2isqrt (example 3.6)

Percentage Accurate: 13.3% → 99.2%

Time: 14.0s

Alternatives: 10

Speedup: 1.8×

Specification

\[x > 1 \land x < 10^{+100}\]

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: 13.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.4%

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

   1. flip-- 13.4%

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

   2. clear-num 13.4%

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

   3. pow1/2 13.4%

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

   4. pow-flip 13.4%

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

   5. metadata-eval 13.4%

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

   6. inv-pow 13.4%

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

   7. sqrt-pow2 13.4%

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

   8. +-commutative 13.4%

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

   9. metadata-eval 13.4%

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

   10. frac-times 13.9%

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

   11. metadata-eval 13.9%

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

   12. add-sqr-sqrt 13.8%

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

   13. frac-times 14.0%

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

   14. metadata-eval 14.0%

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

   15. add-sqr-sqrt 13.8%

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

3. Applied egg-rr 13.8%

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

   1. frac-sub 19.2%

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

   2. *-un-lft-identity 19.2%

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

5. Applied egg-rr 19.2%

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

   1. *-rgt-identity 19.2%

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

   2. associate--l+ 99.3%

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

   3. distribute-lft-in 99.3%

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

   4. *-rgt-identity 99.3%

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

   5. unpow2 99.3%

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.4%

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

   1. flip-- 13.4%

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

   2. clear-num 13.4%

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

   3. pow1/2 13.4%

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

   4. pow-flip 13.4%

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

   5. metadata-eval 13.4%

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

   6. inv-pow 13.4%

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

   7. sqrt-pow2 13.4%

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

   8. +-commutative 13.4%

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

   9. metadata-eval 13.4%

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

   10. frac-times 13.9%

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

   11. metadata-eval 13.9%

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

   12. add-sqr-sqrt 13.8%

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

   13. frac-times 14.0%

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

   14. metadata-eval 14.0%

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

   15. add-sqr-sqrt 13.8%

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

3. Applied egg-rr 13.8%

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

   1. frac-sub 19.2%

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

   2. *-un-lft-identity 19.2%

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

5. Applied egg-rr 19.2%

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

   1. associate-/r* 19.2%

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

   2. *-rgt-identity 19.2%

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

   3. associate--l+ 99.3%

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

   4. +-inverses 99.3%

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

   5. metadata-eval 99.3%

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

   6. +-commutative 99.3%

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 3: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 120000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 120000000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 120000000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 120000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 120000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 120000000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 120000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 120000000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 120000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2e8

   1. Initial program 79.7%

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

      1. *-un-lft-identity 79.7%

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

      2. clear-num 79.7%

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

      3. associate-/r/ 79.7%

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

      4. prod-diff 79.7%

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

      5. *-un-lft-identity 79.7%

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

      6. fma-neg 79.7%

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

      7. *-un-lft-identity 79.7%

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

      8. pow1/2 79.7%

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

      9. pow-flip 80.4%

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

      10. metadata-eval 80.4%

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

      11. pow1/2 80.4%

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

      12. pow-flip 80.6%

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

      13. +-commutative 80.6%

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

      14. metadata-eval 80.6%

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

   3. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 80.6%

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

      2. sub-neg 80.6%

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

      3. fma-udef 80.6%

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

      4. distribute-lft1-in 80.6%

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

      5. metadata-eval 80.6%

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

      6. mul0-lft 80.6%

         \[\leadsto \color{blue}{0} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      7. +-commutative 80.6%

         \[\leadsto 0 + \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}\right)} \]

      8. associate-+r+ 80.6%

         \[\leadsto \color{blue}{\left(0 + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) + {x}^{-0.5}} \]

      9. sub-neg 80.6%

         \[\leadsto \color{blue}{\left(0 - {\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      10. neg-sub0 80.6%

