2isqrt (example 3.6)

Percentage Accurate: 37.8% → 99.7%

Time: 24.8s

Alternatives: 13

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: 37.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(x, x, x\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x} - -0.5 \cdot {x}^{-0.5}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+16)
   (/ (/ 1.0 (fma x x x)) (+ (pow x -0.5) (pow (+ x 1.0) -0.5)))
   (/ (- (/ (* (sqrt (/ 1.0 x)) 0.625) x) (* -0.5 (pow x -0.5))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2e+16) {
		tmp = (1.0 / fma(x, x, x)) / (pow(x, -0.5) + pow((x + 1.0), -0.5));
	} else {
		tmp = (((sqrt((1.0 / x)) * 0.625) / x) - (-0.5 * pow(x, -0.5))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+16)
		tmp = Float64(Float64(1.0 / fma(x, x, x)) / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)));
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / x)) * 0.625) / x) - Float64(-0.5 * (x ^ -0.5))) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+16], N[(N[(1.0 / N[(x * x + x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.625), $MachinePrecision] / x), $MachinePrecision] - N[(-0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e16

   1. Initial program 59.3%

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

      1. flip-- 59.4%

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

      2. div-inv 59.5%

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

      3. frac-times 59.7%

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

      4. metadata-eval 59.7%

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

      5. add-sqr-sqrt 59.9%

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

      6. frac-times 61.0%

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

      7. metadata-eval 61.0%

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

      8. add-sqr-sqrt 61.9%

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

      9. +-commutative 61.9%

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

      10. inv-pow 61.9%

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

      11. sqrt-pow2 61.9%

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

      12. metadata-eval 61.9%

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

      13. pow1/2 61.9%

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

      14. pow-flip 61.9%

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

      15. +-commutative 61.9%

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

      16. metadata-eval 61.9%

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

   4. Applied egg-rr 61.9%

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

      1. associate-*r/ 62.0%

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

      2. *-rgt-identity 62.0%

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

   6. Simplified 62.0%

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

      1. frac-sub 94.0%

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

      2. *-un-lft-identity 94.0%

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

      3. distribute-rgt-in 94.1%

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

      4. *-un-lft-identity 94.1%

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

      5. unpow2 94.1%

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

   8. Applied egg-rr 94.1%

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

      1. *-rgt-identity 94.1%

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

      2. associate--l+ 99.1%

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

      3. +-inverses 99.1%

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

      4. metadata-eval 99.1%

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

      5. +-commutative 99.1%

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

      6. unpow2 99.1%

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

      7. fma-define 99.1%

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

   10. Simplified 99.1%

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

   if 2e16 < x

   1. Initial program 38.0%

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

   3. Taylor expanded in x around inf 84.1%

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

   4. Taylor expanded in x around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. distribute-neg-frac2 0.0%

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

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x}} \]
   7. Step-by-step derivation

      1. pow1/2 99.7%

         \[\leadsto \frac{\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]

      2. inv-pow 99.7%

         \[\leadsto \frac{{\color{blue}{\left({x}^{-1}\right)}}^{0.5} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]

      3. pow-pow 99.8%

         \[\leadsto \frac{\color{blue}{{x}^{\left(-1 \cdot 0.5\right)}} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{{x}^{\color{blue}{-0.5}} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]

      5. *-un-lft-identity 99.8%

         \[\leadsto \frac{\color{blue}{\left(1 \cdot {x}^{-0.5}\right)} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{\color{blue}{\left(1 \cdot {x}^{-0.5}\right)} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]
   9. Step-by-step derivation

      1. *-lft-identity 99.8%

         \[\leadsto \frac{\color{blue}{{x}^{-0.5}} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]

