2isqrt (example 3.6)

Percentage Accurate: 38.8% → 99.7%

Time: 9.1s

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: 38.8% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] / N[(x * N[(N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $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 39.3%

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

   2. div-inv 39.3%

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

   3. *-un-lft-identity 39.3%

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

   4. +-commutative 39.3%

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

   5. *-rgt-identity 39.3%

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

   6. metadata-eval 39.3%

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

   7. frac-times 39.3%

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

   8. associate-*l/ 39.3%

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

   9. *-un-lft-identity 39.3%

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

   10. inv-pow 39.3%

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

   11. sqrt-pow2 39.3%

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

   12. +-commutative 39.3%

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

   13. metadata-eval 39.3%

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

4. Applied egg-rr 39.3%

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

   1. associate-*r/ 39.3%

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

   2. *-rgt-identity 39.3%

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

   3. times-frac 39.3%

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

   4. div-sub 39.2%

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

   5. sub-neg 39.2%

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

   6. *-inverses 39.2%

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

   7. metadata-eval 39.2%

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

   8. /-rgt-identity 39.2%

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

6. Simplified 39.2%

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

   1. flip-+ 39.3%

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

   2. sqrt-undiv 39.3%

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

   3. sqrt-undiv 39.3%

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

   4. add-sqr-sqrt 39.5%

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

   5. metadata-eval 39.5%

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

   6. sqrt-undiv 39.5%

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

8. Applied egg-rr 39.5%

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

9. Taylor expanded in x around 0 99.6%

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

    1. *-commutative 99.6%

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

    2. clear-num 99.6%

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

    3. un-div-inv 99.6%

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

    4. +-commutative 99.6%

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

    5. div-inv 99.6%

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

    6. sub-neg 99.6%

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

    7. +-commutative 99.6%

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

    8. metadata-eval 99.6%

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

    9. clear-num 99.7%

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

    10. /-rgt-identity 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 + \frac{\frac{0.0625 + \frac{-0.0390625}{x}}{x} - 0.125}{x}}{x} \cdot {\left(x + 1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (+ 0.5 (/ (- (/ (+ 0.0625 (/ -0.0390625 x)) x) 0.125) x)) x)
  (pow (+ x 1.0) -0.5)))

C

double code(double x) {
	return ((0.5 + ((((0.0625 + (-0.0390625 / x)) / x) - 0.125) / x)) / x) * pow((x + 1.0), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 + ((((0.0625d0 + ((-0.0390625d0) / x)) / x) - 0.125d0) / x)) / x) * ((x + 1.0d0) ** (-0.5d0))
end function

Java

public static double code(double x) {
	return ((0.5 + ((((0.0625 + (-0.0390625 / x)) / x) - 0.125) / x)) / x) * Math.pow((x + 1.0), -0.5);
}

Python

def code(x):
	return ((0.5 + ((((0.0625 + (-0.0390625 / x)) / x) - 0.125) / x)) / x) * math.pow((x + 1.0), -0.5)

Julia

function code(x)
	return Float64(Float64(Float64(0.5 + Float64(Float64(Float64(Float64(0.0625 + Float64(-0.0390625 / x)) / x) - 0.125) / x)) / x) * (Float64(x + 1.0) ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 + ((((0.0625 + (-0.0390625 / x)) / x) - 0.125) / x)) / x) * ((x + 1.0) ^ -0.5);
end

Wolfram

code[x_] := N[(N[(N[(0.5 + N[(N[(N[(N[(0.0625 + N[(-0.0390625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.125), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $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 39.3%

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

   2. div-inv 39.3%

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

   3. *-un-lft-identity 39.3%

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

   4. +-commutative 39.3%

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

   5. *-rgt-identity 39.3%

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

   6. metadata-eval 39.3%

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

   7. frac-times 39.3%

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

   8. associate-*l/ 39.3%

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

   9. *-un-lft-identity 39.3%

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

   10. inv-pow 39.3%

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

   11. sqrt-pow2 39.3%

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

   12. +-commutative 39.3%

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

   13. metadata-eval 39.3%

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

4. Applied egg-rr 39.3%

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

   1. associate-*r/ 39.3%

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

   2. *-rgt-identity 39.3%

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

   3. times-frac 39.3%

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

   4. div-sub 39.2%

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

   5. sub-neg 39.2%

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

   6. *-inverses 39.2%

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

   7. metadata-eval 39.2%

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

   8. /-rgt-identity 39.2%

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

6. Simplified 39.2%

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

7. Taylor expanded in x around inf 99.3%

   \[\leadsto \color{blue}{\frac{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - \left(0.125 \cdot \frac{1}{x} + 0.0390625 \cdot \frac{1}{{x}^{3}}\right)}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]
8. Step-by-step derivation

