2isqrt (example 3.6)

Percentage Accurate: 38.7% → 99.9%

Time: 13.8s

Alternatives: 7

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: 38.7% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(x - x\right)}{x + {x}^{2}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 0.0)
   (* (pow x -1.5) 0.5)
   (/
    (/ (+ 1.0 (- x x)) (+ x (pow x 2.0)))
    (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = pow(x, -1.5) * 0.5;
	} else {
		tmp = ((1.0 + (x - x)) / (x + pow(x, 2.0))) / (pow(x, -0.5) + pow((1.0 + x), -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 0.0d0) then
        tmp = (x ** (-1.5d0)) * 0.5d0
    else
        tmp = ((1.0d0 + (x - x)) / (x + (x ** 2.0d0))) / ((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 0.0) {
		tmp = Math.pow(x, -1.5) * 0.5;
	} else {
		tmp = ((1.0 + (x - x)) / (x + Math.pow(x, 2.0))) / (Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 0.0:
		tmp = math.pow(x, -1.5) * 0.5
	else:
		tmp = ((1.0 + (x - x)) / (x + math.pow(x, 2.0))) / (math.pow(x, -0.5) + math.pow((1.0 + x), -0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 0.0)
		tmp = Float64((x ^ -1.5) * 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x - x)) / Float64(x + (x ^ 2.0))) / Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0)
		tmp = (x ^ -1.5) * 0.5;
	else
		tmp = ((1.0 + (x - x)) / (x + (x ^ 2.0))) / ((x ^ -0.5) + ((1.0 + x) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(x + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.0

   1. Initial program 39.9%

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

   3. Taylor expanded in x around inf 80.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
   5. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + -1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

      4. *-commutative 99.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}{x} \]

      5. distribute-rgt-out 99.7%

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

      6. metadata-eval 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

      7. *-rgt-identity 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{\color{blue}{x \cdot 1}}}{x} \]

      8. times-frac 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\frac{\sqrt{\frac{1}{x}}}{x} \cdot \frac{0.375}{1}}}{x} \]

      9. metadata-eval 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot \color{blue}{0.375}}{x} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]
   7. Step-by-step derivation

      1. div-sub 99.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} - \frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]

      2. sub-neg 99.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} + \left(-\frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}\right)} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5} + \left(-{x}^{-1.5} \cdot \frac{0.375}{x}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} + \left(-{x}^{-1.5} \cdot \frac{0.375}{x}\right) \]

      2. distribute-rgt-neg-in 100.0%

         \[\leadsto {x}^{-1.5} \cdot 0.5 + \color{blue}{{x}^{-1.5} \cdot \left(-\frac{0.375}{x}\right)} \]

      3. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{{x}^{-1.5} \cdot \left(0.5 + \left(-\frac{0.375}{x}\right)\right)} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \color{blue}{\frac{-0.375}{x}}\right) \]

      5. metadata-eval 100.0%

         \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \frac{\color{blue}{-0.375}}{x}\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{{x}^{-1.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)} \]

   11. Taylor expanded in x around inf 100.0%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 67.7%

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

      1. flip-- 68.1%

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

      2. div-inv 68.1%

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

      3. frac-times 67.4%

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

      4. metadata-eval 67.4%

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

      5. add-sqr-sqrt 67.6%

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

      6. frac-times 68.0%

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

      7. metadata-eval 68.0%

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

      8. add-sqr-sqrt 69.3%

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

      9. +-commutative 69.3%

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

      10. inv-pow 69.3%

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

      11. sqrt-pow2 69.3%

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

      12. metadata-eval 69.3%

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

      13. pow1/2 69.3%

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

      14. pow-flip 69.2%

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

      15. +-commutative 69.2%

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

      16. metadata-eval 69.2%

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

   4. Applied egg-rr 69.2%

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

      1. associate-*r/ 69.2%

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

      2. *-rgt-identity 69.2%

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

   6. Simplified 69.2%

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

      1. frac-sub 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. distribute-rgt-in 99.2%