   10. Simplified 99.8%

       \[\leadsto \frac{\color{blue}{{x}^{-0.5}} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(x, x, x\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x} - -0.5 \cdot {x}^{-0.5}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 4 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \frac{{x}^{-0.5}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}{\frac{-1}{-1 - x} + \frac{-1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ x 1.0)))) 4e-16)
   (* 0.5 (/ (pow x -0.5) (+ x 0.5)))
   (/
    -1.0
    (/
     (+ (pow x -0.5) (pow (+ x 1.0) -0.5))
     (+ (/ -1.0 (- -1.0 x)) (/ -1.0 x))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 4e-16) {
		tmp = 0.5 * (pow(x, -0.5) / (x + 0.5));
	} else {
		tmp = -1.0 / ((pow(x, -0.5) + pow((x + 1.0), -0.5)) / ((-1.0 / (-1.0 - x)) + (-1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((x + 1.0d0)))) <= 4d-16) then
        tmp = 0.5d0 * ((x ** (-0.5d0)) / (x + 0.5d0))
    else
        tmp = (-1.0d0) / (((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0))) / (((-1.0d0) / ((-1.0d0) - x)) + ((-1.0d0) / x)))
    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((x + 1.0)))) <= 4e-16) {
		tmp = 0.5 * (Math.pow(x, -0.5) / (x + 0.5));
	} else {
		tmp = -1.0 / ((Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5)) / ((-1.0 / (-1.0 - x)) + (-1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((x + 1.0)))) <= 4e-16:
		tmp = 0.5 * (math.pow(x, -0.5) / (x + 0.5))
	else:
		tmp = -1.0 / ((math.pow(x, -0.5) + math.pow((x + 1.0), -0.5)) / ((-1.0 / (-1.0 - x)) + (-1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(x + 1.0)))) <= 4e-16)
		tmp = Float64(0.5 * Float64((x ^ -0.5) / Float64(x + 0.5)));
	else
		tmp = Float64(-1.0 / Float64(Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)) / Float64(Float64(-1.0 / Float64(-1.0 - x)) + Float64(-1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 4e-16)
		tmp = 0.5 * ((x ^ -0.5) / (x + 0.5));
	else
		tmp = -1.0 / (((x ^ -0.5) + ((x + 1.0) ^ -0.5)) / ((-1.0 / (-1.0 - x)) + (-1.0 / x)));
	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[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-16], N[(0.5 * N[(N[Power[x, -0.5], $MachinePrecision] / N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $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))))) < 3.9999999999999999e-16

   1. Initial program 37.7%

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

      1. frac-sub 37.8%

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

      2. *-un-lft-identity 37.8%

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

      3. +-commutative 37.8%

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

      4. *-rgt-identity 37.8%

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

      5. sqrt-unprod 37.8%

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

      6. +-commutative 37.8%

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

   4. Applied egg-rr 37.8%

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

   5. Taylor expanded in x around inf 84.2%

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

      1. *-commutative 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in x around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   10. Simplified 99.4%

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

       1. times-frac 99.3%

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

       2. inv-pow 99.3%

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

       3. sqrt-pow1 99.4%

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

       4. metadata-eval 99.4%

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

       5. frac-times 99.4%

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

       6. *-commutative 99.4%

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

       7. *-un-lft-identity 99.4%

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

       8. times-frac 99.4%

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

       9. metadata-eval 99.4%

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

   12. Applied egg-rr 99.4%

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

       1. distribute-rgt-in 99.4%

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

       2. *-lft-identity 99.4%

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

       3. associate-*l/ 99.4%

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

       4. associate-/l* 99.4%

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

       5. *-inverses 99.4%

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

       6. metadata-eval 99.4%

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

   14. Simplified 99.4%

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

   if 3.9999999999999999e-16 < (-.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 80.2%