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

      11. +-commutative 80.6%

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

      12. sub-neg 80.6%

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

   5. Simplified 80.6%

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

   if 1.2e8 < x

   1. Initial program 6.9%

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

      1. flip-- 6.8%

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

      2. clear-num 6.8%

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

      3. pow1/2 6.8%

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

      4. pow-flip 6.8%

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

      5. metadata-eval 6.8%

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

      6. inv-pow 6.8%

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

      7. sqrt-pow2 6.8%

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

      8. +-commutative 6.8%

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

      9. metadata-eval 6.8%

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

      10. frac-times 7.4%

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

      11. metadata-eval 7.4%

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

      12. add-sqr-sqrt 7.3%

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

      13. frac-times 7.4%

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

      14. metadata-eval 7.4%

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

      15. add-sqr-sqrt 7.0%

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

   3. Applied egg-rr 7.0%

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

   4. Taylor expanded in x around inf 97.9%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 120000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 4: 22.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot \sqrt{\frac{1}{x}}}{\frac{1}{x} - \frac{1}{1 + x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 7.2e+15)
   (/ 1.0 (/ (* 2.0 (sqrt (/ 1.0 x))) (- (/ 1.0 x) (/ 1.0 (+ 1.0 x)))))
   (sqrt (/ 1.0 (pow x 3.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 7.2e+15) {
		tmp = 1.0 / ((2.0 * sqrt((1.0 / x))) / ((1.0 / x) - (1.0 / (1.0 + x))));
	} else {
		tmp = sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 7.2d+15) then
        tmp = 1.0d0 / ((2.0d0 * sqrt((1.0d0 / x))) / ((1.0d0 / x) - (1.0d0 / (1.0d0 + x))))
    else
        tmp = sqrt((1.0d0 / (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 7.2e+15) {
		tmp = 1.0 / ((2.0 * Math.sqrt((1.0 / x))) / ((1.0 / x) - (1.0 / (1.0 + x))));
	} else {
		tmp = Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 7.2e+15:
		tmp = 1.0 / ((2.0 * math.sqrt((1.0 / x))) / ((1.0 / x) - (1.0 / (1.0 + x))))
	else:
		tmp = math.sqrt((1.0 / math.pow(x, 3.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.2e+15)
		tmp = Float64(1.0 / Float64(Float64(2.0 * sqrt(Float64(1.0 / x))) / Float64(Float64(1.0 / x) - Float64(1.0 / Float64(1.0 + x)))));
	else
		tmp = sqrt(Float64(1.0 / (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 7.2e+15)
		tmp = 1.0 / ((2.0 * sqrt((1.0 / x))) / ((1.0 / x) - (1.0 / (1.0 + x))));
	else
		tmp = sqrt((1.0 / (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 7.2e+15], N[(1.0 / N[(N[(2.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2e15

   1. Initial program 64.6%

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

      1. flip-- 64.5%

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

      2. clear-num 64.5%

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

      3. pow1/2 64.5%

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

      4. pow-flip 64.5%

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

      5. metadata-eval 64.5%

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

      6. inv-pow 64.5%

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

      7. sqrt-pow2 64.5%

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

      8. +-commutative 64.5%

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

      9. metadata-eval 64.5%

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

      10. frac-times 64.7%

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

      11. metadata-eval 64.7%

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

      12. add-sqr-sqrt 65.3%

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

      13. frac-times 65.7%

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

      14. metadata-eval 65.7%

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

      15. add-sqr-sqrt 66.9%

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

   3. Applied egg-rr 66.9%

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

   4. Taylor expanded in x around inf 43.8%

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

   if 7.2e15 < x

   1. Initial program 3.9%

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

      1. sub-neg 3.9%

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

      2. flip-+ 3.9%

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

      3. frac-times 4.5%

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

      4. metadata-eval 4.5%

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

      5. add-sqr-sqrt 4.3%

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

      6. distribute-neg-frac 4.3%

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

      7. metadata-eval 4.3%

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

      8. +-commutative 4.3%

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

      9. distribute-neg-frac 4.3%

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

      10. metadata-eval 4.3%

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

      11. +-commutative 4.3%

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

      12. pow1/2 4.3%

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

      13. pow-flip 4.3%

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

      14. metadata-eval 4.3%

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

   3. Applied egg-rr 4.3%

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

   4. Taylor expanded in x around inf 18.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{3}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot \sqrt{\frac{1}{x}}}{\frac{1}{x} - \frac{1}{1 + x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 5: 93.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.4%

