2isqrt (example 3.6)

Percentage Accurate: 37.8% → 99.2%

Time: 14.1s

Alternatives: 10

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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{hypot}\left({x}^{-0.25}, {\left(x + 1\right)}^{-0.25}\right) \cdot \mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (* (hypot (pow x -0.25) (pow (+ x 1.0) -0.25)) (hypot x (sqrt x))) -2.0))

C

double code(double x) {
	return pow((hypot(pow(x, -0.25), pow((x + 1.0), -0.25)) * hypot(x, sqrt(x))), -2.0);
}

Java

public static double code(double x) {
	return Math.pow((Math.hypot(Math.pow(x, -0.25), Math.pow((x + 1.0), -0.25)) * Math.hypot(x, Math.sqrt(x))), -2.0);
}

Python

def code(x):
	return math.pow((math.hypot(math.pow(x, -0.25), math.pow((x + 1.0), -0.25)) * math.hypot(x, math.sqrt(x))), -2.0)

Julia

function code(x)
	return Float64(hypot((x ^ -0.25), (Float64(x + 1.0) ^ -0.25)) * hypot(x, sqrt(x))) ^ -2.0
end

MATLAB

function tmp = code(x)
	tmp = (hypot((x ^ -0.25), ((x + 1.0) ^ -0.25)) * hypot(x, sqrt(x))) ^ -2.0;
end

Wolfram

code[x_] := N[Power[N[(N[Sqrt[N[Power[x, -0.25], $MachinePrecision] ^ 2 + N[Power[N[(x + 1.0), $MachinePrecision], -0.25], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[x ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 38.7%

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

   1. sub-neg 38.7%

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

   2. flip-+ 38.6%

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

   3. frac-times 19.4%

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

   4. metadata-eval 19.4%

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

   5. add-sqr-sqrt 17.0%

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

   6. distribute-neg-frac 17.0%

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

   7. metadata-eval 17.0%

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

   8. +-commutative 17.0%

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

   9. distribute-neg-frac 17.0%

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

   10. metadata-eval 17.0%

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

   11. +-commutative 17.0%

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

   12. pow1/2 17.0%

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

   13. pow-flip 17.0%

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

   14. metadata-eval 17.0%

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

3. Applied egg-rr 17.0%

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

   1. associate-*r/ 21.5%

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

   2. associate-*l/ 21.5%

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

   3. metadata-eval 21.5%

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

   4. associate-/l/ 26.9%

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

   5. rem-square-sqrt 38.8%

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

   6. sub-neg 38.8%

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

   7. distribute-neg-frac 38.8%

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

   8. metadata-eval 38.8%

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

   9. sub-neg 38.8%

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

   10. distribute-neg-frac 38.8%

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

   11. metadata-eval 38.8%

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

5. Simplified 38.8%

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

   1. frac-add 41.0%

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

   2. *-un-lft-identity 41.0%

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

   3. distribute-rgt-in 41.0%

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

   4. *-un-lft-identity 41.0%

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

   5. pow2 41.0%

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

7. Applied egg-rr 41.0%

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

   1. *-commutative 41.0%

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

   2. neg-mul-1 41.0%

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

   3. sub-neg 41.0%

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

   4. associate--l+ 81.7%

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

9. Simplified 81.7%

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

    1. add-sqr-sqrt 81.5%

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

11. Applied egg-rr 99.2%

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

    1. unpow2 99.2%

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

    2. associate-/l/ 99.2%

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

13. Simplified 99.2%

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

15. Applied egg-rr 37.5%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left({x}^{-0.25}, {\left(1 + x\right)}^{-0.25}\right) \cdot \mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-2}\right)} - 1} \]
16. Step-by-step derivation

    1. expm1-def 99.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left({x}^{-0.25}, {\left(1 + x\right)}^{-0.25}\right) \cdot \mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-2}\right)\right)} \]

    2. expm1-log1p 99.2%

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

17. Simplified 99.2%

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

18. Final simplification 99.2%

    \[\leadsto {\left(\mathsf{hypot}\left({x}^{-0.25}, {\left(x + 1\right)}^{-0.25}\right) \cdot \mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-2} \]

Alternative 2: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1}{\left(-{\left(x + 1\right)}^{-0.5}\right) - {x}^{-0.5}}}{x \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ (/ 1.0 (- (- (pow (+ x 1.0) -0.5)) (pow x -0.5))) (* x (- -1.0 x)))
   (pow x -1.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (1.0 / (-pow((x + 1.0), -0.5) - pow(x, -0.5))) / (x * (-1.0 - x));
	} else {
		tmp = pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.35d+154) then
        tmp = (1.0d0 / (-((x + 1.0d0) ** (-0.5d0)) - (x ** (-0.5d0)))) / (x * ((-1.0d0) - x))
    else
        tmp = x ** (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (1.0 / (-Math.pow((x + 1.0), -0.5) - Math.pow(x, -0.5))) / (x * (-1.0 - x));
	} else {
		tmp = Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.35e+154:
		tmp = (1.0 / (-math.pow((x + 1.0), -0.5) - math.pow(x, -0.5))) / (x * (-1.0 - x))
	else:
		tmp = math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(1.0 / Float64(Float64(-(Float64(x + 1.0) ^ -0.5)) - (x ^ -0.5))) / Float64(x * Float64(-1.0 - x)));
	else
		tmp = x ^ -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.35e+154)
		tmp = (1.0 / (-((x + 1.0) ^ -0.5) - (x ^ -0.5))) / (x * (-1.0 - x));
	else
		tmp = x ^ -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(1.0 / N[((-N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]) - N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x, -1.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.8%