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

      1. flip-- 79.9%

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

      2. clear-num 80.0%

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

      3. inv-pow 80.0%

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

      4. sqrt-pow2 80.0%

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

      5. metadata-eval 80.0%

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

      6. pow1/2 80.0%

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

      7. pow-flip 80.0%

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

      8. +-commutative 80.0%

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

      9. metadata-eval 80.0%

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

      10. frac-times 80.7%

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

      11. metadata-eval 80.7%

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

      12. add-sqr-sqrt 81.4%

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

      13. frac-times 82.3%

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

      14. metadata-eval 82.3%

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

      15. add-sqr-sqrt 83.5%

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

   4. Applied egg-rr 83.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 4 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \frac{{x}^{-0.5}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}{\frac{-1}{-1 - x} + \frac{-1}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 4 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \frac{{x}^{-0.5}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ x 1.0)))) 4e-16)
   (* 0.5 (/ (pow x -0.5) (+ x 0.5)))
   (- (pow x -0.5) (pow (+ x 1.0) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 4e-16) {
		tmp = 0.5 * (pow(x, -0.5) / (x + 0.5));
	} else {
		tmp = pow(x, -0.5) - pow((x + 1.0), -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((x + 1.0d0)))) <= 4d-16) then
        tmp = 0.5d0 * ((x ** (-0.5d0)) / (x + 0.5d0))
    else
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-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((x + 1.0)))) <= 4e-16) {
		tmp = 0.5 * (Math.pow(x, -0.5) / (x + 0.5));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((x + 1.0)))) <= 4e-16:
		tmp = 0.5 * (math.pow(x, -0.5) / (x + 0.5))
	else:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(x + 1.0)))) <= 4e-16)
		tmp = Float64(0.5 * Float64((x ^ -0.5) / Float64(x + 0.5)));
	else
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 4e-16)
		tmp = 0.5 * ((x ^ -0.5) / (x + 0.5));
	else
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -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[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-16], N[(0.5 * N[(N[Power[x, -0.5], $MachinePrecision] / N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $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))))) < 3.9999999999999999e-16

   1. Initial program 37.7%

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

      1. frac-sub 37.8%

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

      2. *-un-lft-identity 37.8%

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

      3. +-commutative 37.8%

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

      4. *-rgt-identity 37.8%

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

      5. sqrt-unprod 37.8%

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

      6. +-commutative 37.8%

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

   4. Applied egg-rr 37.8%

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

   5. Taylor expanded in x around inf 84.2%

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

      1. *-commutative 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in x around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   10. Simplified 99.4%

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

       1. times-frac 99.3%

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

       2. inv-pow 99.3%

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

       3. sqrt-pow1 99.4%

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

       4. metadata-eval 99.4%

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

       5. frac-times 99.4%

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

       6. *-commutative 99.4%

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

       7. *-un-lft-identity 99.4%

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

       8. times-frac 99.4%

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

       9. metadata-eval 99.4%

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

   12. Applied egg-rr 99.4%

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

       1. distribute-rgt-in 99.4%

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

       2. *-lft-identity 99.4%

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

       3. associate-*l/ 99.4%

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

       4. associate-/l* 99.4%

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

       5. *-inverses 99.4%

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

       6. metadata-eval 99.4%

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

   14. Simplified 99.4%

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

   if 3.9999999999999999e-16 < (-.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 80.2%

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

      1. sub-neg 80.2%

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

      2. inv-pow 80.2%

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

      3. sqrt-pow2 80.6%

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

      4. metadata-eval 80.6%

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

      5. distribute-neg-frac 80.6%

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

      6. metadata-eval 80.6%

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

      7. +-commutative 80.6%

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

   4. Applied egg-rr 80.6%

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

      1. *-rgt-identity 80.6%

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

      2. cancel-sign-sub 80.6%

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

      3. distribute-lft-neg-in 80.6%

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

      4. *-rgt-identity 80.6%

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

      5. distribute-neg-frac 80.6%

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

      6. metadata-eval 80.6%

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

      7. unpow1/2 80.6%

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

      8. exp-to-pow 79.2%

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

      9. log1p-undefine 79.2%

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

      10. *-commutative 79.2%

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

      11. exp-neg 80.2%

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

      12. *-commutative 80.2%

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

      13. distribute-rgt-neg-in 80.2%

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

      14. log1p-undefine 80.2%

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

      15. metadata-eval 80.2%

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

      16. exp-to-pow 83.4%

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

   6. Simplified 83.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 4 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \frac{{x}^{-0.5}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x \cdot \left(x + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x} - -0.5 \cdot {x}^{-0.5}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+22)
   (/ (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))) (sqrt (* x (+ x 1.0))))
   (/ (- (/ (* (sqrt (/ 1.0 x)) 0.625) x) (* -0.5 (pow x -0.5))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 5e+22) {
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) / sqrt((x * (x + 1.0)));
	} else {
		tmp = (((sqrt((1.0 / x)) * 0.625) / x) - (-0.5 * pow(x, -0.5))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5d+22) then
        tmp = (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) / sqrt((x * (x + 1.0d0)))
    else
        tmp = (((sqrt((1.0d0 / x)) * 0.625d0) / x) - ((-0.5d0) * (x ** (-0.5d0)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e+22) {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) / Math.sqrt((x * (x + 1.0)));
	} else {
		tmp = (((Math.sqrt((1.0 / x)) * 0.625) / x) - (-0.5 * Math.pow(x, -0.5))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e+22:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) / math.sqrt((x * (x + 1.0)))
	else:
		tmp = (((math.sqrt((1.0 / x)) * 0.625) / x) - (-0.5 * math.pow(x, -0.5))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+22)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) / sqrt(Float64(x * Float64(x + 1.0))));
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / x)) * 0.625) / x) - Float64(-0.5 * (x ^ -0.5))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e+22)
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) / sqrt((x * (x + 1.0)));
	else
		tmp = (((sqrt((1.0 / x)) * 0.625) / x) - (-0.5 * (x ^ -0.5))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e+22], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.625), $MachinePrecision] / x), $MachinePrecision] - N[(-0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999996e22

