2isqrt (example 3.6)

Percentage Accurate: 38.5% → 99.6%

Time: 15.8s

Alternatives: 8

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. frac-sub 37.2%

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

   2. div-inv 37.2%

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

   3. *-un-lft-identity 37.2%

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

   4. +-commutative 37.2%

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

   5. *-rgt-identity 37.2%

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

   6. metadata-eval 37.2%

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

   7. frac-times 37.2%

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

   8. associate-*l/ 37.2%

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

   9. *-un-lft-identity 37.2%

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

   10. inv-pow 37.2%

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

   11. sqrt-pow2 37.2%

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

   12. +-commutative 37.2%

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

   13. metadata-eval 37.2%

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

4. Applied egg-rr 37.2%

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

   1. *-commutative 37.2%

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

   2. /-rgt-identity 37.2%

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

   3. times-frac 37.2%

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

   4. *-commutative 37.2%

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

   5. times-frac 37.2%

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

   6. div-sub 37.1%

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

   7. sub-neg 37.1%

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

   8. *-inverses 37.1%

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

   9. metadata-eval 37.1%

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

6. Simplified 37.1%

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

   1. flip-+ 37.1%

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

   2. frac-times 37.1%

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

8. Applied egg-rr 37.3%

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

   1. *-commutative 37.3%

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

   2. *-rgt-identity 37.3%

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

   3. associate-/l* 37.3%

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

   4. +-commutative 37.3%

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

   5. +-commutative 37.3%

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

10. Simplified 37.3%

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

11. Taylor expanded in x around 0 99.6%

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

12. Final simplification 99.6%

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

Alternative 2: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. frac-sub 37.2%

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

   2. div-inv 37.2%

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

   3. *-un-lft-identity 37.2%

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

   4. +-commutative 37.2%

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

   5. *-rgt-identity 37.2%

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

   6. metadata-eval 37.2%

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

   7. frac-times 37.2%

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

   8. associate-*l/ 37.2%

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

   9. *-un-lft-identity 37.2%

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

   10. inv-pow 37.2%

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

   11. sqrt-pow2 37.2%

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

   12. +-commutative 37.2%

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

   13. metadata-eval 37.2%

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

4. Applied egg-rr 37.2%

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

   1. *-commutative 37.2%

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

   2. /-rgt-identity 37.2%

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

   3. times-frac 37.2%

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

   4. *-commutative 37.2%

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

   5. times-frac 37.2%

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

   6. div-sub 37.1%

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

   7. sub-neg 37.1%

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

   8. *-inverses 37.1%

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

   9. metadata-eval 37.1%

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

6. Simplified 37.1%

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

7. Taylor expanded in x around inf 99.2%

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

   1. associate--r+ 99.2%

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

   2. +-commutative 99.2%

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

   3. associate--l+ 99.2%

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

   4. sub-neg 99.2%

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

   5. associate-*r/ 99.2%

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

   6. metadata-eval 99.2%

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

   7. distribute-neg-frac 99.2%

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

   8. metadata-eval 99.2%

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

   9. associate-*r/ 99.2%

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

   10. metadata-eval 99.2%

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

9. Simplified 99.2%

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

10. Taylor expanded in x around -inf 99.2%

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

11. Final simplification 99.2%

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

Alternative 3: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return pow((x + 1.0), -0.5) * ((0.5 - ((0.125 + (0.0625 * (-1.0 / 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 * ((-1.0d0) / 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 * (-1.0 / x))) / x)) / x);
}

Python

def code(x):
	return math.pow((x + 1.0), -0.5) * ((0.5 - ((0.125 + (0.0625 * (-1.0 / 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 * Float64(-1.0 / x))) / x)) / x))
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) ^ -0.5) * ((0.5 - ((0.125 + (0.0625 * (-1.0 / 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 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.1%