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

      1. flip-- 9.6%

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

      2. clear-num 9.6%

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

      3. pow1/2 9.6%

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

      4. pow-flip 9.6%

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

      5. metadata-eval 9.6%

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

      6. inv-pow 9.6%

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

      7. sqrt-pow2 9.6%

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

      8. +-commutative 9.6%

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

      9. metadata-eval 9.6%

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

      10. frac-times 10.1%

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

      11. metadata-eval 10.1%

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

      12. add-sqr-sqrt 10.1%

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

      13. frac-times 9.8%

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

      14. metadata-eval 9.8%

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

      15. add-sqr-sqrt 10.1%

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

   3. Applied egg-rr 10.1%

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

      1. frac-2neg 10.1%

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

      2. metadata-eval 10.1%

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

      3. frac-sub 14.6%

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

   5. Applied egg-rr 14.6%

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

      1. *-commutative 14.6%

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

      2. neg-mul-1 14.6%

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

      3. *-rgt-identity 14.6%

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

      4. *-lft-identity 14.6%

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

      5. neg-sub0 14.6%

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

      6. associate--l- 14.6%

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

      7. associate-+l+ 99.2%

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

      8. *-rgt-identity 99.2%

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

      9. sub-neg 99.2%

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

      10. +-inverses 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. distribute-neg-in 99.2%

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

      14. metadata-eval 99.2%

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

      15. unsub-neg 99.2%

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

   7. Simplified 99.2%

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

      1. inv-pow 99.2%

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

      2. associate-/r/ 99.3%

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

      3. unpow-prod-down 99.3%

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

      4. div-inv 99.3%

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

      5. metadata-eval 99.3%

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

      6. inv-pow 99.3%

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

   9. Applied egg-rr 99.3%

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

       1. associate-*r/ 99.3%

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

       2. *-rgt-identity 99.3%

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

       3. unpow-1 99.3%

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

       4. *-commutative 99.3%

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

       5. neg-mul-1 99.3%

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

       6. neg-sub0 99.3%

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

       7. +-commutative 99.3%

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

       8. associate--r+ 99.3%

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

       9. neg-sub0 99.3%

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

   11. Simplified 99.3%

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

   if 1.35000000000000003e154 < x

   1. Initial program 65.4%

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

      1. sub-neg 65.4%

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

      2. flip-+ 65.4%

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

      3. frac-times 28.0%

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

      4. metadata-eval 28.0%

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

      5. add-sqr-sqrt 23.5%

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

      6. distribute-neg-frac 23.5%

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

      7. metadata-eval 23.5%

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

      8. +-commutative 23.5%

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

      9. distribute-neg-frac 23.5%

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

      10. metadata-eval 23.5%

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

      11. +-commutative 23.5%

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

      12. pow1/2 23.5%

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

      13. pow-flip 23.5%

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

      14. metadata-eval 23.5%

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

   3. Applied egg-rr 23.5%

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

      1. associate-*r/ 32.2%

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

      2. associate-*l/ 32.2%

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

      3. metadata-eval 32.2%

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

      4. associate-/l/ 42.7%

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

      5. rem-square-sqrt 65.4%

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

      6. sub-neg 65.4%

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

      7. distribute-neg-frac 65.4%

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

      8. metadata-eval 65.4%

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

      9. sub-neg 65.4%

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

      10. distribute-neg-frac 65.4%

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

      11. metadata-eval 65.4%

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

   5. Simplified 65.4%

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

      1. frac-add 65.4%

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

      2. *-un-lft-identity 65.4%

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

      3. distribute-rgt-in 65.4%

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

      4. *-un-lft-identity 65.4%

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

      5. pow2 65.4%

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

   7. Applied egg-rr 65.4%

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

      1. *-commutative 65.4%

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

      2. neg-mul-1 65.4%

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

      3. sub-neg 65.4%

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

      4. associate--l+ 65.4%

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

   9. Simplified 65.4%

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

   10. Taylor expanded in x around inf 65.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{3}}}} \]
   11. Step-by-step derivation

       1. metadata-eval 65.4%

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

       2. cube-div 65.4%

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

   12. Simplified 65.4%

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

       1. expm1-log1p-u 65.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(\frac{1}{x}\right)}^{3}}\right)\right)} \]

       2. expm1-udef 65.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{{\left(\frac{1}{x}\right)}^{3}}\right)} - 1} \]

       3. sqrt-pow1 65.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{x}\right)}^{\left(\frac{3}{2}\right)}}\right)} - 1 \]

       4. inv-pow 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left({x}^{-1}\right)}}^{\left(\frac{3}{2}\right)}\right)} - 1 \]

       5. pow-pow 65.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(-1 \cdot \frac{3}{2}\right)}}\right)} - 1 \]

       6. metadata-eval 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({x}^{\left(-1 \cdot \color{blue}{1.5}\right)}\right)} - 1 \]

       7. metadata-eval 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({x}^{\color{blue}{-1.5}}\right)} - 1 \]

   14. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({x}^{-1.5}\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 69.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-1.5}\right)\right)} \]

       2. expm1-log1p 69.0%

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

   16. Simplified 69.0%

       \[\leadsto \color{blue}{{x}^{-1.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1}{\left(-{\left(x + 1\right)}^{-0.5}\right) - {x}^{-0.5}}}{x \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \]