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

      4. *-un-lft-identity 99.2%

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

      5. unpow2 99.2%

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

   8. Applied egg-rr 99.2%

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

      1. *-rgt-identity 99.2%

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

      2. associate--l+ 99.7%

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

   10. Simplified 99.7%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(x - x\right)}{x + {x}^{2}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 0.0)
   (* (pow x -1.5) 0.5)
   (/ (/ 1.0 (* x (+ 1.0 x))) (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = pow(x, -1.5) * 0.5;
	} else {
		tmp = (1.0 / (x * (1.0 + x))) / (pow(x, -0.5) + pow((1.0 + x), -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 0.0d0) then
        tmp = (x ** (-1.5d0)) * 0.5d0
    else
        tmp = (1.0d0 / (x * (1.0d0 + x))) / ((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 0.0) {
		tmp = Math.pow(x, -1.5) * 0.5;
	} else {
		tmp = (1.0 / (x * (1.0 + x))) / (Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 0.0:
		tmp = math.pow(x, -1.5) * 0.5
	else:
		tmp = (1.0 / (x * (1.0 + x))) / (math.pow(x, -0.5) + math.pow((1.0 + x), -0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 0.0)
		tmp = Float64((x ^ -1.5) * 0.5);
	else
		tmp = Float64(Float64(1.0 / Float64(x * Float64(1.0 + x))) / Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0)
		tmp = (x ^ -1.5) * 0.5;
	else
		tmp = (1.0 / (x * (1.0 + x))) / ((x ^ -0.5) + ((1.0 + x) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(1.0 / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.0

   1. Initial program 39.9%

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

   3. Taylor expanded in x around inf 80.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
   5. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + -1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

      4. *-commutative 99.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}{x} \]

      5. distribute-rgt-out 99.7%

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

      6. metadata-eval 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

      7. *-rgt-identity 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{\color{blue}{x \cdot 1}}}{x} \]

      8. times-frac 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\frac{\sqrt{\frac{1}{x}}}{x} \cdot \frac{0.375}{1}}}{x} \]

      9. metadata-eval 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot \color{blue}{0.375}}{x} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]
   7. Step-by-step derivation

      1. div-sub 99.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} - \frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]

      2. sub-neg 99.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} + \left(-\frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}\right)} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5} + \left(-{x}^{-1.5} \cdot \frac{0.375}{x}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} + \left(-{x}^{-1.5} \cdot \frac{0.375}{x}\right) \]

      2. distribute-rgt-neg-in 100.0%

         \[\leadsto {x}^{-1.5} \cdot 0.5 + \color{blue}{{x}^{-1.5} \cdot \left(-\frac{0.375}{x}\right)} \]

      3. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{{x}^{-1.5} \cdot \left(0.5 + \left(-\frac{0.375}{x}\right)\right)} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \color{blue}{\frac{-0.375}{x}}\right) \]

      5. metadata-eval 100.0%

         \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \frac{\color{blue}{-0.375}}{x}\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{{x}^{-1.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)} \]

   11. Taylor expanded in x around inf 100.0%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 67.7%

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

      1. flip-- 68.1%

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

      2. div-inv 68.1%

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

      3. frac-times 67.4%

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

      4. metadata-eval 67.4%

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

      5. add-sqr-sqrt 67.6%

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

      6. frac-times 68.0%

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

      7. metadata-eval 68.0%

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

      8. add-sqr-sqrt 69.3%

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

      9. +-commutative 69.3%

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

      10. inv-pow 69.3%

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

      11. sqrt-pow2 69.3%

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

      12. metadata-eval 69.3%

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

      13. pow1/2 69.3%

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

      14. pow-flip 69.2%

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

      15. +-commutative 69.2%

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

      16. metadata-eval 69.2%

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

   4. Applied egg-rr 69.2%

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

      1. associate-*r/ 69.2%

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

      2. *-rgt-identity 69.2%

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

   6. Simplified 69.2%

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

      1. frac-sub 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. distribute-rgt-in 99.2%

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

      4. *-un-lft-identity 99.2%

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

      5. unpow2 99.2%

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

   8. Applied egg-rr 99.2%

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

      1. *-rgt-identity 99.2%

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

      2. associate--l+ 99.7%

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

      3. +-inverses 99.7%

         \[\leadsto \frac{\frac{1 + \color{blue}{0}}{x + {x}^{2}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

