrt[x], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. add-cbrt-cube 7.9%

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

   2. pow1/3 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}}} \]

   3. add-sqr-sqrt 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{\left(x + 1\right)} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

   4. pow1 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{{\left(x + 1\right)}^{1}} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

   5. pow1/2 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\left(x + 1\right)}^{1} \cdot \color{blue}{{\left(x + 1\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

   6. pow-prod-up 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\color{blue}{\left({\left(x + 1\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

   7. +-commutative 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\color{blue}{\left(1 + x\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}} \]

   8. metadata-eval 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\left(1 + x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

4. Applied egg-rr 5.8%

   \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. unpow1/3 7.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]

6. Simplified 7.8%

   \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]
7. Step-by-step derivation

   1. pow1/3 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

   2. pow-pow 38.4%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left(1 + x\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}} \]

   3. metadata-eval 38.4%

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

   4. pow1/2 38.4%

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

   5. frac-sub 38.5%

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

   6. *-un-lft-identity 38.5%

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

   7. *-rgt-identity 38.5%

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

   8. sqrt-prod 38.5%

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

   9. div-inv 38.5%

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

   10. flip-- 38.9%

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

   11. frac-times 38.9%

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

8. Applied egg-rr 99.1%

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

   1. *-rgt-identity 99.1%

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

   2. +-inverses 99.1%

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

   3. metadata-eval 99.1%

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

   4. *-commutative 99.1%

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

10. Simplified 99.1%

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

11. Final simplification 99.1%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) / (x + 0.5);
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 + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. frac-sub 38.5%

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

   2. *-un-lft-identity 38.5%

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

   3. +-commutative 38.5%

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

   4. *-rgt-identity 38.5%

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

   5. sqrt-unprod 38.5%

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

   6. +-commutative 38.5%

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

4. Applied egg-rr 38.5%

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

5. Taylor expanded in x around inf 38.1%

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

   1. +-commutative 38.1%

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

7. Simplified 38.1%

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

   1. flip-- 38.4%

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

   2. div-inv 38.4%

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

   3. add-sqr-sqrt 22.1%

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

   4. add-sqr-sqrt 40.7%

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

   5. associate--l+ 99.0%

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

9. Applied egg-rr 99.0%

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

    1. +-inverses 99.0%

       \[\leadsto \frac{\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{x + 0.5} \]

    2. metadata-eval 99.0%

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

    3. *-lft-identity 99.0%

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

11. Simplified 99.0%

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

12. Final simplification 99.0%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.4%

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

   1. add-cbrt-cube 7.9%

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

   2. pow1/3 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}}} \]

   3. add-sqr-sqrt 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{\left(x + 1\right)} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

   4. pow1 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{{\left(x + 1\right)}^{1}} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

   5. pow1/2 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\left(x + 1\right)}^{1} \cdot \color{blue}{{\left(x + 1\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

   6. pow-prod-up 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\color{blue}{\left({\left(x + 1\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

   7. +-commutative 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\color{blue}{\left(1 + x\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}} \]

   8. metadata-eval 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\left(1 + x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

4. Applied egg-rr 5.8%

   \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. unpow1/3 7.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]

6. Simplified 7.8%

   \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]
7. Step-by-step derivation

   1. pow1/3 5.8%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

   2. pow-pow 38.4%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left(1 + x\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}} \]

   3. metadata-eval 38.4%

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

   4. pow1/2 38.4%

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

   5. frac-sub 38.5%

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

   6. *-un-lft-identity 38.5%

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

   7. *-rgt-identity 38.5%

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

   8. sqrt-prod 38.5%

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

   9. div-inv 38.5%

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

   10. flip-- 38.9%

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

   11. frac-times 38.9%

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

8. Applied egg-rr 99.1%

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

   1. *-rgt-identity 99.1%

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

   2. +-inverses 99.1%

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

   3. metadata-eval 99.1%

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

   4. *-commutative 99.1%

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

10. Simplified 99.1%

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

    1. expm1-log1p-u 99.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)\right)} \]

    2. expm1-udef 37.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)} - 1} \]

    3. associate-/r* 37.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right)}}{\sqrt{1 + x} + \sqrt{x}}}\right)} - 1 \]

    4. +-commutative 37.4%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right)}}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\right)} - 1 \]

12. Applied egg-rr 37.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right)}}{\sqrt{x} + \sqrt{1 + x}}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 99.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right)}}{\sqrt{x} + \sqrt{1 + x}}\right)\right)} \]

    2. expm1-log1p 99.7%

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

14. Simplified 99.7%

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

15. Taylor expanded in x around inf 97.9%

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

16. Final simplification 97.9%

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

Alternative 5: 65.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (sqrt (/ 1.0 (pow x 3.0)))))