Alternative 3: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-1}{x \cdot \left(-1 - x\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ (/ -1.0 (* x (- -1.0 x))) (+ (pow x -0.5) (pow (+ x 1.0) -0.5)))
   (pow x -1.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (-1.0 / (x * (-1.0 - x))) / (pow(x, -0.5) + pow((x + 1.0), -0.5));
	} else {
		tmp = pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.35d+154) then
        tmp = ((-1.0d0) / (x * ((-1.0d0) - x))) / ((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0)))
    else
        tmp = x ** (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (-1.0 / (x * (-1.0 - x))) / (Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5));
	} else {
		tmp = Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.35e+154:
		tmp = (-1.0 / (x * (-1.0 - x))) / (math.pow(x, -0.5) + math.pow((x + 1.0), -0.5))
	else:
		tmp = math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(-1.0 / Float64(x * Float64(-1.0 - x))) / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)));
	else
		tmp = x ^ -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.35e+154)
		tmp = (-1.0 / (x * (-1.0 - x))) / ((x ^ -0.5) + ((x + 1.0) ^ -0.5));
	else
		tmp = x ^ -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(-1.0 / N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x, -1.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.8%

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

      1. sub-neg 9.8%

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

      2. flip-+ 9.6%

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

      3. frac-times 10.1%

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

      4. metadata-eval 10.1%

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

      5. add-sqr-sqrt 10.1%

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

      6. distribute-neg-frac 10.1%

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

      7. metadata-eval 10.1%

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

      8. +-commutative 10.1%

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

      9. distribute-neg-frac 10.1%

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

      10. metadata-eval 10.1%

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

      11. +-commutative 10.1%

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

      12. pow1/2 10.1%

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

      13. pow-flip 10.1%

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

      14. metadata-eval 10.1%

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

   3. Applied egg-rr 10.1%

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

      1. associate-*r/ 10.0%

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

      2. associate-*l/ 10.0%

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

      3. metadata-eval 10.0%

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

      4. associate-/l/ 9.8%

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

      5. rem-square-sqrt 10.1%

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

      6. sub-neg 10.1%

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

      7. distribute-neg-frac 10.1%

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

      8. metadata-eval 10.1%

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

      9. sub-neg 10.1%

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

      10. distribute-neg-frac 10.1%

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

      11. metadata-eval 10.1%

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

   5. Simplified 10.1%

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

      1. frac-add 14.6%

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

      2. *-un-lft-identity 14.6%

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

      3. distribute-rgt-in 14.7%

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

      4. *-un-lft-identity 14.7%

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

      5. pow2 14.7%

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

   7. Applied egg-rr 14.7%

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

      1. *-commutative 14.7%

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

      2. neg-mul-1 14.7%

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

      3. sub-neg 14.7%

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

      4. associate--l+ 99.3%

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

   9. Simplified 99.3%

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

       1. add-sqr-sqrt 98.9%

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

   11. Applied egg-rr 98.6%

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

       1. unpow2 98.6%

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

       2. associate-/l/ 98.7%

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

   13. Simplified 98.7%

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left({x}^{-0.25}, {\left(1 + x\right)}^{-0.25}\right) \cdot \mathsf{hypot}\left(x, \sqrt{x}\right)}\right)}^{2}} \]
   14. Step-by-step derivation

       1. metadata-eval 98.7%

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

       2. hypot-udef 98.6%

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

       3. pow-sqr 98.6%

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

       4. metadata-eval 98.6%

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

       5. pow-sqr 98.7%

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

       6. metadata-eval 98.7%

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

       7. hypot-udef 98.7%

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

       8. add-sqr-sqrt 98.7%

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

       9. fma-udef 98.7%

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

       10. sqrt-prod 98.8%

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

       11. sqrt-div 99.0%

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

   15. Applied egg-rr 99.3%

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

       1. associate-/r* 99.3%

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

       2. fma-udef 99.3%

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

       3. distribute-lft1-in 99.3%

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

       4. +-commutative 99.3%

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

       5. metadata-eval 99.3%

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

       6. associate--r- 99.3%

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

       7. neg-sub0 99.3%

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

       8. *-commutative 99.3%

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

       9. distribute-rgt-neg-out 99.3%

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

       10. remove-double-neg 99.3%

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

       11. associate-*r/ 99.3%

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

       12. associate-*l/ 99.3%

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

       13. associate-*r/ 99.3%

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

       14. *-rgt-identity 99.3%

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

   17. Simplified 99.3%

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

   if 1.35000000000000003e154 < x

   1. Initial program 65.4%

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

      1. sub-neg 65.4%

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

      2. flip-+ 65.4%

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

      3. frac-times 28.0%

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

      4. metadata-eval 28.0%

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

      5. add-sqr-sqrt 23.5%

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

      6. distribute-neg-frac 23.5%

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

      7. metadata-eval 23.5%

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

      8. +-commutative 23.5%

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

      9. distribute-neg-frac 23.5%

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

      10. metadata-eval 23.5%

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

      11. +-commutative 23.5%

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

      12. pow1/2 23.5%

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

      13. pow-flip 23.5%

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

      14. metadata-eval 23.5%

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

   3. Applied egg-rr 23.5%

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

      1. associate-*r/ 32.2%

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

      2. associate-*l/ 32.2%

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

      3. metadata-eval 32.2%

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

      4. associate-/l/ 42.7%

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

      5. rem-square-sqrt 65.4%

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

      6. sub-neg 65.4%

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

      7. distribute-neg-frac 65.4%

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

      8. metadata-eval 65.4%

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

      9. sub-neg 65.4%

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

      10. distribute-neg-frac 65.4%

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

      11. metadata-eval 65.4%

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

   5. Simplified 65.4%

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

      1. frac-add 65.4%

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

      2. *-un-lft-identity 65.4%

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

      3. distribute-rgt-in 65.4%

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

      4. *-un-lft-identity 65.4%

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

      5. pow2 65.4%

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

   7. Applied egg-rr 65.4%

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

      1. *-commutative 65.4%

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

      2. neg-mul-1 65.4%

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

      3. sub-neg 65.4%

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

      4. associate--l+ 65.4%

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

   9. Simplified 65.4%

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

   10. Taylor expanded in x around inf 65.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{3}}}} \]
   11. Step-by-step derivation