      4. metadata-eval 99.7%

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

      5. *-rgt-identity 99.7%

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

      6. unpow2 99.7%

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

      7. distribute-lft-out 99.7%

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

   10. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot \left(1 + x\right)}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{x}}\\ \mathbf{if}\;t\_0 + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;{x}^{-1.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt x))))
   (if (<= (+ t_0 (/ -1.0 (sqrt (+ 1.0 x)))) 2e-12)
     (* (pow x -1.5) (+ 0.5 (/ -0.375 x)))
     (- t_0 (pow (+ 1.0 x) -0.5)))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(x);
	double tmp;
	if ((t_0 + (-1.0 / sqrt((1.0 + x)))) <= 2e-12) {
		tmp = pow(x, -1.5) * (0.5 + (-0.375 / x));
	} else {
		tmp = t_0 - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / sqrt(x)
    if ((t_0 + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 2d-12) then
        tmp = (x ** (-1.5d0)) * (0.5d0 + ((-0.375d0) / x))
    else
        tmp = t_0 - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.sqrt(x);
	double tmp;
	if ((t_0 + (-1.0 / Math.sqrt((1.0 + x)))) <= 2e-12) {
		tmp = Math.pow(x, -1.5) * (0.5 + (-0.375 / x));
	} else {
		tmp = t_0 - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / math.sqrt(x)
	tmp = 0
	if (t_0 + (-1.0 / math.sqrt((1.0 + x)))) <= 2e-12:
		tmp = math.pow(x, -1.5) * (0.5 + (-0.375 / x))
	else:
		tmp = t_0 - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(x))
	tmp = 0.0
	if (Float64(t_0 + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-12)
		tmp = Float64((x ^ -1.5) * Float64(0.5 + Float64(-0.375 / x)));
	else
		tmp = Float64(t_0 - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / sqrt(x);
	tmp = 0.0;
	if ((t_0 + (-1.0 / sqrt((1.0 + x)))) <= 2e-12)
		tmp = (x ^ -1.5) * (0.5 + (-0.375 / x));
	else
		tmp = t_0 - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-12], N[(N[Power[x, -1.5], $MachinePrecision] * N[(0.5 + N[(-0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 1.99999999999999996e-12

   1. Initial program 40.1%

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

   3. Taylor expanded in x around inf 81.3%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
   5. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + -1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

      4. *-commutative 99.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}{x} \]

      5. distribute-rgt-out 99.7%

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

      6. metadata-eval 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

      7. *-rgt-identity 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{\color{blue}{x \cdot 1}}}{x} \]

      8. times-frac 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\frac{\sqrt{\frac{1}{x}}}{x} \cdot \frac{0.375}{1}}}{x} \]

      9. metadata-eval 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot \color{blue}{0.375}}{x} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]
   7. Step-by-step derivation

      1. div-sub 99.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} - \frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]

      2. sub-neg 99.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} + \left(-\frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}\right)} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5} + \left(-{x}^{-1.5} \cdot \frac{0.375}{x}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} + \left(-{x}^{-1.5} \cdot \frac{0.375}{x}\right) \]

      2. distribute-rgt-neg-in 100.0%

         \[\leadsto {x}^{-1.5} \cdot 0.5 + \color{blue}{{x}^{-1.5} \cdot \left(-\frac{0.375}{x}\right)} \]

      3. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{{x}^{-1.5} \cdot \left(0.5 + \left(-\frac{0.375}{x}\right)\right)} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \color{blue}{\frac{-0.375}{x}}\right) \]

      5. metadata-eval 100.0%

         \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \frac{\color{blue}{-0.375}}{x}\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{{x}^{-1.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)} \]

   if 1.99999999999999996e-12 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 90.2%

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

      1. frac-2neg 90.2%

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

      2. metadata-eval 90.2%

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

      3. div-inv 90.2%

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

      4. frac-2neg 90.2%

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

      5. metadata-eval 90.2%

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

      6. div-inv 90.2%

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

      7. distribute-neg-frac2 90.2%

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

      8. prod-diff 90.2%

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

      9. distribute-neg-frac 90.2%

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

      10. metadata-eval 90.2%

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

      11. +-commutative 90.2%

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

   4. Applied egg-rr 91.4%

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

      1. +-commutative 91.4%

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

      2. fma-undefine 91.4%

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

      3. mul-1-neg 91.4%

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

      4. distribute-neg-out 91.4%

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

      5. unsub-neg 91.4%

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

   6. Simplified 91.4%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} - {\left(1 + x\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;{x}^{-1.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x}} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 235000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 235000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (* (pow x -1.5) (+ 0.5 (/ -0.375 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 235000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = pow(x, -1.5) * (0.5 + (-0.375 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 235000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = (x ** (-1.5d0)) * (0.5d0 + ((-0.375d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 235000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = Math.pow(x, -1.5) * (0.5 + (-0.375 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 235000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = math.pow(x, -1.5) * (0.5 + (-0.375 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 235000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64((x ^ -1.5) * Float64(0.5 + Float64(-0.375 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 235000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = (x ^ -1.5) * (0.5 + (-0.375 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 235000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.5], $MachinePrecision] * N[(0.5 + N[(-0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 235000