C

double code(double x) {
	return 0.5 * sqrt((1.0 / pow(x, 3.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
end function

Java

public static double code(double x) {
	return 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
}

Python

def code(x):
	return 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))

Julia

function code(x)
	return Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
end

Wolfram

code[x_] := N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. flip-- 38.4%

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

   2. frac-times 20.6%

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

   3. metadata-eval 20.6%

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

   4. add-sqr-sqrt 19.2%

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

   5. frac-times 24.4%

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

   6. metadata-eval 24.4%

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

   7. add-sqr-sqrt 38.5%

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

   8. +-commutative 38.5%

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

   9. pow1/2 38.5%

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

   10. pow-flip 38.5%

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

   11. metadata-eval 38.5%

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

   12. inv-pow 38.5%

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

   13. sqrt-pow2 38.5%

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

   14. +-commutative 38.5%

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

   15. metadata-eval 38.5%

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

4. Applied egg-rr 38.5%

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

5. Taylor expanded in x around inf 66.1%

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

6. Final simplification 66.1%

   \[\leadsto 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} \]
7. Add Preprocessing

Alternative 6: 37.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.4%

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

   1. flip-- 38.4%

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

   2. frac-times 20.6%

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

   3. metadata-eval 20.6%

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

   4. add-sqr-sqrt 19.2%

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

   5. frac-times 24.4%

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

   6. metadata-eval 24.4%

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

   7. add-sqr-sqrt 38.5%

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

   8. +-commutative 38.5%

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

   9. pow1/2 38.5%

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

   10. pow-flip 38.5%

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

   11. metadata-eval 38.5%

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

   12. inv-pow 38.5%

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

   13. sqrt-pow2 38.5%

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

   14. +-commutative 38.5%

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

   15. metadata-eval 38.5%

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

4. Applied egg-rr 38.5%

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

5. Taylor expanded in x around inf 37.7%

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

6. Final simplification 37.7%

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

Alternative 7: 37.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.4%

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

   1. frac-sub 38.5%

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

   2. *-un-lft-identity 38.5%

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

   3. +-commutative 38.5%

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

   4. *-rgt-identity 38.5%

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

   5. sqrt-unprod 38.5%

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

   6. +-commutative 38.5%

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

4. Applied egg-rr 38.5%

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

5. Taylor expanded in x around 0 37.7%

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

6. Final simplification 37.7%

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

Alternative 8: 36.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.4%

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

   1. expm1-log1p-u 38.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)\right)} - \frac{1}{\sqrt{x + 1}} \]

   2. expm1-udef 5.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]

   3. pow1/2 5.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{x}^{0.5}}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   4. pow-flip 5.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(-0.5\right)}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   5. metadata-eval 5.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

4. Applied egg-rr 5.2%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]
5. Step-by-step derivation

   1. expm1-def 31.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} - \frac{1}{\sqrt{x + 1}} \]

   2. expm1-log1p 31.2%

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

6. Simplified 31.2%

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

7. Taylor expanded in x around inf 5.6%

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

   1. inv-pow 5.6%

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

   2. sqrt-pow1 5.6%

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

   3. metadata-eval 5.6%

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

   4. expm1-log1p-u 5.6%

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

   5. expm1-udef 36.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1} \]

   6. log1p-udef 36.4%

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

   7. add-exp-log 36.4%

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

9. Applied egg-rr 36.4%

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

10. Final simplification 36.4%

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

Alternative 9: 7.8% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / (x + 0.5)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.4%

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

   1. frac-sub 38.5%

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

   2. *-un-lft-identity 38.5%

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

   3. +-commutative 38.5%

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

   4. *-rgt-identity 38.5%

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

   5. sqrt-unprod 38.5%

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

   6. +-commutative 38.5%

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

4. Applied egg-rr 38.5%

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

5. Taylor expanded in x around inf 38.1%

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

   1. +-commutative 38.1%

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

7. Simplified 38.1%

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

8. Taylor expanded in x around 0 7.7%

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

9. Final simplification 7.7%

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

Alternative 10: 4.6% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 38.4%

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

   1. frac-sub 38.5%

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

   2. *-un-lft-identity 38.5%

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

   3. +-commutative 38.5%

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

   4. *-rgt-identity 38.5%

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

   5. sqrt-unprod 38.5%

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

   6. +-commutative 38.5%

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

4. Applied egg-rr 38.5%

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

5. Taylor expanded in x around inf 38.1%

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

   1. +-commutative 38.1%

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

7. Simplified 38.1%

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

8. Taylor expanded in x around 0 4.6%

   \[\leadsto \color{blue}{2} \]

9. Final simplification 4.6%

   \[\leadsto 2 \]
10. Add Preprocessing

Developer target: 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 2024031 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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