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

   1. frac-sub 37.2%

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

   2. div-inv 37.2%

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

   3. *-un-lft-identity 37.2%

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

   4. +-commutative 37.2%

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

   5. *-rgt-identity 37.2%

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

   6. metadata-eval 37.2%

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

   7. frac-times 37.2%

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

   8. associate-*l/ 37.2%

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

   9. *-un-lft-identity 37.2%

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

   10. inv-pow 37.2%

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

   11. sqrt-pow2 37.2%

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

   12. +-commutative 37.2%

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

   13. metadata-eval 37.2%

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

4. Applied egg-rr 37.2%

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

   1. *-commutative 37.2%

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

   2. /-rgt-identity 37.2%

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

   3. times-frac 37.2%

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

   4. *-commutative 37.2%

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

   5. times-frac 37.2%

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

   6. div-sub 37.1%

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

   7. sub-neg 37.1%

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

   8. *-inverses 37.1%

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

   9. metadata-eval 37.1%

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

6. Simplified 37.1%

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

7. Taylor expanded in x around inf 99.2%

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

   1. associate--r+ 99.2%

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

   2. +-commutative 99.2%

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

   3. associate--l+ 99.2%

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

   4. sub-neg 99.2%

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

   5. associate-*r/ 99.2%

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

   6. metadata-eval 99.2%

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

   7. distribute-neg-frac 99.2%

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

   8. metadata-eval 99.2%

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

   9. associate-*r/ 99.2%

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

   10. metadata-eval 99.2%

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

9. Simplified 99.2%

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

10. Taylor expanded in x around -inf 98.9%

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

11. Final simplification 98.9%

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

Alternative 4: 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 37.1%

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

   1. frac-sub 37.2%

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

   2. div-inv 37.2%

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

   3. *-un-lft-identity 37.2%

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

   4. +-commutative 37.2%

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

   5. *-rgt-identity 37.2%

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

   6. metadata-eval 37.2%

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

   7. frac-times 37.2%

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

   8. associate-*l/ 37.2%

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

   9. *-un-lft-identity 37.2%

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

   10. inv-pow 37.2%

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

   11. sqrt-pow2 37.2%

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

   12. +-commutative 37.2%

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

   13. metadata-eval 37.2%

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

4. Applied egg-rr 37.2%

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

   1. *-commutative 37.2%

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

   2. /-rgt-identity 37.2%

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

   3. times-frac 37.2%

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

   4. *-commutative 37.2%

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

   5. times-frac 37.2%

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

   6. div-sub 37.1%

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

   7. sub-neg 37.1%

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

   8. *-inverses 37.1%

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

   9. metadata-eval 37.1%

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

6. Simplified 37.1%

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

7. Taylor expanded in x around inf 98.5%

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

   1. associate-*r/ 98.5%

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

   2. metadata-eval 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 5: 37.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.0390625}{{x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 8.5e+122) (pow x -0.5) (/ -0.0390625 (pow x 4.0))))

C

double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = pow(x, -0.5);
	} else {
		tmp = -0.0390625 / pow(x, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.5d+122) then
        tmp = x ** (-0.5d0)
    else
        tmp = (-0.0390625d0) / (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = Math.pow(x, -0.5);
	} else {
		tmp = -0.0390625 / Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.5e+122:
		tmp = math.pow(x, -0.5)
	else:
		tmp = -0.0390625 / math.pow(x, 4.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.5e+122)
		tmp = x ^ -0.5;
	else
		tmp = Float64(-0.0390625 / (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.5e+122)
		tmp = x ^ -0.5;
	else
		tmp = -0.0390625 / (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.5e+122], N[Power[x, -0.5], $MachinePrecision], N[(-0.0390625 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.50000000000000003e122

   1. Initial program 12.0%

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

      1. add-sqr-sqrt 12.5%

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

      2. sqrt-unprod 12.0%

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

      3. frac-times 12.2%

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

      4. metadata-eval 12.2%

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

      5. add-sqr-sqrt 12.2%

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

      6. +-commutative 12.2%

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

   4. Applied egg-rr 12.2%

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

   5. Taylor expanded in x around 0 7.5%

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

      1. rem-exp-log 7.5%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{e^{\log x}}}} \]