   1. associate--l+ 99.3%

      \[\leadsto \frac{\color{blue}{0.5 + \left(\frac{0.0625}{{x}^{2}} - \left(0.125 \cdot \frac{1}{x} + 0.0390625 \cdot \frac{1}{{x}^{3}}\right)\right)}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   2. associate-*r/ 99.3%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \left(\color{blue}{\frac{0.125 \cdot 1}{x}} + 0.0390625 \cdot \frac{1}{{x}^{3}}\right)\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   3. metadata-eval 99.3%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \left(\frac{\color{blue}{0.125}}{x} + 0.0390625 \cdot \frac{1}{{x}^{3}}\right)\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   4. associate-*r/ 99.3%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \left(\frac{0.125}{x} + \color{blue}{\frac{0.0390625 \cdot 1}{{x}^{3}}}\right)\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   5. metadata-eval 99.3%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \left(\frac{0.125}{x} + \frac{\color{blue}{0.0390625}}{{x}^{3}}\right)\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

9. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \left(\frac{0.125}{x} + \frac{0.0390625}{{x}^{3}}\right)\right)}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]

10. Taylor expanded in x around -inf 99.3%

    \[\leadsto \frac{\color{blue}{0.5 + -1 \cdot \frac{0.125 + -1 \cdot \frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}}{x}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]
11. Step-by-step derivation

    1. mul-1-neg 99.3%

       \[\leadsto \frac{0.5 + \color{blue}{\left(-\frac{0.125 + -1 \cdot \frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}}{x}\right)}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    2. unsub-neg 99.3%

       \[\leadsto \frac{\color{blue}{0.5 - \frac{0.125 + -1 \cdot \frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}}{x}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    3. mul-1-neg 99.3%

       \[\leadsto \frac{0.5 - \frac{0.125 + \color{blue}{\left(-\frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}\right)}}{x}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    4. unsub-neg 99.3%

       \[\leadsto \frac{0.5 - \frac{\color{blue}{0.125 - \frac{0.0625 - 0.0390625 \cdot \frac{1}{x}}{x}}}{x}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    5. sub-neg 99.3%

       \[\leadsto \frac{0.5 - \frac{0.125 - \frac{\color{blue}{0.0625 + \left(-0.0390625 \cdot \frac{1}{x}\right)}}{x}}{x}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    6. associate-*r/ 99.3%

       \[\leadsto \frac{0.5 - \frac{0.125 - \frac{0.0625 + \left(-\color{blue}{\frac{0.0390625 \cdot 1}{x}}\right)}{x}}{x}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    7. metadata-eval 99.3%

       \[\leadsto \frac{0.5 - \frac{0.125 - \frac{0.0625 + \left(-\frac{\color{blue}{0.0390625}}{x}\right)}{x}}{x}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    8. distribute-neg-frac 99.3%

       \[\leadsto \frac{0.5 - \frac{0.125 - \frac{0.0625 + \color{blue}{\frac{-0.0390625}{x}}}{x}}{x}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

    9. metadata-eval 99.3%

       \[\leadsto \frac{0.5 - \frac{0.125 - \frac{0.0625 + \frac{\color{blue}{-0.0390625}}{x}}{x}}{x}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

12. Simplified 99.3%

    \[\leadsto \frac{\color{blue}{0.5 - \frac{0.125 - \frac{0.0625 + \frac{-0.0390625}{x}}{x}}{x}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

13. Final simplification 99.3%

    \[\leadsto \frac{0.5 + \frac{\frac{0.0625 + \frac{-0.0390625}{x}}{x} - 0.125}{x}}{x} \cdot {\left(x + 1\right)}^{-0.5} \]
14. Add Preprocessing

Alternative 3: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ {\left(x + 1\right)}^{-0.5} \cdot \frac{\frac{1}{x}}{2 + \frac{0.5 + \frac{-0.125}{x}}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow (+ x 1.0) -0.5) (/ (/ 1.0 x) (+ 2.0 (/ (+ 0.5 (/ -0.125 x)) x)))))

C

double code(double x) {
	return pow((x + 1.0), -0.5) * ((1.0 / x) / (2.0 + ((0.5 + (-0.125 / x)) / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + 1.0d0) ** (-0.5d0)) * ((1.0d0 / x) / (2.0d0 + ((0.5d0 + ((-0.125d0) / x)) / x)))
end function