   1. Initial program 90.2%

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

      1. sub-neg 90.2%

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

      2. inv-pow 90.2%

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

      3. sqrt-pow2 90.2%

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

      4. metadata-eval 90.2%

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

      5. distribute-neg-frac 90.2%

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

      6. metadata-eval 90.2%

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

      7. +-commutative 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. *-rgt-identity 90.2%

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

      2. cancel-sign-sub 90.2%

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

      3. distribute-lft-neg-in 90.2%

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

      4. *-rgt-identity 90.2%

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

      5. distribute-neg-frac 90.2%

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

      6. metadata-eval 90.2%

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

      7. unpow1/2 90.2%

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

      8. exp-to-pow 89.2%

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

      9. log1p-undefine 89.0%

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

      10. *-commutative 89.0%

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

      11. exp-neg 89.3%

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

      12. *-commutative 89.3%

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

      13. distribute-rgt-neg-in 89.3%

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

      14. log1p-undefine 89.6%

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

      15. metadata-eval 89.6%

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

      16. exp-to-pow 91.3%

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

   6. Simplified 91.3%

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

   if 235000 < x

   1. Initial program 40.1%

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

   3. Taylor expanded in x around inf 81.3%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
   5. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + -1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

      4. *-commutative 99.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}{x} \]

      5. distribute-rgt-out 99.7%

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

      6. metadata-eval 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

      7. *-rgt-identity 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{\color{blue}{x \cdot 1}}}{x} \]

      8. times-frac 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\frac{\sqrt{\frac{1}{x}}}{x} \cdot \frac{0.375}{1}}}{x} \]

      9. metadata-eval 99.7%

         \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot \color{blue}{0.375}}{x} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]
   7. Step-by-step derivation

      1. div-sub 99.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} - \frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]

      2. sub-neg 99.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} + \left(-\frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}\right)} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5} + \left(-{x}^{-1.5} \cdot \frac{0.375}{x}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} + \left(-{x}^{-1.5} \cdot \frac{0.375}{x}\right) \]

      2. distribute-rgt-neg-in 100.0%

         \[\leadsto {x}^{-1.5} \cdot 0.5 + \color{blue}{{x}^{-1.5} \cdot \left(-\frac{0.375}{x}\right)} \]

      3. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{{x}^{-1.5} \cdot \left(0.5 + \left(-\frac{0.375}{x}\right)\right)} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \color{blue}{\frac{-0.375}{x}}\right) \]

      5. metadata-eval 100.0%

         \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \frac{\color{blue}{-0.375}}{x}\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{{x}^{-1.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 235000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {x}^{-1.5} \cdot \left(0.5 + \frac{-0.375}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow x -1.5) (+ 0.5 (/ -0.375 x))))

C

double code(double x) {
	return pow(x, -1.5) * (0.5 + (-0.375 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-1.5d0)) * (0.5d0 + ((-0.375d0) / x))
end function

Java

public static double code(double x) {
	return Math.pow(x, -1.5) * (0.5 + (-0.375 / x));
}

Python

def code(x):
	return math.pow(x, -1.5) * (0.5 + (-0.375 / x))

Julia

function code(x)
	return Float64((x ^ -1.5) * Float64(0.5 + Float64(-0.375 / x)))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -1.5) * (0.5 + (-0.375 / x));
end

Wolfram

code[x_] := N[(N[Power[x, -1.5], $MachinePrecision] * N[(0.5 + N[(-0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.9%

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

3. Taylor expanded in x around inf 79.9%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 97.6%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
5. Step-by-step derivation

   1. +-commutative 97.6%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + -1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   2. mul-1-neg 97.6%

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

   3. unsub-neg 97.6%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   4. *-commutative 97.6%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}{x} \]

   5. distribute-rgt-out 97.6%

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

   6. metadata-eval 97.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

   7. *-rgt-identity 97.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{\color{blue}{x \cdot 1}}}{x} \]