   1. Initial program 49.6%

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

      1. frac-sub 51.7%

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

      2. *-un-lft-identity 51.7%

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

      3. +-commutative 51.7%

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

      4. *-rgt-identity 51.7%

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

      5. sqrt-unprod 51.7%

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

      6. +-commutative 51.7%

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

   4. Applied egg-rr 51.7%

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

      1. flip-- 68.0%

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

      2. add-sqr-sqrt 58.8%

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

      3. add-sqr-sqrt 78.2%

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

   6. Applied egg-rr 78.2%

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

      1. associate--l+ 99.2%

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

      2. +-inverses 99.2%

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

      3. metadata-eval 99.2%

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

      4. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   if 4.9999999999999996e22 < x

   1. Initial program 38.6%

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

   3. Taylor expanded in x around inf 83.8%

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

   4. Taylor expanded in x around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. distribute-neg-frac2 0.0%

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

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x}} \]
   7. Step-by-step derivation

      1. pow1/2 99.7%

         \[\leadsto \frac{\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]

      2. inv-pow 99.7%

         \[\leadsto \frac{{\color{blue}{\left({x}^{-1}\right)}}^{0.5} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]

      3. pow-pow 99.8%

         \[\leadsto \frac{\color{blue}{{x}^{\left(-1 \cdot 0.5\right)}} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{{x}^{\color{blue}{-0.5}} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]

      5. *-un-lft-identity 99.8%

         \[\leadsto \frac{\color{blue}{\left(1 \cdot {x}^{-0.5}\right)} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{\color{blue}{\left(1 \cdot {x}^{-0.5}\right)} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]
   9. Step-by-step derivation

      1. *-lft-identity 99.8%

         \[\leadsto \frac{\color{blue}{{x}^{-0.5}} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]

   10. Simplified 99.8%

       \[\leadsto \frac{\color{blue}{{x}^{-0.5}} \cdot -0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x}}{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x \cdot \left(x + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{1}{x}} \cdot 0.625}{x} - -0.5 \cdot {x}^{-0.5}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 75000000:\\ \;\;\;\;\frac{\sqrt{x + 1} - \sqrt{x}}{\sqrt{x \cdot \left(x + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{x}^{-0.5}}{x + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 75000000.0)
   (/ (- (sqrt (+ x 1.0)) (sqrt x)) (sqrt (* x (+ x 1.0))))
   (* 0.5 (/ (pow x -0.5) (+ x 0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 75000000.0) {
		tmp = (sqrt((x + 1.0)) - sqrt(x)) / sqrt((x * (x + 1.0)));
	} else {
		tmp = 0.5 * (pow(x, -0.5) / (x + 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 75000000.0d0) then
        tmp = (sqrt((x + 1.0d0)) - sqrt(x)) / sqrt((x * (x + 1.0d0)))
    else
        tmp = 0.5d0 * ((x ** (-0.5d0)) / (x + 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 75000000.0) {
		tmp = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) / Math.sqrt((x * (x + 1.0)));
	} else {
		tmp = 0.5 * (Math.pow(x, -0.5) / (x + 0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 75000000.0:
		tmp = (math.sqrt((x + 1.0)) - math.sqrt(x)) / math.sqrt((x * (x + 1.0)))
	else:
		tmp = 0.5 * (math.pow(x, -0.5) / (x + 0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 75000000.0)
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) / sqrt(Float64(x * Float64(x + 1.0))));
	else
		tmp = Float64(0.5 * Float64((x ^ -0.5) / Float64(x + 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 75000000.0)
		tmp = (sqrt((x + 1.0)) - sqrt(x)) / sqrt((x * (x + 1.0)));
	else
		tmp = 0.5 * ((x ^ -0.5) / (x + 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 75000000.0], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Power[x, -0.5], $MachinePrecision] / N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5e7