Java

public static double code(double x) {
	return Math.pow((x + 1.0), -0.5) * ((1.0 / x) / (2.0 + ((0.5 + (-0.125 / x)) / x)));
}

Python

def code(x):
	return math.pow((x + 1.0), -0.5) * ((1.0 / x) / (2.0 + ((0.5 + (-0.125 / x)) / x)))

Julia

function code(x)
	return Float64((Float64(x + 1.0) ^ -0.5) * Float64(Float64(1.0 / x) / Float64(2.0 + Float64(Float64(0.5 + Float64(-0.125 / x)) / x))))
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) ^ -0.5) * ((1.0 / x) / (2.0 + ((0.5 + (-0.125 / x)) / x)));
end

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(2.0 + N[(N[(0.5 + N[(-0.125 / x), $MachinePrecision]), $MachinePrecision] / 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 39.3%

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

   2. div-inv 39.3%

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

   3. *-un-lft-identity 39.3%

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

   4. +-commutative 39.3%

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

   5. *-rgt-identity 39.3%

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

   6. metadata-eval 39.3%

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

   7. frac-times 39.3%

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

   8. associate-*l/ 39.3%

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

   9. *-un-lft-identity 39.3%

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

   10. inv-pow 39.3%

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

   11. sqrt-pow2 39.3%

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

   12. +-commutative 39.3%

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

   13. metadata-eval 39.3%

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

4. Applied egg-rr 39.3%

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

   1. associate-*r/ 39.3%

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

   2. *-rgt-identity 39.3%

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

   3. times-frac 39.3%

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

   4. div-sub 39.2%

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

   5. sub-neg 39.2%

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

   6. *-inverses 39.2%

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

   7. metadata-eval 39.2%

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

   8. /-rgt-identity 39.2%

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

6. Simplified 39.2%

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

   1. flip-+ 39.3%

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

   2. sqrt-undiv 39.3%

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

   3. sqrt-undiv 39.3%

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

   4. add-sqr-sqrt 39.5%

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

   5. metadata-eval 39.5%

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

   6. sqrt-undiv 39.5%

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

8. Applied egg-rr 39.5%

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

9. Taylor expanded in x around 0 99.6%

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

10. Taylor expanded in x around inf 99.2%

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

    1. associate--l+ 99.2%

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

    2. associate-*r/ 99.2%

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

    3. metadata-eval 99.2%

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

    4. unpow2 99.2%

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

    5. associate-/l/ 99.2%

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

    6. div-sub 99.2%

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

    7. sub-neg 99.2%

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

    8. distribute-neg-frac2 99.2%

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

    9. neg-mul-1 99.2%

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

    10. rem-square-sqrt 0.0%

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

    11. unpow2 0.0%

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

    12. *-commutative 0.0%

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

    13. metadata-eval 0.0%

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

    14. associate-*r/ 0.0%

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

12. Simplified 99.2%

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

13. Final simplification 99.2%

    \[\leadsto {\left(x + 1\right)}^{-0.5} \cdot \frac{\frac{1}{x}}{2 + \frac{0.5 + \frac{-0.125}{x}}{x}} \]
14. Add Preprocessing

Alternative 4: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(0.5 + N[(N[(-0.125 + N[(0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $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 39.3%

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

   2. div-inv 39.3%

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

   3. *-un-lft-identity 39.3%

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

   4. +-commutative 39.3%

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

   5. *-rgt-identity 39.3%

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

   6. metadata-eval 39.3%

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

   7. frac-times 39.3%

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

   8. associate-*l/ 39.3%

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

   9. *-un-lft-identity 39.3%

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

   10. inv-pow 39.3%

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

   11. sqrt-pow2 39.3%

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

   12. +-commutative 39.3%

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

   13. metadata-eval 39.3%

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

4. Applied egg-rr 39.3%

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

   1. associate-*r/ 39.3%

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

   2. *-rgt-identity 39.3%

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

   3. times-frac 39.3%

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

   4. div-sub 39.2%

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

   5. sub-neg 39.2%

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

   6. *-inverses 39.2%

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

   7. metadata-eval 39.2%

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

   8. /-rgt-identity 39.2%

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

6. Simplified 39.2%

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

7. Taylor expanded in x around inf 99.3%

   \[\leadsto \color{blue}{\frac{\left(0.5 + \frac{0.0625}{{x}^{2}}\right) - \left(0.125 \cdot \frac{1}{x} + 0.0390625 \cdot \frac{1}{{x}^{3}}\right)}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]
8. Step-by-step derivation

