2isqrt (example 3.6)

Percentage Accurate: 37.8% → 99.2%

Time: 16.8s

Alternatives: 12

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. Add Preprocessing
3. 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%

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

4. Applied egg-rr 17.0%

   \[\leadsto \color{blue}{\frac{\frac{1}{x} - \frac{-1}{\sqrt{1 + x}} \cdot \frac{-1}{\sqrt{1 + x}}}{{x}^{-0.5} - \frac{-1}{\sqrt{1 + x}}}} \]
5. 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%

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

6. Simplified 38.8%

   \[\leadsto \color{blue}{\frac{\frac{1}{x} + \frac{-1}{1 + x}}{{x}^{-0.5} + \frac{1}{\sqrt{1 + x}}}} \]
7. 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%

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

8. Applied egg-rr 41.0%

   \[\leadsto \frac{\color{blue}{\frac{\left(1 + x\right) + x \cdot -1}{x + {x}^{2}}}}{{x}^{-0.5} + \frac{1}{\sqrt{1 + x}}} \]
9. 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%

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

10. Simplified 81.7%

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

    1. add-sqr-sqrt 81.5%

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

12. Applied egg-rr 99.2%

    \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}}{\mathsf{hypot}\left({x}^{-0.25}, {\left(1 + x\right)}^{-0.25}\right)} \cdot \frac{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}}{\mathsf{hypot}\left({x}^{-0.25}, {\left(1 + x\right)}^{-0.25}\right)}} \]
13. 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%

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

14. Simplified 99.2%

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

16. Applied egg-rr 37.5%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left({x}^{-0.25}, {\left(1 + x\right)}^{-0.25}\right) \cdot \mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-2}\right)} - 1} \]
17. 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%

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

18. Simplified 99.2%

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

19. Final simplification 99.2%

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

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(x, \sqrt{\frac{1}{x}} \cdot 1.5, 2 \cdot \left|{x}^{1.5}\right|\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (fma x (* (sqrt (/ 1.0 x)) 1.5) (* 2.0 (fabs (pow x 1.5))))))

C

double code(double x) {
	return 1.0 / fma(x, (sqrt((1.0 / x)) * 1.5), (2.0 * fabs(pow(x, 1.5))));
}

Julia

function code(x)
	return Float64(1.0 / fma(x, Float64(sqrt(Float64(1.0 / x)) * 1.5), Float64(2.0 * abs((x ^ 1.5)))))
end

Wolfram

code[x_] := N[(1.0 / N[(x * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 1.5), $MachinePrecision] + N[(2.0 * N[Abs[N[Power[x, 1.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. 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%

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

4. Applied egg-rr 38.8%

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

5. Taylor expanded in x around inf 65.5%

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

   1. +-commutative 65.5%

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

   2. fma-def 65.5%

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

   3. distribute-rgt-out-- 65.5%

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

   4. metadata-eval 65.5%

      \[\leadsto \frac{1}{\mathsf{fma}\left(x, \sqrt{\frac{1}{x}} \cdot \color{blue}{1.5}, 2 \cdot \sqrt{{x}^{3}}\right)} \]

   5. sqr-pow 65.5%

      \[\leadsto \frac{1}{\mathsf{fma}\left(x, \sqrt{\frac{1}{x}} \cdot 1.5, 2 \cdot \sqrt{\color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}\right)} \]

   6. rem-sqrt-square 98.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(x, \sqrt{\frac{1}{x}} \cdot 1.5, 2 \cdot \color{blue}{\left|{x}^{\left(\frac{3}{2}\right)}\right|}\right)} \]

   7. metadata-eval 98.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(x, \sqrt{\frac{1}{x}} \cdot 1.5, 2 \cdot \left|{x}^{\color{blue}{1.5}}\right|\right)} \]

7. Simplified 98.0%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \sqrt{\frac{1}{x}} \cdot 1.5, 2 \cdot \left|{x}^{1.5}\right|\right)}} \]

8. Final simplification 98.0%

   \[\leadsto \frac{1}{\mathsf{fma}\left(x, \sqrt{\frac{1}{x}} \cdot 1.5, 2 \cdot \left|{x}^{1.5}\right|\right)} \]
9. Add Preprocessing