      2. exp-neg 7.5%

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

      3. unpow1/2 7.5%

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

      4. exp-prod 7.5%

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

      5. distribute-lft-neg-out 7.5%

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

      6. distribute-rgt-neg-in 7.5%

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

      7. metadata-eval 7.5%

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

      8. exp-to-pow 7.5%

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

   7. Simplified 7.5%

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

   if 8.50000000000000003e122 < x

   1. Initial program 53.8%

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

      1. frac-sub 53.8%

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

      2. div-inv 53.8%

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

      3. *-un-lft-identity 53.8%

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

      4. +-commutative 53.8%

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

      5. *-rgt-identity 53.8%

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

      6. metadata-eval 53.8%

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

      7. frac-times 53.8%

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

      8. associate-*l/ 53.8%

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

      9. *-un-lft-identity 53.8%

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

      10. inv-pow 53.8%

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

      11. sqrt-pow2 53.8%

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

      12. +-commutative 53.8%

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

      13. metadata-eval 53.8%

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

   4. Applied egg-rr 53.8%

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

      1. *-commutative 53.8%

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

      2. /-rgt-identity 53.8%

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

      3. times-frac 53.8%

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

      4. *-commutative 53.8%

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

      5. times-frac 53.8%

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

      6. div-sub 53.8%

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

      7. sub-neg 53.8%

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

      8. *-inverses 53.8%

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

      9. metadata-eval 53.8%

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

   6. Simplified 53.8%

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

   7. Taylor expanded in x around inf 99.7%

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

      1. associate--r+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l+ 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-*r/ 99.7%

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

      6. metadata-eval 99.7%

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

      7. distribute-neg-frac 99.7%

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

      8. metadata-eval 99.7%

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

      9. associate-*r/ 99.7%

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

      10. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in x around 0 53.8%

       \[\leadsto \color{blue}{\frac{-0.0390625}{{x}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.0390625}{{x}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. frac-sub 37.2%

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

   2. div-inv 37.2%

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

   3. *-un-lft-identity 37.2%

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

   4. +-commutative 37.2%

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

   5. *-rgt-identity 37.2%

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

   6. metadata-eval 37.2%

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

   7. frac-times 37.2%

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

   8. associate-*l/ 37.2%

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

   9. *-un-lft-identity 37.2%

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

   10. inv-pow 37.2%

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

   11. sqrt-pow2 37.2%

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

   12. +-commutative 37.2%

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

   13. metadata-eval 37.2%

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

4. Applied egg-rr 37.2%

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

   1. *-commutative 37.2%

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

   2. /-rgt-identity 37.2%

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

   3. times-frac 37.2%

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

   4. *-commutative 37.2%

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

   5. times-frac 37.2%

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

   6. div-sub 37.1%

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

   7. sub-neg 37.1%

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

   8. *-inverses 37.1%

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

   9. metadata-eval 37.1%

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

6. Simplified 37.1%

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

7. Taylor expanded in x around inf 97.5%

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

8. Final simplification 97.5%

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

Alternative 7: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

3. Taylor expanded in x around inf 80.3%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 98.5%

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

6. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, {x}^{-1.5}, 0.5 \cdot \left({x}^{-0.5} - {x}^{-1.5}\right)\right)}{x}} \]

7. Taylor expanded in x around inf 97.4%

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

8. Final simplification 97.4%

   \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]
9. Add Preprocessing

Alternative 8: 5.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 37.1%

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

   1. add-sqr-sqrt 23.6%

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

   2. sqrt-unprod 37.1%

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

   3. frac-times 32.8%

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

   4. metadata-eval 32.8%

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

   5. add-sqr-sqrt 29.1%

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

   6. +-commutative 29.1%

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

4. Applied egg-rr 29.1%

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

5. Taylor expanded in x around 0 5.7%

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

   1. rem-exp-log 5.7%

      \[\leadsto \sqrt{\frac{1}{\color{blue}{e^{\log x}}}} \]

   2. exp-neg 5.7%

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

   3. unpow1/2 5.7%

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

   4. exp-prod 5.7%

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

   5. distribute-lft-neg-out 5.7%

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

   6. distribute-rgt-neg-in 5.7%

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

   7. metadata-eval 5.7%

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

   8. exp-to-pow 5.7%

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

7. Simplified 5.7%

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

8. Final simplification 5.7%

   \[\leadsto {x}^{-0.5} \]
9. 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 2024074 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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