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

   1. flip-- 13.4%

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

   2. clear-num 13.4%

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

   3. pow1/2 13.4%

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

   4. pow-flip 13.4%

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

   5. metadata-eval 13.4%

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

   6. inv-pow 13.4%

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

   7. sqrt-pow2 13.4%

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

   8. +-commutative 13.4%

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

   9. metadata-eval 13.4%

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

   10. frac-times 13.9%

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

   11. metadata-eval 13.9%

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

   12. add-sqr-sqrt 13.8%

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

   13. frac-times 14.0%

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

   14. metadata-eval 14.0%

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

   15. add-sqr-sqrt 13.8%

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

3. Applied egg-rr 13.8%

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

4. Taylor expanded in x around inf 92.6%

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

5. Final simplification 92.6%

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

Alternative 6: 14.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot \sqrt{\frac{1}{x}}}{\frac{1}{x} - \frac{1}{1 + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + {x}^{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 7.2e+15)
   (/ 1.0 (/ (* 2.0 (sqrt (/ 1.0 x))) (- (/ 1.0 x) (/ 1.0 (+ 1.0 x)))))
   (/ 1.0 (+ x (pow x 0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 7.2e+15) {
		tmp = 1.0 / ((2.0 * sqrt((1.0 / x))) / ((1.0 / x) - (1.0 / (1.0 + x))));
	} else {
		tmp = 1.0 / (x + pow(x, 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 7.2d+15) then
        tmp = 1.0d0 / ((2.0d0 * sqrt((1.0d0 / x))) / ((1.0d0 / x) - (1.0d0 / (1.0d0 + x))))
    else
        tmp = 1.0d0 / (x + (x ** 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 7.2e+15) {
		tmp = 1.0 / ((2.0 * Math.sqrt((1.0 / x))) / ((1.0 / x) - (1.0 / (1.0 + x))));
	} else {
		tmp = 1.0 / (x + Math.pow(x, 0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 7.2e+15:
		tmp = 1.0 / ((2.0 * math.sqrt((1.0 / x))) / ((1.0 / x) - (1.0 / (1.0 + x))))
	else:
		tmp = 1.0 / (x + math.pow(x, 0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.2e+15)
		tmp = Float64(1.0 / Float64(Float64(2.0 * sqrt(Float64(1.0 / x))) / Float64(Float64(1.0 / x) - Float64(1.0 / Float64(1.0 + x)))));
	else
		tmp = Float64(1.0 / Float64(x + (x ^ 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 7.2e+15)
		tmp = 1.0 / ((2.0 * sqrt((1.0 / x))) / ((1.0 / x) - (1.0 / (1.0 + x))));
	else
		tmp = 1.0 / (x + (x ^ 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 7.2e+15], N[(1.0 / N[(N[(2.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x + N[Power[x, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2e15

   1. Initial program 64.6%

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

      1. flip-- 64.5%

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

      2. clear-num 64.5%

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

      3. pow1/2 64.5%

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

      4. pow-flip 64.5%

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

      5. metadata-eval 64.5%

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

      6. inv-pow 64.5%

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

      7. sqrt-pow2 64.5%

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

      8. +-commutative 64.5%

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

      9. metadata-eval 64.5%

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

      10. frac-times 64.7%

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

      11. metadata-eval 64.7%

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

      12. add-sqr-sqrt 65.3%

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

      13. frac-times 65.7%

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

      14. metadata-eval 65.7%

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

      15. add-sqr-sqrt 66.9%

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

   3. Applied egg-rr 66.9%

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

   4. Taylor expanded in x around inf 43.8%

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

   if 7.2e15 < x

   1. Initial program 3.9%

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

      1. sub-neg 3.9%

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

      2. flip-+ 3.9%

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

      3. frac-times 4.5%

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

      4. metadata-eval 4.5%

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

      5. add-sqr-sqrt 4.3%

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

      6. distribute-neg-frac 4.3%

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

      7. metadata-eval 4.3%

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

      8. +-commutative 4.3%

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

      9. distribute-neg-frac 4.3%

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

      10. metadata-eval 4.3%

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

      11. +-commutative 4.3%

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

      12. pow1/2 4.3%

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

      13. pow-flip 4.3%

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

      14. metadata-eval 4.3%

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

   3. Applied egg-rr 4.3%

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

   4. Taylor expanded in x around 0 8.7%

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

      1. distribute-rgt-in 8.7%

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

      2. *-lft-identity 8.7%

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

      3. pow-plus 8.7%

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

      4. metadata-eval 8.7%

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

   6. Simplified 8.7%

      \[\leadsto \color{blue}{\frac{1}{x + {x}^{0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot \sqrt{\frac{1}{x}}}{\frac{1}{x} - \frac{1}{1 + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + {x}^{0.5}}\\ \end{array} \]