       1. metadata-eval 65.4%

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

       2. cube-div 65.4%

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

   12. Simplified 65.4%

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

       1. expm1-log1p-u 65.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(\frac{1}{x}\right)}^{3}}\right)\right)} \]

       2. expm1-udef 65.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{{\left(\frac{1}{x}\right)}^{3}}\right)} - 1} \]

       3. sqrt-pow1 65.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{x}\right)}^{\left(\frac{3}{2}\right)}}\right)} - 1 \]

       4. inv-pow 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left({x}^{-1}\right)}}^{\left(\frac{3}{2}\right)}\right)} - 1 \]

       5. pow-pow 65.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(-1 \cdot \frac{3}{2}\right)}}\right)} - 1 \]

       6. metadata-eval 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({x}^{\left(-1 \cdot \color{blue}{1.5}\right)}\right)} - 1 \]

       7. metadata-eval 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({x}^{\color{blue}{-1.5}}\right)} - 1 \]

   14. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({x}^{-1.5}\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 69.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-1.5}\right)\right)} \]

       2. expm1-log1p 69.0%

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

   16. Simplified 69.0%

       \[\leadsto \color{blue}{{x}^{-1.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-1}{x \cdot \left(-1 - x\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \]

Alternative 4: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot \left(x \cdot \left(x - -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 1.0 (* (+ (pow x -0.5) (pow (+ x 1.0) -0.5)) (* x (- x -1.0))))
   (pow x -1.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / ((pow(x, -0.5) + pow((x + 1.0), -0.5)) * (x * (x - -1.0)));
	} else {
		tmp = pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.35d+154) then
        tmp = 1.0d0 / (((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0))) * (x * (x - (-1.0d0))))
    else
        tmp = x ** (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / ((Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5)) * (x * (x - -1.0)));
	} else {
		tmp = Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.35e+154:
		tmp = 1.0 / ((math.pow(x, -0.5) + math.pow((x + 1.0), -0.5)) * (x * (x - -1.0)))
	else:
		tmp = math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(1.0 / Float64(Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)) * Float64(x * Float64(x - -1.0))));
	else
		tmp = x ^ -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.35e+154)
		tmp = 1.0 / (((x ^ -0.5) + ((x + 1.0) ^ -0.5)) * (x * (x - -1.0)));
	else
		tmp = x ^ -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(1.0 / N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x, -1.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.8%

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

      1. flip-- 9.6%

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

      2. clear-num 9.6%

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

      3. pow1/2 9.6%

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

      4. pow-flip 9.6%

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

      5. metadata-eval 9.6%

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

      6. inv-pow 9.6%

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

      7. sqrt-pow2 9.6%

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

      8. +-commutative 9.6%

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

      9. metadata-eval 9.6%

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

      10. frac-times 10.1%

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

      11. metadata-eval 10.1%

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

      12. add-sqr-sqrt 10.1%

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

      13. frac-times 9.8%

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

      14. metadata-eval 9.8%

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

      15. add-sqr-sqrt 10.1%

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

   3. Applied egg-rr 10.1%

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

      1. frac-2neg 10.1%

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

      2. metadata-eval 10.1%

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

      3. frac-sub 14.6%

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

   5. Applied egg-rr 14.6%

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

      1. *-commutative 14.6%

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

      2. neg-mul-1 14.6%

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

      3. *-rgt-identity 14.6%

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

      4. *-lft-identity 14.6%

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

      5. neg-sub0 14.6%

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

      6. associate--l- 14.6%

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

      7. associate-+l+ 99.2%

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

      8. *-rgt-identity 99.2%

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

      9. sub-neg 99.2%

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

      10. +-inverses 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. distribute-neg-in 99.2%

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

      14. metadata-eval 99.2%

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

      15. unsub-neg 99.2%

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

   7. Simplified 99.2%

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

      1. expm1-log1p-u 92.0%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{-1}{x \cdot \left(-1 - x\right)}}\right)\right)}} \]

      2. expm1-udef 92.0%

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

      3. pow1 92.0%

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

      4. pow1 92.0%

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

      5. div-inv 92.0%

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

      6. frac-2neg 92.0%

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

      7. metadata-eval 92.0%

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

      8. remove-double-div 92.0%

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

      9. distribute-rgt-neg-in 92.0%

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

   9. Applied egg-rr 92.0%

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

       1. expm1-def 92.0%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(x \cdot \left(-\left(-1 - x\right)\right)\right)\right)\right)}} \]