Alternative 3: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+69}:\\ \;\;\;\;\frac{\frac{1 + \left(x - x\right)}{x + {x}^{2}}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left|{x}^{1.5}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+69)
   (/
    (/ (+ 1.0 (- x x)) (+ x (pow x 2.0)))
    (+ (pow x -0.5) (pow (+ x 1.0) -0.5)))
   (/ 1.0 (* 2.0 (fabs (pow x 1.5))))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+69) {
		tmp = ((1.0 + (x - x)) / (x + pow(x, 2.0))) / (pow(x, -0.5) + pow((x + 1.0), -0.5));
	} else {
		tmp = 1.0 / (2.0 * fabs(pow(x, 1.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d+69) then
        tmp = ((1.0d0 + (x - x)) / (x + (x ** 2.0d0))) / ((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0)))
    else
        tmp = 1.0d0 / (2.0d0 * abs((x ** 1.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e+69) {
		tmp = ((1.0 + (x - x)) / (x + Math.pow(x, 2.0))) / (Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5));
	} else {
		tmp = 1.0 / (2.0 * Math.abs(Math.pow(x, 1.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e+69:
		tmp = ((1.0 + (x - x)) / (x + math.pow(x, 2.0))) / (math.pow(x, -0.5) + math.pow((x + 1.0), -0.5))
	else:
		tmp = 1.0 / (2.0 * math.fabs(math.pow(x, 1.5)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+69)
		tmp = Float64(Float64(Float64(1.0 + Float64(x - x)) / Float64(x + (x ^ 2.0))) / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)));
	else
		tmp = Float64(1.0 / Float64(2.0 * abs((x ^ 1.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e+69)
		tmp = ((1.0 + (x - x)) / (x + (x ^ 2.0))) / ((x ^ -0.5) + ((x + 1.0) ^ -0.5));
	else
		tmp = 1.0 / (2.0 * abs((x ^ 1.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e+69], N[(N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(x + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 * N[Abs[N[Power[x, 1.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0000000000000001e69

   1. Initial program 17.0%

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

      1. sub-neg 17.0%

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

      2. flip-+ 16.5%

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

      3. frac-times 18.0%

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

      4. metadata-eval 18.0%

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

      5. add-sqr-sqrt 18.2%

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

      6. distribute-neg-frac 18.2%

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

      7. metadata-eval 18.2%

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

      8. +-commutative 18.2%

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

      9. distribute-neg-frac 18.2%

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

      10. metadata-eval 18.2%

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

      11. +-commutative 18.2%

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

      12. pow1/2 18.2%

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

      13. pow-flip 18.2%

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

      14. metadata-eval 18.2%

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

   4. Applied egg-rr 18.2%

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

      1. associate-*r/ 18.2%

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

      2. associate-*l/ 18.2%

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

      3. metadata-eval 18.2%

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

      4. associate-/l/ 17.5%

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

      5. rem-square-sqrt 17.7%

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

      6. sub-neg 17.7%

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

      7. distribute-neg-frac 17.7%

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

      8. metadata-eval 17.7%

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

      9. sub-neg 17.7%

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

      10. distribute-neg-frac 17.7%

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

      11. metadata-eval 17.7%

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

   6. Simplified 17.7%

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

      1. frac-add 28.7%

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

      2. *-un-lft-identity 28.7%

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

      3. distribute-rgt-in 28.8%

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

      4. *-un-lft-identity 28.8%

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

      5. pow2 28.8%

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

   8. Applied egg-rr 28.8%

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

      1. *-commutative 28.8%

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

      2. neg-mul-1 28.8%

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

      3. sub-neg 28.8%

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

      4. associate--l+ 99.3%

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

   10. Simplified 99.3%

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

   12. Applied egg-rr 39.3%

       \[\leadsto \frac{\frac{1 + \left(x - x\right)}{x + {x}^{2}}}{{x}^{-0.5} + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - 1\right)}} \]
   13. Step-by-step derivation

       1. expm1-def 99.3%

          \[\leadsto \frac{\frac{1 + \left(x - x\right)}{x + {x}^{2}}}{{x}^{-0.5} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)}} \]

       2. expm1-log1p 99.3%

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

   14. Simplified 99.3%

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

   if 1.0000000000000001e69 < x

   1. Initial program 44.0%

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

      1. flip-- 44.0%

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

      2. clear-num 44.0%

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

      3. pow1/2 44.0%

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

      4. pow-flip 44.0%

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

      5. metadata-eval 44.0%

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

      6. inv-pow 44.0%

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

      7. sqrt-pow2 44.0%

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

      8. +-commutative 44.0%

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

      9. metadata-eval 44.0%

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

      10. frac-times 19.7%

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

      11. metadata-eval 19.7%

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

      12. add-sqr-sqrt 16.8%

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

      13. frac-times 29.2%

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

      14. metadata-eval 29.2%

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

      15. add-sqr-sqrt 44.0%

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

   4. Applied egg-rr 44.0%

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

   5. Taylor expanded in x around inf 57.9%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
   6. Step-by-step derivation

      1. sqr-pow 57.9%

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

      2. rem-sqrt-square 98.3%

         \[\leadsto \frac{1}{2 \cdot \color{blue}{\left|{x}^{\left(\frac{3}{2}\right)}\right|}} \]

      3. metadata-eval 98.3%

         \[\leadsto \frac{1}{2 \cdot \left|{x}^{\color{blue}{1.5}}\right|} \]