Alternative 7: 14.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\frac{{x}^{-0.5} \cdot 2}{\frac{1}{x} - \frac{1}{1 + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + {x}^{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 7.2e+15)
   (/ 1.0 (/ (* (pow x -0.5) 2.0) (- (/ 1.0 x) (/ 1.0 (+ 1.0 x)))))
   (/ 1.0 (+ x (pow x 0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 7.2e+15) {
		tmp = 1.0 / ((pow(x, -0.5) * 2.0) / ((1.0 / x) - (1.0 / (1.0 + x))));
	} else {
		tmp = 1.0 / (x + pow(x, 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 7.2d+15) then
        tmp = 1.0d0 / (((x ** (-0.5d0)) * 2.0d0) / ((1.0d0 / x) - (1.0d0 / (1.0d0 + x))))
    else
        tmp = 1.0d0 / (x + (x ** 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 7.2e+15) {
		tmp = 1.0 / ((Math.pow(x, -0.5) * 2.0) / ((1.0 / x) - (1.0 / (1.0 + x))));
	} else {
		tmp = 1.0 / (x + Math.pow(x, 0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 7.2e+15:
		tmp = 1.0 / ((math.pow(x, -0.5) * 2.0) / ((1.0 / x) - (1.0 / (1.0 + x))))
	else:
		tmp = 1.0 / (x + math.pow(x, 0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.2e+15)
		tmp = Float64(1.0 / Float64(Float64((x ^ -0.5) * 2.0) / Float64(Float64(1.0 / x) - Float64(1.0 / Float64(1.0 + x)))));
	else
		tmp = Float64(1.0 / Float64(x + (x ^ 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 7.2e+15)
		tmp = 1.0 / (((x ^ -0.5) * 2.0) / ((1.0 / x) - (1.0 / (1.0 + x))));
	else
		tmp = 1.0 / (x + (x ^ 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 7.2e+15], N[(1.0 / N[(N[(N[Power[x, -0.5], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x + N[Power[x, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2e15

   1. Initial program 64.6%

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

      1. flip-- 64.5%

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

      2. clear-num 64.5%

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

      3. pow1/2 64.5%

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

      4. pow-flip 64.5%

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

      5. metadata-eval 64.5%

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

      6. inv-pow 64.5%

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

      7. sqrt-pow2 64.5%

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

      8. +-commutative 64.5%

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

      9. metadata-eval 64.5%

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

      10. frac-times 64.7%

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

      11. metadata-eval 64.7%

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

      12. add-sqr-sqrt 65.3%

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

      13. frac-times 65.7%

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

      14. metadata-eval 65.7%

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

      15. add-sqr-sqrt 66.9%

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

   3. Applied egg-rr 66.9%

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

   4. Taylor expanded in x around inf 43.8%

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

      1. *-commutative 43.8%

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

      2. rem-exp-log 43.8%

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

      3. exp-neg 43.8%

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

      4. unpow1/2 43.8%

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

      5. exp-prod 43.8%

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

      6. distribute-lft-neg-out 43.8%

         \[\leadsto \frac{1}{\frac{e^{\color{blue}{-\log x \cdot 0.5}} \cdot 2}{\frac{1}{x} - \frac{1}{1 + x}}} \]

      7. distribute-rgt-neg-in 43.8%

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

      8. metadata-eval 43.8%

         \[\leadsto \frac{1}{\frac{e^{\log x \cdot \color{blue}{-0.5}} \cdot 2}{\frac{1}{x} - \frac{1}{1 + x}}} \]

      9. exp-to-pow 43.8%

         \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{-0.5}} \cdot 2}{\frac{1}{x} - \frac{1}{1 + x}}} \]