       2. expm1-log1p 99.3%

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

   11. Simplified 99.3%

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

   if 1.35000000000000003e154 < x

   1. Initial program 65.4%

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

      1. sub-neg 65.4%

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

      2. flip-+ 65.4%

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

      3. frac-times 28.0%

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

      4. metadata-eval 28.0%

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

      5. add-sqr-sqrt 23.5%

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

      6. distribute-neg-frac 23.5%

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

      7. metadata-eval 23.5%

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

      8. +-commutative 23.5%

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

      9. distribute-neg-frac 23.5%

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

      10. metadata-eval 23.5%

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

      11. +-commutative 23.5%

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

      12. pow1/2 23.5%

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

      13. pow-flip 23.5%

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

      14. metadata-eval 23.5%

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

   3. Applied egg-rr 23.5%

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

      1. associate-*r/ 32.2%

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

      2. associate-*l/ 32.2%

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

      3. metadata-eval 32.2%

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

      4. associate-/l/ 42.7%

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

      5. rem-square-sqrt 65.4%

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

      6. sub-neg 65.4%

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

      7. distribute-neg-frac 65.4%

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

      8. metadata-eval 65.4%

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

      9. sub-neg 65.4%

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

      10. distribute-neg-frac 65.4%

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

      11. metadata-eval 65.4%

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

   5. Simplified 65.4%

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

      1. frac-add 65.4%

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

      2. *-un-lft-identity 65.4%

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

      3. distribute-rgt-in 65.4%

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

      4. *-un-lft-identity 65.4%

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

      5. pow2 65.4%

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

   7. Applied egg-rr 65.4%

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

      1. *-commutative 65.4%

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

      2. neg-mul-1 65.4%

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

      3. sub-neg 65.4%

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

      4. associate--l+ 65.4%

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

   9. Simplified 65.4%

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

   10. Taylor expanded in x around inf 65.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{3}}}} \]
   11. Step-by-step derivation

       1. metadata-eval 65.4%

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

       2. cube-div 65.4%

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

   12. Simplified 65.4%

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

       1. expm1-log1p-u 65.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(\frac{1}{x}\right)}^{3}}\right)\right)} \]

       2. expm1-udef 65.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{{\left(\frac{1}{x}\right)}^{3}}\right)} - 1} \]

       3. sqrt-pow1 65.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{x}\right)}^{\left(\frac{3}{2}\right)}}\right)} - 1 \]

       4. inv-pow 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left({x}^{-1}\right)}}^{\left(\frac{3}{2}\right)}\right)} - 1 \]

       5. pow-pow 65.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(-1 \cdot \frac{3}{2}\right)}}\right)} - 1 \]

       6. metadata-eval 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({x}^{\left(-1 \cdot \color{blue}{1.5}\right)}\right)} - 1 \]

       7. metadata-eval 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({x}^{\color{blue}{-1.5}}\right)} - 1 \]

   14. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({x}^{-1.5}\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 69.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-1.5}\right)\right)} \]

       2. expm1-log1p 69.0%

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

   16. Simplified 69.0%

       \[\leadsto \color{blue}{{x}^{-1.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot \left(x \cdot \left(x - -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \]

Alternative 5: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.7%

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

   1. flip-- 38.6%

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

   2. clear-num 38.6%

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

   3. pow1/2 38.6%

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

   4. pow-flip 38.6%

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

   5. metadata-eval 38.6%

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

   6. inv-pow 38.6%

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

   7. sqrt-pow2 38.6%

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

   8. +-commutative 38.6%

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

   9. metadata-eval 38.6%

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

   10. frac-times 19.4%

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

   11. metadata-eval 19.4%

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

   12. add-sqr-sqrt 17.0%

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

   13. frac-times 26.9%

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

   14. metadata-eval 26.9%

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

   15. add-sqr-sqrt 38.8%

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

3. Applied egg-rr 38.8%

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

   1. frac-2neg 38.8%

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

   2. metadata-eval 38.8%

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

   3. frac-sub 41.0%

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

5. Applied egg-rr 41.0%

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

   1. *-commutative 41.0%

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

   2. neg-mul-1 41.0%

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

   3. *-rgt-identity 41.0%

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

   4. *-lft-identity 41.0%

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

   5. neg-sub0 41.0%

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

   6. associate--l- 41.0%

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

   7. associate-+l+ 81.6%

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

   8. *-rgt-identity 81.6%

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

   9. sub-neg 81.6%

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

   10. +-inverses 81.6%

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

   11. metadata-eval 81.6%

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

   12. metadata-eval 81.6%

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

   13. distribute-neg-in 81.6%

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

   14. metadata-eval 81.6%

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

   15. unsub-neg 81.6%

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

7. Simplified 81.6%

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

   1. associate-/r/ 81.7%

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

   2. *-commutative 81.7%

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

   3. associate-*r* 98.3%

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

   4. div-inv 98.3%

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

   5. metadata-eval 98.3%

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

9. Applied egg-rr 98.3%

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

10. Final simplification 98.3%

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

Alternative 6: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left(\sqrt{\frac{1}{x}} \cdot 2\right) \cdot \mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 1.0 (* (* (sqrt (/ 1.0 x)) 2.0) (fma x x x)))
   (pow x -1.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / ((sqrt((1.0 / x)) * 2.0) * fma(x, x, x));
	} else {
		tmp = pow(x, -1.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(1.0 / Float64(Float64(sqrt(Float64(1.0 / x)) * 2.0) * fma(x, x, x)));
	else
		tmp = x ^ -1.5;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(1.0 / N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x, -1.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.8%