   1. Initial program 80.2%

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

      1. frac-sub 83.5%

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

      2. *-un-lft-identity 83.5%

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

      3. +-commutative 83.5%

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

      4. *-rgt-identity 83.5%

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

      5. sqrt-unprod 83.5%

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

      6. +-commutative 83.5%

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

   4. Applied egg-rr 83.5%

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

   if 7.5e7 < x

   1. Initial program 37.7%

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

      1. frac-sub 37.8%

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

      2. *-un-lft-identity 37.8%

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

      3. +-commutative 37.8%

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

      4. *-rgt-identity 37.8%

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

      5. sqrt-unprod 37.8%

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

      6. +-commutative 37.8%

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

   4. Applied egg-rr 37.8%

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

   5. Taylor expanded in x around inf 84.2%

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

      1. *-commutative 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in x around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   10. Simplified 99.4%

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

       1. times-frac 99.3%

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

       2. inv-pow 99.3%

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

       3. sqrt-pow1 99.4%

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

       4. metadata-eval 99.4%

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

       5. frac-times 99.4%

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

       6. *-commutative 99.4%

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

       7. *-un-lft-identity 99.4%

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

       8. times-frac 99.4%

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

       9. metadata-eval 99.4%

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

   12. Applied egg-rr 99.4%

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

       1. distribute-rgt-in 99.4%

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

       2. *-lft-identity 99.4%

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

       3. associate-*l/ 99.4%

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

       4. associate-/l* 99.4%

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

       5. *-inverses 99.4%

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

       6. metadata-eval 99.4%

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

   14. Simplified 99.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 75000000:\\ \;\;\;\;\frac{\sqrt{x + 1} - \sqrt{x}}{\sqrt{x \cdot \left(x + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{x}^{-0.5}}{x + 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

   1. frac-sub 39.8%

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

   2. *-un-lft-identity 39.8%

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

   3. +-commutative 39.8%

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

   4. *-rgt-identity 39.8%

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

   5. sqrt-unprod 39.8%

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

   6. +-commutative 39.8%

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

4. Applied egg-rr 39.8%

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

5. Taylor expanded in x around inf 82.3%

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

   1. *-commutative 82.3%

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

7. Simplified 82.3%

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

8. Taylor expanded in x around inf 96.8%

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

   1. associate-*r/ 96.8%

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

   2. metadata-eval 96.8%

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

10. Simplified 96.8%

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

    1. times-frac 96.8%

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

    2. inv-pow 96.8%

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

    3. sqrt-pow1 96.9%

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

    4. metadata-eval 96.9%

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

    5. frac-times 96.9%

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

    6. *-commutative 96.9%

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

    7. *-un-lft-identity 96.9%

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

    8. times-frac 96.9%

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

    9. metadata-eval 96.9%

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

12. Applied egg-rr 96.9%

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

    1. distribute-rgt-in 96.9%

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

    2. *-lft-identity 96.9%

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

    3. associate-*l/ 96.9%

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

    4. associate-/l* 96.9%

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

    5. *-inverses 96.9%

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

    6. metadata-eval 96.9%

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

14. Simplified 96.9%

    \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{-0.5}}{x + 0.5}} \]
15. Add Preprocessing