   8. times-frac 97.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\frac{\sqrt{\frac{1}{x}}}{x} \cdot \frac{0.375}{1}}}{x} \]

   9. metadata-eval 97.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot \color{blue}{0.375}}{x} \]

6. Simplified 97.6%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]
7. Step-by-step derivation

   1. div-sub 97.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} - \frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]

   2. sub-neg 97.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} + \left(-\frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}\right)} \]

8. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5} + \left(-{x}^{-1.5} \cdot \frac{0.375}{x}\right)} \]
9. Step-by-step derivation

   1. *-commutative 97.8%

      \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} + \left(-{x}^{-1.5} \cdot \frac{0.375}{x}\right) \]

   2. distribute-rgt-neg-in 97.8%

      \[\leadsto {x}^{-1.5} \cdot 0.5 + \color{blue}{{x}^{-1.5} \cdot \left(-\frac{0.375}{x}\right)} \]

   3. distribute-lft-out 97.8%

      \[\leadsto \color{blue}{{x}^{-1.5} \cdot \left(0.5 + \left(-\frac{0.375}{x}\right)\right)} \]

   4. distribute-neg-frac 97.8%

      \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \color{blue}{\frac{-0.375}{x}}\right) \]

   5. metadata-eval 97.8%

      \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \frac{\color{blue}{-0.375}}{x}\right) \]

10. Simplified 97.8%

    \[\leadsto \color{blue}{{x}^{-1.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)} \]

11. Final simplification 97.8%

    \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \frac{-0.375}{x}\right) \]
12. Add Preprocessing

Alternative 6: 98.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (pow x -1.5) 0.5))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64((x ^ -1.5) * 0.5)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.9%

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

3. Taylor expanded in x around inf 79.9%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 97.6%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
5. Step-by-step derivation

   1. +-commutative 97.6%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + -1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   2. mul-1-neg 97.6%

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

   3. unsub-neg 97.6%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   4. *-commutative 97.6%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}{x} \]

   5. distribute-rgt-out 97.6%

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

   6. metadata-eval 97.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

   7. *-rgt-identity 97.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{\color{blue}{x \cdot 1}}}{x} \]

   8. times-frac 97.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\frac{\sqrt{\frac{1}{x}}}{x} \cdot \frac{0.375}{1}}}{x} \]

   9. metadata-eval 97.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot \color{blue}{0.375}}{x} \]

6. Simplified 97.6%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]
7. Step-by-step derivation

   1. div-sub 97.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} - \frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]

   2. sub-neg 97.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} + \left(-\frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}\right)} \]

8. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5} + \left(-{x}^{-1.5} \cdot \frac{0.375}{x}\right)} \]
9. Step-by-step derivation

   1. *-commutative 97.8%

      \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} + \left(-{x}^{-1.5} \cdot \frac{0.375}{x}\right) \]

   2. distribute-rgt-neg-in 97.8%

      \[\leadsto {x}^{-1.5} \cdot 0.5 + \color{blue}{{x}^{-1.5} \cdot \left(-\frac{0.375}{x}\right)} \]

   3. distribute-lft-out 97.8%

      \[\leadsto \color{blue}{{x}^{-1.5} \cdot \left(0.5 + \left(-\frac{0.375}{x}\right)\right)} \]

   4. distribute-neg-frac 97.8%

      \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \color{blue}{\frac{-0.375}{x}}\right) \]

   5. metadata-eval 97.8%

      \[\leadsto {x}^{-1.5} \cdot \left(0.5 + \frac{\color{blue}{-0.375}}{x}\right) \]

10. Simplified 97.8%

    \[\leadsto \color{blue}{{x}^{-1.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)} \]

11. Taylor expanded in x around inf 96.6%

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

12. Final simplification 96.6%

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

Alternative 7: 35.9% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 41.9%

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

   1. sub-neg 41.9%

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

   2. +-commutative 41.9%

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

   3. add-cube-cbrt 14.3%

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

   4. distribute-lft-neg-in 14.3%

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

   5. fma-define 9.2%

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

4. Applied egg-rr 9.3%

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

5. Taylor expanded in x around inf 37.3%

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

   1. distribute-rgt1-in 37.3%

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

   2. metadata-eval 37.3%

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

   3. mul0-lft 37.3%

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

7. Simplified 37.3%

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

8. Final simplification 37.3%

   \[\leadsto 0 \]
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 2024077 
(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)))))