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

      1. sub-neg 9.8%

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

      2. flip-+ 9.6%

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

      3. frac-times 10.1%

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

      4. metadata-eval 10.1%

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

      5. add-sqr-sqrt 10.1%

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

      6. distribute-neg-frac 10.1%

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

      7. metadata-eval 10.1%

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

      8. +-commutative 10.1%

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

      9. distribute-neg-frac 10.1%

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

      10. metadata-eval 10.1%

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

      11. +-commutative 10.1%

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

      12. pow1/2 10.1%

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

      13. pow-flip 10.1%

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

      14. metadata-eval 10.1%

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

   3. Applied egg-rr 10.1%

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

      1. associate-*r/ 10.0%

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

      2. associate-*l/ 10.0%

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

      3. metadata-eval 10.0%

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

      4. associate-/l/ 9.8%

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

      5. rem-square-sqrt 10.1%

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

      6. sub-neg 10.1%

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

      7. distribute-neg-frac 10.1%

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

      8. metadata-eval 10.1%

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

      9. sub-neg 10.1%

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

      10. distribute-neg-frac 10.1%

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

      11. metadata-eval 10.1%

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

   5. Simplified 10.1%

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

      1. frac-add 14.6%

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

      2. *-un-lft-identity 14.6%

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

      3. distribute-rgt-in 14.7%

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

      4. *-un-lft-identity 14.7%

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

      5. pow2 14.7%

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

   7. Applied egg-rr 14.7%

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

      1. *-commutative 14.7%

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

      2. neg-mul-1 14.7%

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

      3. sub-neg 14.7%

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

      4. associate--l+ 99.3%

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

   9. Simplified 99.3%

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

       1. *-un-lft-identity 99.3%

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

       2. associate-/l/ 99.3%

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

       3. +-inverses 99.3%

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

       4. metadata-eval 99.3%

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

       5. inv-pow 99.3%

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

       6. sqrt-pow2 99.3%

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

       7. metadata-eval 99.3%

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

       8. +-commutative 99.3%

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

       9. unpow2 99.3%

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

       10. fma-def 99.3%

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

   11. Applied egg-rr 99.3%

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

   12. Taylor expanded in x around inf 96.0%

       \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(2 \cdot \sqrt{\frac{1}{x}}\right)} \cdot \mathsf{fma}\left(x, x, x\right)} \]
   13. Step-by-step derivation

       1. *-commutative 95.9%

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

   14. Simplified 96.0%

       \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot 2\right)} \cdot \mathsf{fma}\left(x, x, x\right)} \]

   if 1.35000000000000003e154 < x

   1. Initial program 65.4%

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

      1. sub-neg 65.4%

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

      2. flip-+ 65.4%

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

      3. frac-times 28.0%

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

      4. metadata-eval 28.0%

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

      5. add-sqr-sqrt 23.5%

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

      6. distribute-neg-frac 23.5%

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

      7. metadata-eval 23.5%

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

      8. +-commutative 23.5%

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

      9. distribute-neg-frac 23.5%

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

      10. metadata-eval 23.5%

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

      11. +-commutative 23.5%

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

      12. pow1/2 23.5%

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

      13. pow-flip 23.5%

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

      14. metadata-eval 23.5%

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

   3. Applied egg-rr 23.5%

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

      1. associate-*r/ 32.2%

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

      2. associate-*l/ 32.2%

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

      3. metadata-eval 32.2%

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

      4. associate-/l/ 42.7%

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

      5. rem-square-sqrt 65.4%

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

      6. sub-neg 65.4%

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

      7. distribute-neg-frac 65.4%

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

      8. metadata-eval 65.4%

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

      9. sub-neg 65.4%

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

      10. distribute-neg-frac 65.4%

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

      11. metadata-eval 65.4%

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

   5. Simplified 65.4%

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

      1. frac-add 65.4%

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

      2. *-un-lft-identity 65.4%

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

      3. distribute-rgt-in 65.4%

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

      4. *-un-lft-identity 65.4%

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

      5. pow2 65.4%

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

   7. Applied egg-rr 65.4%

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

      1. *-commutative 65.4%

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

      2. neg-mul-1 65.4%

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

      3. sub-neg 65.4%

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

      4. associate--l+ 65.4%

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

   9. Simplified 65.4%

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

   10. Taylor expanded in x around inf 65.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{3}}}} \]
   11. Step-by-step derivation

       1. metadata-eval 65.4%

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

       2. cube-div 65.4%

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

   12. Simplified 65.4%

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

       1. expm1-log1p-u 65.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(\frac{1}{x}\right)}^{3}}\right)\right)} \]

       2. expm1-udef 65.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{{\left(\frac{1}{x}\right)}^{3}}\right)} - 1} \]

       3. sqrt-pow1 65.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{x}\right)}^{\left(\frac{3}{2}\right)}}\right)} - 1 \]

       4. inv-pow 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left({x}^{-1}\right)}}^{\left(\frac{3}{2}\right)}\right)} - 1 \]

       5. pow-pow 65.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(-1 \cdot \frac{3}{2}\right)}}\right)} - 1 \]

       6. metadata-eval 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({x}^{\left(-1 \cdot \color{blue}{1.5}\right)}\right)} - 1 \]

       7. metadata-eval 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({x}^{\color{blue}{-1.5}}\right)} - 1 \]

   14. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({x}^{-1.5}\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 69.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-1.5}\right)\right)} \]

       2. expm1-log1p 69.0%

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

   16. Simplified 69.0%

       \[\leadsto \color{blue}{{x}^{-1.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left(\sqrt{\frac{1}{x}} \cdot 2\right) \cdot \mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \]