Alternative 7: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

   1. frac-sub 39.8%

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

   2. *-un-lft-identity 39.8%

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

   3. +-commutative 39.8%

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

   4. *-rgt-identity 39.8%

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

   5. sqrt-unprod 39.8%

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

   6. +-commutative 39.8%

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

4. Applied egg-rr 39.8%

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

5. Taylor expanded in x around inf 82.3%

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

   1. *-commutative 82.3%

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

7. Simplified 82.3%

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

8. Taylor expanded in x around inf 96.7%

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

Alternative 8: 96.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

   1. frac-sub 39.8%

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

   2. *-un-lft-identity 39.8%

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

   3. +-commutative 39.8%

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

   4. *-rgt-identity 39.8%

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

   5. sqrt-unprod 39.8%

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

   6. +-commutative 39.8%

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

4. Applied egg-rr 39.8%

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

5. Taylor expanded in x around inf 82.3%

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

   1. *-commutative 82.3%

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

7. Simplified 82.3%

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

8. Taylor expanded in x around inf 96.8%

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

   1. associate-*r/ 96.8%

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

   2. metadata-eval 96.8%

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

10. Simplified 96.8%

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

    1. associate-/l* 96.6%

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

    2. inv-pow 96.6%

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

    3. sqrt-pow1 96.7%

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

    4. metadata-eval 96.7%

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

    5. *-un-lft-identity 96.7%

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

    6. metadata-eval 96.7%

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

    7. sqrt-pow1 96.6%

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

    8. inv-pow 96.6%

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

    9. sqrt-div 96.6%

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

    10. metadata-eval 96.6%

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

    11. frac-times 95.4%

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

    12. metadata-eval 95.4%

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

12. Applied egg-rr 95.4%

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

    1. *-lft-identity 95.4%

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

    2. distribute-rgt-in 95.4%

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

    3. *-lft-identity 95.4%

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

    4. associate-*l/ 95.4%

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

    5. associate-/l* 95.4%

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

    6. *-inverses 95.4%

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

    7. metadata-eval 95.4%

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

14. Simplified 95.4%

    \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x} \cdot \left(x + 0.5\right)}} \]
15. Add Preprocessing

Alternative 9: 43.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -1.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -1.5], $MachinePrecision]

Derivation

1. Initial program 39.6%

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

3. Taylor expanded in x around 0 5.7%

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

   1. pow1/2 5.7%

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

   2. pow-to-exp 5.7%

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

   3. log-rec 5.7%

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

5. Applied egg-rr 5.7%

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

7. Applied egg-rr 44.4%

   \[\leadsto \color{blue}{{x}^{-1.5}} \]
8. Add Preprocessing

Alternative 10: 37.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

   1. frac-sub 39.8%

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

   2. *-un-lft-identity 39.8%

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

   3. +-commutative 39.8%

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

   4. *-rgt-identity 39.8%

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

   5. sqrt-unprod 39.8%

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

   6. +-commutative 39.8%

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

4. Applied egg-rr 39.8%

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

5. Taylor expanded in x around inf 82.3%

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

   1. *-commutative 82.3%

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

7. Simplified 82.3%

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

9. Applied egg-rr 37.6%

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

    1. unpow-1 37.6%

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

    2. associate-/r* 37.6%

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

11. Simplified 37.6%

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

12. Final simplification 37.6%

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

Alternative 11: 6.4% accurate, 26.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 5.5e+153) 0.25 (/ -0.125 x)))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e+153) {
		tmp = 0.25;
	} else {
		tmp = -0.125 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.5d+153) then
        tmp = 0.25d0
    else
        tmp = (-0.125d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e+153) {
		tmp = 0.25;
	} else {
		tmp = -0.125 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e+153:
		tmp = 0.25
	else:
		tmp = -0.125 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e+153)
		tmp = 0.25;
	else
		tmp = Float64(-0.125 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e+153)
		tmp = 0.25;
	else
		tmp = -0.125 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e+153], 0.25, N[(-0.125 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5000000000000003e153

   1. Initial program 12.4%

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

      1. frac-sub 12.7%

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

      2. *-un-lft-identity 12.7%

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

      3. +-commutative 12.7%

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

      4. *-rgt-identity 12.7%

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

      5. sqrt-unprod 12.7%

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

      6. +-commutative 12.7%

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

   4. Applied egg-rr 12.7%

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

   5. Taylor expanded in x around inf 94.0%

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

      1. *-commutative 94.0%

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

   7. Simplified 94.0%

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

   9. Applied egg-rr 6.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{x + x}} \]
   10. Step-by-step derivation

       1. associate-*r/ 6.2%

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

       2. count-2 6.2%

          \[\leadsto \frac{0.5 \cdot x}{\color{blue}{2 \cdot x}} \]

       3. times-frac 6.2%

          \[\leadsto \color{blue}{\frac{0.5}{2} \cdot \frac{x}{x}} \]

       4. metadata-eval 6.2%

          \[\leadsto \color{blue}{0.25} \cdot \frac{x}{x} \]

       5. *-inverses 6.2%

          \[\leadsto 0.25 \cdot \color{blue}{1} \]

       6. metadata-eval 6.2%

          \[\leadsto \color{blue}{0.25} \]

   11. Simplified 6.2%

       \[\leadsto \color{blue}{0.25} \]

   if 5.5000000000000003e153 < x

   1. Initial program 69.5%

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

      1. frac-sub 69.5%

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

      2. *-un-lft-identity 69.5%

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

      3. +-commutative 69.5%

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

      4. *-rgt-identity 69.5%

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

      5. sqrt-unprod 69.5%

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

      6. +-commutative 69.5%

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

   4. Applied egg-rr 69.5%

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

   5. Taylor expanded in x around inf 69.5%

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

      1. *-commutative 69.5%

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

   7. Simplified 69.5%

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

   9. Applied egg-rr 7.4%

      \[\leadsto \color{blue}{\frac{-0.5}{x + x} \cdot \frac{x}{x + x}} \]
   10. Step-by-step derivation