   6. Simplified 43.8%

      \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{-0.5} \cdot 2}}{\frac{1}{x} - \frac{1}{1 + x}}} \]

   if 7.2e15 < x

   1. Initial program 3.9%

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

      1. sub-neg 3.9%

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

      2. flip-+ 3.9%

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

      3. frac-times 4.5%

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

      4. metadata-eval 4.5%

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

      5. add-sqr-sqrt 4.3%

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

      6. distribute-neg-frac 4.3%

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

      7. metadata-eval 4.3%

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

      8. +-commutative 4.3%

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

      9. distribute-neg-frac 4.3%

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

      10. metadata-eval 4.3%

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

      11. +-commutative 4.3%

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

      12. pow1/2 4.3%

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

      13. pow-flip 4.3%

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

      14. metadata-eval 4.3%

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

   3. Applied egg-rr 4.3%

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

   4. Taylor expanded in x around 0 8.7%

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

      1. distribute-rgt-in 8.7%

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

      2. *-lft-identity 8.7%

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

      3. pow-plus 8.7%

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

      4. metadata-eval 8.7%

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

   6. Simplified 8.7%

      \[\leadsto \color{blue}{\frac{1}{x + {x}^{0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\frac{{x}^{-0.5} \cdot 2}{\frac{1}{x} - \frac{1}{1 + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + {x}^{0.5}}\\ \end{array} \]

Alternative 8: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.4%

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

   1. sub-neg 13.4%

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

   2. flip-+ 13.4%

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

   3. frac-times 13.9%

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

   4. metadata-eval 13.9%

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

   5. add-sqr-sqrt 13.8%

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

   6. distribute-neg-frac 13.8%

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

   7. metadata-eval 13.8%

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

   8. +-commutative 13.8%

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

   9. distribute-neg-frac 13.8%

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

   10. metadata-eval 13.8%

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

   11. +-commutative 13.8%

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

   12. pow1/2 13.8%

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

   13. pow-flip 13.8%

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

   14. metadata-eval 13.8%

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

3. Applied egg-rr 13.8%

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

4. Taylor expanded in x around 0 9.4%

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

   1. distribute-rgt-in 9.4%

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

   2. *-lft-identity 9.4%

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

   3. pow-plus 9.4%

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

   4. metadata-eval 9.4%

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

6. Simplified 9.4%

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

7. Final simplification 9.4%

   \[\leadsto \frac{1}{x + {x}^{0.5}} \]

Alternative 9: 7.9% 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 13.4%

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

   1. inv-pow 13.4%

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

   2. add-sqr-sqrt 13.6%

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

   3. unpow-prod-down 13.9%

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

   4. pow1/2 13.9%

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

   5. sqrt-pow1 13.7%

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

   6. metadata-eval 13.7%

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

   7. pow1/2 13.7%

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

   8. sqrt-pow1 13.7%

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

   9. metadata-eval 13.7%

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

3. Applied egg-rr 13.7%

   \[\leadsto \color{blue}{{\left({x}^{0.25}\right)}^{-1} \cdot {\left({x}^{0.25}\right)}^{-1}} - \frac{1}{\sqrt{x + 1}} \]
4. Step-by-step derivation

   1. pow-sqr 13.5%

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

   2. metadata-eval 13.5%

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

5. Simplified 13.5%

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

6. Taylor expanded in x around inf 7.9%

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

   1. pow1/2 7.9%

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

   2. inv-pow 7.9%

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

   3. pow-pow 7.9%

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

   4. metadata-eval 7.9%

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

   5. expm1-log1p-u 7.9%

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

   6. expm1-udef 6.2%

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

8. Applied egg-rr 6.2%

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

   1. expm1-def 7.9%

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

   2. expm1-log1p 7.9%

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

10. Simplified 7.9%

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

11. Final simplification 7.9%

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

Alternative 10: 1.9% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

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

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 13.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

2. Taylor expanded in x around 0 1.9%

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

3. Taylor expanded in x around inf 1.9%

   \[\leadsto \color{blue}{-1} \]

4. Final simplification 1.9%

   \[\leadsto -1 \]

Developer target: 99.4% 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 2023333 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+100))

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

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