Alternative 7: 82.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{1}{x}} \cdot 2}{\frac{-1}{x \cdot \left(-1 - x\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 1.0 (/ (* (sqrt (/ 1.0 x)) 2.0) (/ -1.0 (* x (- -1.0 x)))))
   (pow x -1.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / ((sqrt((1.0 / x)) * 2.0) / (-1.0 / (x * (-1.0 - x))));
	} else {
		tmp = pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.35d+154) then
        tmp = 1.0d0 / ((sqrt((1.0d0 / x)) * 2.0d0) / ((-1.0d0) / (x * ((-1.0d0) - x))))
    else
        tmp = x ** (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / ((Math.sqrt((1.0 / x)) * 2.0) / (-1.0 / (x * (-1.0 - x))));
	} else {
		tmp = Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.35e+154:
		tmp = 1.0 / ((math.sqrt((1.0 / x)) * 2.0) / (-1.0 / (x * (-1.0 - x))))
	else:
		tmp = math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(1.0 / Float64(Float64(sqrt(Float64(1.0 / x)) * 2.0) / Float64(-1.0 / Float64(x * Float64(-1.0 - x)))));
	else
		tmp = x ^ -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.35e+154)
		tmp = 1.0 / ((sqrt((1.0 / x)) * 2.0) / (-1.0 / (x * (-1.0 - x))));
	else
		tmp = x ^ -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(1.0 / N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] / N[(-1.0 / N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x, -1.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.8%

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

      1. flip-- 9.6%

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

      2. clear-num 9.6%

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

      3. pow1/2 9.6%

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

      4. pow-flip 9.6%

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

      5. metadata-eval 9.6%

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

      6. inv-pow 9.6%

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

      7. sqrt-pow2 9.6%

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

      8. +-commutative 9.6%

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

      9. metadata-eval 9.6%

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

      10. frac-times 10.1%

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

      11. metadata-eval 10.1%

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

      12. add-sqr-sqrt 10.1%

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

      13. frac-times 9.8%

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

      14. metadata-eval 9.8%

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

      15. add-sqr-sqrt 10.1%

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

   3. Applied egg-rr 10.1%

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

      1. frac-2neg 10.1%

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

      2. metadata-eval 10.1%

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

      3. frac-sub 14.6%

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

   5. Applied egg-rr 14.6%

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

      1. *-commutative 14.6%

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

      2. neg-mul-1 14.6%

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

      3. *-rgt-identity 14.6%

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

      4. *-lft-identity 14.6%

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

      5. neg-sub0 14.6%

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

      6. associate--l- 14.6%

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

      7. associate-+l+ 99.2%

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

      8. *-rgt-identity 99.2%

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

      9. sub-neg 99.2%

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

      10. +-inverses 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. distribute-neg-in 99.2%

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

      14. metadata-eval 99.2%

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

      15. unsub-neg 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around inf 95.9%

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

      1. *-commutative 95.9%

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

   10. Simplified 95.9%

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

   if 1.35000000000000003e154 < x

   1. Initial program 65.4%

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

      1. sub-neg 65.4%

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

      2. flip-+ 65.4%

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

      3. frac-times 28.0%

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

      4. metadata-eval 28.0%

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

      5. add-sqr-sqrt 23.5%

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

      6. distribute-neg-frac 23.5%

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

      7. metadata-eval 23.5%

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

      8. +-commutative 23.5%

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

      9. distribute-neg-frac 23.5%

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

      10. metadata-eval 23.5%

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

      11. +-commutative 23.5%

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

      12. pow1/2 23.5%

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

      13. pow-flip 23.5%

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

      14. metadata-eval 23.5%

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

   3. Applied egg-rr 23.5%

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

      1. associate-*r/ 32.2%

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

      2. associate-*l/ 32.2%

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

      3. metadata-eval 32.2%

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

      4. associate-/l/ 42.7%

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

      5. rem-square-sqrt 65.4%

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

      6. sub-neg 65.4%

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

      7. distribute-neg-frac 65.4%

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

      8. metadata-eval 65.4%

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

      9. sub-neg 65.4%

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

      10. distribute-neg-frac 65.4%

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

      11. metadata-eval 65.4%

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

   5. Simplified 65.4%

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

      1. frac-add 65.4%

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

      2. *-un-lft-identity 65.4%

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

      3. distribute-rgt-in 65.4%

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

      4. *-un-lft-identity 65.4%

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

      5. pow2 65.4%

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

   7. Applied egg-rr 65.4%

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

      1. *-commutative 65.4%

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

      2. neg-mul-1 65.4%

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

      3. sub-neg 65.4%

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

      4. associate--l+ 65.4%

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

   9. Simplified 65.4%

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

   10. Taylor expanded in x around inf 65.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{3}}}} \]
   11. Step-by-step derivation

       1. metadata-eval 65.4%

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

       2. cube-div 65.4%

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

   12. Simplified 65.4%

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

       1. expm1-log1p-u 65.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(\frac{1}{x}\right)}^{3}}\right)\right)} \]

       2. expm1-udef 65.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{{\left(\frac{1}{x}\right)}^{3}}\right)} - 1} \]

       3. sqrt-pow1 65.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{x}\right)}^{\left(\frac{3}{2}\right)}}\right)} - 1 \]

       4. inv-pow 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left({x}^{-1}\right)}}^{\left(\frac{3}{2}\right)}\right)} - 1 \]

       5. pow-pow 65.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(-1 \cdot \frac{3}{2}\right)}}\right)} - 1 \]

       6. metadata-eval 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({x}^{\left(-1 \cdot \color{blue}{1.5}\right)}\right)} - 1 \]

       7. metadata-eval 65.4%

          \[\leadsto e^{\mathsf{log1p}\left({x}^{\color{blue}{-1.5}}\right)} - 1 \]