       1. associate-*l/ 7.4%

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

       2. count-2 7.4%

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

       3. associate-/r* 7.4%

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

       4. associate-*r/ 7.4%

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

       5. count-2 7.4%

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

       6. times-frac 7.4%

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

       7. metadata-eval 7.4%

          \[\leadsto \frac{\frac{\color{blue}{-0.25} \cdot \frac{x}{x}}{2}}{x} \]

       8. *-inverses 7.4%

          \[\leadsto \frac{\frac{-0.25 \cdot \color{blue}{1}}{2}}{x} \]

       9. metadata-eval 7.4%

          \[\leadsto \frac{\frac{\color{blue}{-0.25}}{2}}{x} \]

       10. metadata-eval 7.4%

           \[\leadsto \frac{\color{blue}{-0.125}}{x} \]

   11. Simplified 7.4%

       \[\leadsto \color{blue}{\frac{-0.125}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 7.9% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.25 x))

C

double code(double x) {
	return 0.25 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.25d0 / x
end function

Java

public static double code(double x) {
	return 0.25 / x;
}

Python

def code(x):
	return 0.25 / x

Julia

function code(x)
	return Float64(0.25 / x)
end

MATLAB

function tmp = code(x)
	tmp = 0.25 / x;
end

Wolfram

code[x_] := N[(0.25 / x), $MachinePrecision]

Derivation

1. Initial program 39.6%

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

   1. frac-sub 39.8%

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

   2. *-un-lft-identity 39.8%

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

   3. +-commutative 39.8%

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

   4. *-rgt-identity 39.8%

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

   5. sqrt-unprod 39.8%

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

   6. +-commutative 39.8%

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

4. Applied egg-rr 39.8%

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

5. Taylor expanded in x around inf 82.3%

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

   1. *-commutative 82.3%

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

7. Simplified 82.3%

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

9. Applied egg-rr 8.2%

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

    1. distribute-neg-frac 8.2%

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

    2. metadata-eval 8.2%

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

    3. count-2 8.2%

       \[\leadsto \frac{0.5}{\color{blue}{2 \cdot x}} \]

    4. associate-/r* 8.2%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{2}}{x}} \]

    5. metadata-eval 8.2%

       \[\leadsto \frac{\color{blue}{0.25}}{x} \]

11. Simplified 8.2%

    \[\leadsto \color{blue}{\frac{0.25}{x}} \]
12. Add Preprocessing

Alternative 13: 4.7% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.25)

C

double code(double x) {
	return 0.25;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.25d0
end function

Java

public static double code(double x) {
	return 0.25;
}

Python

def code(x):
	return 0.25

Julia

function code(x)
	return 0.25
end

MATLAB

function tmp = code(x)
	tmp = 0.25;
end

Wolfram

code[x_] := 0.25

Derivation

1. Initial program 39.6%

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

   1. frac-sub 39.8%

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

   2. *-un-lft-identity 39.8%

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

   3. +-commutative 39.8%

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

   4. *-rgt-identity 39.8%

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

   5. sqrt-unprod 39.8%

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

   6. +-commutative 39.8%

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

4. Applied egg-rr 39.8%

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

5. Taylor expanded in x around inf 82.3%

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

   1. *-commutative 82.3%

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

7. Simplified 82.3%

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

9. Applied egg-rr 4.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{x}{x + x}} \]
10. Step-by-step derivation

    1. associate-*r/ 4.8%

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

    2. count-2 4.8%

       \[\leadsto \frac{0.5 \cdot x}{\color{blue}{2 \cdot x}} \]

    3. times-frac 4.8%

       \[\leadsto \color{blue}{\frac{0.5}{2} \cdot \frac{x}{x}} \]

    4. metadata-eval 4.8%

       \[\leadsto \color{blue}{0.25} \cdot \frac{x}{x} \]

    5. *-inverses 4.8%

       \[\leadsto 0.25 \cdot \color{blue}{1} \]

    6. metadata-eval 4.8%

       \[\leadsto \color{blue}{0.25} \]

11. Simplified 4.8%

    \[\leadsto \color{blue}{0.25} \]
12. 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 2024090 
(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)))))