   1. associate--l+ 99.3%

      \[\leadsto \frac{\color{blue}{0.5 + \left(\frac{0.0625}{{x}^{2}} - \left(0.125 \cdot \frac{1}{x} + 0.0390625 \cdot \frac{1}{{x}^{3}}\right)\right)}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   2. associate-*r/ 99.3%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \left(\color{blue}{\frac{0.125 \cdot 1}{x}} + 0.0390625 \cdot \frac{1}{{x}^{3}}\right)\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   3. metadata-eval 99.3%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \left(\frac{\color{blue}{0.125}}{x} + 0.0390625 \cdot \frac{1}{{x}^{3}}\right)\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   4. associate-*r/ 99.3%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \left(\frac{0.125}{x} + \color{blue}{\frac{0.0390625 \cdot 1}{{x}^{3}}}\right)\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

   5. metadata-eval 99.3%

      \[\leadsto \frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \left(\frac{0.125}{x} + \frac{\color{blue}{0.0390625}}{{x}^{3}}\right)\right)}{x} \cdot {\left(1 + x\right)}^{-0.5} \]

9. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.0625}{{x}^{2}} - \left(\frac{0.125}{x} + \frac{0.0390625}{{x}^{3}}\right)\right)}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]

10. Taylor expanded in x around inf 99.2%

    \[\leadsto \frac{0.5 + \color{blue}{\frac{0.0625 \cdot \frac{1}{x} - 0.125}{x}}}{x} \cdot {\left(1 + x\right)}^{-0.5} \]
11. Step-by-step derivation

    1. sub-neg 99.2%

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

    2. associate-*r/ 99.2%

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

    3. metadata-eval 99.2%

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

    4. metadata-eval 99.2%

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

12. Simplified 99.2%

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

13. Final simplification 99.2%

    \[\leadsto {\left(x + 1\right)}^{-0.5} \cdot \frac{0.5 + \frac{-0.125 + \frac{0.0625}{x}}{x}}{x} \]
14. Add Preprocessing

Alternative 5: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(0.5 - N[(0.125 / x), $MachinePrecision]), $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 39.3%

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

   2. div-inv 39.3%

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

   3. *-un-lft-identity 39.3%

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

   4. +-commutative 39.3%

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

   5. *-rgt-identity 39.3%

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

   6. metadata-eval 39.3%

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

   7. frac-times 39.3%

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

   8. associate-*l/ 39.3%

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

   9. *-un-lft-identity 39.3%

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

   10. inv-pow 39.3%

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

   11. sqrt-pow2 39.3%

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

   12. +-commutative 39.3%

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

   13. metadata-eval 39.3%

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

4. Applied egg-rr 39.3%

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

   1. associate-*r/ 39.3%

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

   2. *-rgt-identity 39.3%

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

   3. times-frac 39.3%

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

   4. div-sub 39.2%

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

   5. sub-neg 39.2%

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

   6. *-inverses 39.2%

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

   7. metadata-eval 39.2%

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

   8. /-rgt-identity 39.2%

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

6. Simplified 39.2%

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

7. Taylor expanded in x around inf 98.9%

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

   1. associate-*r/ 98.9%

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

   2. metadata-eval 98.9%

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

9. Simplified 98.9%

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

    1. associate-*l/ 98.9%

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

    2. +-commutative 98.9%

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

11. Applied egg-rr 98.9%

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

12. Final simplification 98.9%

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

Alternative 6: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(0.5 + N[(-0.125 / x), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] / x), $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 39.3%