   14. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({x}^{-1.5}\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 69.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-1.5}\right)\right)} \]

       2. expm1-log1p 69.0%

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

   16. Simplified 69.0%

       \[\leadsto \color{blue}{{x}^{-1.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{1}{x}} \cdot 2}{\frac{-1}{x \cdot \left(-1 - x\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \]

Alternative 8: 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 38.7%

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

   1. sub-neg 38.7%

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

   2. flip-+ 38.6%

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

   3. frac-times 19.4%

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

   4. metadata-eval 19.4%

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

   5. add-sqr-sqrt 17.0%

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

   6. distribute-neg-frac 17.0%

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

   7. metadata-eval 17.0%

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

   8. +-commutative 17.0%

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

   9. distribute-neg-frac 17.0%

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

   10. metadata-eval 17.0%

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

   11. +-commutative 17.0%

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

   12. pow1/2 17.0%

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

   13. pow-flip 17.0%

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

   14. metadata-eval 17.0%

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

3. Applied egg-rr 17.0%

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

   1. associate-*r/ 21.5%

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

   2. associate-*l/ 21.5%

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

   3. metadata-eval 21.5%

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

   4. associate-/l/ 26.9%

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

   5. rem-square-sqrt 38.8%

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

   6. sub-neg 38.8%

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

   7. distribute-neg-frac 38.8%

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

   8. metadata-eval 38.8%

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

   9. sub-neg 38.8%

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

   10. distribute-neg-frac 38.8%

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

   11. metadata-eval 38.8%

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

5. Simplified 38.8%

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

   1. frac-add 41.0%

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

   2. *-un-lft-identity 41.0%

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

   3. distribute-rgt-in 41.0%

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

   4. *-un-lft-identity 41.0%

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

   5. pow2 41.0%

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

7. Applied egg-rr 41.0%

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

   1. *-commutative 41.0%

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

   2. neg-mul-1 41.0%

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

   3. sub-neg 41.0%

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

   4. associate--l+ 81.7%

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

9. Simplified 81.7%

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

10. Taylor expanded in x around inf 40.7%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{3}}}} \]
11. Step-by-step derivation

    1. metadata-eval 40.7%

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

    2. cube-div 41.1%

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

12. Simplified 41.1%

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

    1. expm1-log1p-u 41.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(\frac{1}{x}\right)}^{3}}\right)\right)} \]

    2. expm1-udef 36.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{{\left(\frac{1}{x}\right)}^{3}}\right)} - 1} \]

    3. sqrt-pow1 36.6%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{x}\right)}^{\left(\frac{3}{2}\right)}}\right)} - 1 \]

    4. inv-pow 36.6%

       \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left({x}^{-1}\right)}}^{\left(\frac{3}{2}\right)}\right)} - 1 \]

    5. pow-pow 36.6%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(-1 \cdot \frac{3}{2}\right)}}\right)} - 1 \]

    6. metadata-eval 36.6%

       \[\leadsto e^{\mathsf{log1p}\left({x}^{\left(-1 \cdot \color{blue}{1.5}\right)}\right)} - 1 \]

    7. metadata-eval 36.6%

       \[\leadsto e^{\mathsf{log1p}\left({x}^{\color{blue}{-1.5}}\right)} - 1 \]

14. Applied egg-rr 36.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({x}^{-1.5}\right)} - 1} \]
15. Step-by-step derivation

    1. expm1-def 44.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-1.5}\right)\right)} \]

    2. expm1-log1p 44.9%

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

16. Simplified 44.9%

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

17. Final simplification 44.9%

    \[\leadsto {x}^{-1.5} \]

Alternative 9: 7.8% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.7%

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

   1. sub-neg 38.7%

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

   2. flip-+ 38.6%

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

   3. frac-times 19.4%

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

   4. metadata-eval 19.4%

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

   5. add-sqr-sqrt 17.0%

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

   6. distribute-neg-frac 17.0%

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

   7. metadata-eval 17.0%

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

   8. +-commutative 17.0%

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

   9. distribute-neg-frac 17.0%

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

   10. metadata-eval 17.0%

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

   11. +-commutative 17.0%

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

   12. pow1/2 17.0%

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

   13. pow-flip 17.0%

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

   14. metadata-eval 17.0%

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

3. Applied egg-rr 17.0%

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

   1. associate-*r/ 21.5%

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

   2. associate-*l/ 21.5%

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

   3. metadata-eval 21.5%

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

   4. associate-/l/ 26.9%

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

   5. rem-square-sqrt 38.8%

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

   6. sub-neg 38.8%

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

   7. distribute-neg-frac 38.8%

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

   8. metadata-eval 38.8%

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

   9. sub-neg 38.8%

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

   10. distribute-neg-frac 38.8%

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

   11. metadata-eval 38.8%

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

5. Simplified 38.8%

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

6. Taylor expanded in x around 0 7.7%

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

   1. distribute-rgt-in 7.7%

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

   2. *-lft-identity 7.7%

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

   3. pow-plus 7.7%

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

   4. metadata-eval 7.7%

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

8. Simplified 7.7%

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

9. Taylor expanded in x around inf 7.7%

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

10. Final simplification 7.7%

    \[\leadsto \frac{1}{x} \]

Alternative 10: 2.5% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 38.7%

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

2. Taylor expanded in x around 0 2.6%

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

3. Taylor expanded in x around inf 2.6%

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

4. Final simplification 2.6%

   \[\leadsto -1 \]

Developer target: 98.3% 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 2024024 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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