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

   2. div-inv 39.3%

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

   3. *-un-lft-identity 39.3%

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

   4. +-commutative 39.3%

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

   5. *-rgt-identity 39.3%

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

   6. metadata-eval 39.3%

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

   7. frac-times 39.3%

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

   8. associate-*l/ 39.3%

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

   9. *-un-lft-identity 39.3%

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

   10. inv-pow 39.3%

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

   11. sqrt-pow2 39.3%

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

   12. +-commutative 39.3%

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

   13. metadata-eval 39.3%

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

4. Applied egg-rr 39.3%

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

   1. associate-*r/ 39.3%

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

   2. *-rgt-identity 39.3%

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

   3. times-frac 39.3%

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

   4. div-sub 39.2%

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

   5. sub-neg 39.2%

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

   6. *-inverses 39.2%

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

   7. metadata-eval 39.2%

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

   8. /-rgt-identity 39.2%

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

6. Simplified 39.2%

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

7. Taylor expanded in x around inf 98.9%

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

   1. associate-*r/ 98.9%

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

   2. metadata-eval 98.9%

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

9. Simplified 98.9%

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

    1. clear-num 98.8%

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

    2. inv-pow 98.8%

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

11. Applied egg-rr 98.8%

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

    1. unpow-1 98.8%

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

    2. sub-neg 98.8%

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

    3. distribute-neg-frac 98.8%

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

    4. metadata-eval 98.8%

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

13. Simplified 98.8%

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

    1. associate-*l/ 98.9%

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

    2. *-un-lft-identity 98.9%

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

    3. clear-num 97.2%

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

    4. +-commutative 97.2%

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

15. Applied egg-rr 97.2%

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

    1. associate-/r/ 98.8%

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

    2. associate-/r/ 98.8%

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

    3. associate-*l/ 98.9%

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

    4. *-lft-identity 98.9%

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

    5. *-commutative 98.9%

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

    6. *-lft-identity 98.9%

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

    7. associate-*l/ 98.8%

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

    8. associate-*r* 98.8%

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

    9. *-commutative 98.8%

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

    10. associate-*l/ 98.9%

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

    11. *-lft-identity 98.9%

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

    12. +-commutative 98.9%

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

17. Simplified 98.9%

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

18. Final simplification 98.9%

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

Alternative 7: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(0.5 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $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 39.3%

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

   2. div-inv 39.3%

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

   3. *-un-lft-identity 39.3%

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

   4. +-commutative 39.3%

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

   5. *-rgt-identity 39.3%

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

   6. metadata-eval 39.3%

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

   7. frac-times 39.3%

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

   8. associate-*l/ 39.3%

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

   9. *-un-lft-identity 39.3%

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

   10. inv-pow 39.3%

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

   11. sqrt-pow2 39.3%

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

   12. +-commutative 39.3%

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

   13. metadata-eval 39.3%

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

4. Applied egg-rr 39.3%

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

   1. associate-*r/ 39.3%

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

   2. *-rgt-identity 39.3%

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

   3. times-frac 39.3%

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

   4. div-sub 39.2%

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

   5. sub-neg 39.2%

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

   6. *-inverses 39.2%

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

   7. metadata-eval 39.2%

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

   8. /-rgt-identity 39.2%

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

6. Simplified 39.2%

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

7. Taylor expanded in x around inf 98.9%

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

   1. associate-*r/ 98.9%

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

   2. metadata-eval 98.9%

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

9. Simplified 98.9%

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

10. Final simplification 98.9%

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

Alternative 8: 97.9% accurate, 2.0× speedup

Math

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

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 39.3%

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

   1. add-exp-log 7.9%

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

   2. log-rec 7.9%

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

   3. pow1/2 7.9%

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

   4. log-pow 7.9%

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

   5. +-commutative 7.9%

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

   6. log1p-define 7.9%

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

4. Applied egg-rr 7.9%

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

5. Taylor expanded in x around inf 7.0%

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

   1. +-commutative 7.0%

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

   2. distribute-lft-neg-in 7.0%

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

   3. metadata-eval 7.0%

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

   4. *-commutative 7.0%

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

   5. exp-to-pow 38.4%

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

   6. unpow1/2 38.4%

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

   7. associate--l+ 93.7%

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

7. Simplified 67.6%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{{x}^{-3}} + 0} \]
8. Step-by-step derivation

   1. +-rgt-identity 67.6%

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

   2. *-commutative 67.6%

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

   3. sqrt-pow1 97.8%

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

   4. metadata-eval 97.8%

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

9. Applied egg-rr 97.8%

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

10. Final simplification 97.8%

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

Alternative 9: 35.9% 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 39.3%

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

   1. add-exp-log 7.9%

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

   2. log-rec 7.9%

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

   3. pow1/2 7.9%

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

   4. log-pow 7.9%

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

   5. +-commutative 7.9%

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

   6. log1p-define 7.9%

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

4. Applied egg-rr 7.9%

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

5. Taylor expanded in x around inf 5.0%

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

   1. distribute-lft-neg-in 5.0%

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

   2. metadata-eval 5.0%

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

   3. *-commutative 5.0%

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

   4. exp-to-pow 36.2%

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

   5. unpow1/2 36.2%

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

   6. +-inverses 36.2%

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

7. Simplified 36.2%

   \[\leadsto \color{blue}{0} \]
8. 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]

Developer Target 2: 38.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024180 
(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))))))

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

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