   7. Simplified 98.3%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|{x}^{1.5}\right|}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+69}:\\ \;\;\;\;\frac{\frac{1 + \left(x - x\right)}{x + {x}^{2}}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left|{x}^{1.5}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{1}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left|{x}^{1.5}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6e+23)
   (/ (/ 1.0 (+ (pow x -0.5) (pow (+ x 1.0) -0.5))) (fma x x x))
   (/ 1.0 (* 2.0 (fabs (pow x 1.5))))))

C

double code(double x) {
	double tmp;
	if (x <= 6e+23) {
		tmp = (1.0 / (pow(x, -0.5) + pow((x + 1.0), -0.5))) / fma(x, x, x);
	} else {
		tmp = 1.0 / (2.0 * fabs(pow(x, 1.5)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 6e+23)
		tmp = Float64(Float64(1.0 / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5))) / fma(x, x, x));
	else
		tmp = Float64(1.0 / Float64(2.0 * abs((x ^ 1.5))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6e+23], N[(N[(1.0 / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 * N[Abs[N[Power[x, 1.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.0000000000000002e23

   1. Initial program 42.6%

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

      1. sub-neg 42.6%

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

      2. flip-+ 41.3%

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

      3. frac-times 43.9%

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

      4. metadata-eval 43.9%

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

      5. add-sqr-sqrt 44.3%

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

      6. distribute-neg-frac 44.3%

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

      7. metadata-eval 44.3%

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

      8. +-commutative 44.3%

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

      9. distribute-neg-frac 44.3%

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

      10. metadata-eval 44.3%

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

      11. +-commutative 44.3%

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

      12. pow1/2 44.3%

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

      13. pow-flip 44.3%

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

      14. metadata-eval 44.3%

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

   4. Applied egg-rr 44.3%

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

      1. associate-*r/ 44.4%

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

      2. associate-*l/ 44.4%

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

      3. metadata-eval 44.4%

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

      4. associate-/l/ 43.2%

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

      5. rem-square-sqrt 44.8%

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

      6. sub-neg 44.8%

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

      7. distribute-neg-frac 44.8%

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

      8. metadata-eval 44.8%

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

      9. sub-neg 44.8%

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

      10. distribute-neg-frac 44.8%

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

      11. metadata-eval 44.8%

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

   6. Simplified 44.8%

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

      1. frac-add 77.2%

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

      2. *-un-lft-identity 77.2%

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

      3. distribute-rgt-in 77.5%

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

      4. *-un-lft-identity 77.5%

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

      5. pow2 77.5%

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

   8. Applied egg-rr 77.5%

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

      1. *-commutative 77.5%

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

      2. neg-mul-1 77.5%

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

      3. sub-neg 77.5%

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

      4. associate--l+ 99.2%

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

   10. Simplified 99.2%

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

       1. expm1-log1p-u 99.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 + \left(x - x\right)}{x + {x}^{2}}}{{x}^{-0.5} + \frac{1}{\sqrt{1 + x}}}\right)\right)} \]

       2. expm1-udef 25.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1 + \left(x - x\right)}{x + {x}^{2}}}{{x}^{-0.5} + \frac{1}{\sqrt{1 + x}}}\right)} - 1} \]

   12. Applied egg-rr 25.1%

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

       1. expm1-def 98.9%

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

       2. expm1-log1p 98.9%

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

       3. associate-/r* 99.2%

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

   14. Simplified 99.2%

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

   if 6.0000000000000002e23 < x

   1. Initial program 38.4%

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

      1. flip-- 38.4%

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

      2. clear-num 38.4%

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

      3. pow1/2 38.4%

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

      4. pow-flip 38.4%

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

      5. metadata-eval 38.4%

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

      6. inv-pow 38.4%

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

      7. sqrt-pow2 38.4%

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

      8. +-commutative 38.4%

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

      9. metadata-eval 38.4%

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

      10. frac-times 17.6%

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

      11. metadata-eval 17.6%

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

      12. add-sqr-sqrt 15.1%

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

      13. frac-times 25.7%

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

      14. metadata-eval 25.7%

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

      15. add-sqr-sqrt 38.4%

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

   4. Applied egg-rr 38.4%

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

   5. Taylor expanded in x around inf 63.6%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
   6. Step-by-step derivation

      1. sqr-pow 63.6%

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

      2. rem-sqrt-square 98.4%

         \[\leadsto \frac{1}{2 \cdot \color{blue}{\left|{x}^{\left(\frac{3}{2}\right)}\right|}} \]

      3. metadata-eval 98.4%

         \[\leadsto \frac{1}{2 \cdot \left|{x}^{\color{blue}{1.5}}\right|} \]

   7. Simplified 98.4%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|{x}^{1.5}\right|}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{1}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left|{x}^{1.5}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+59}:\\ \;\;\;\;\frac{1}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \cdot \frac{1}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left|{x}^{1.5}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+59)
   (* (/ 1.0 (+ (pow x -0.5) (pow (+ x 1.0) -0.5))) (/ 1.0 (* x (+ x 1.0))))
   (/ 1.0 (* 2.0 (fabs (pow x 1.5))))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+59) {
		tmp = (1.0 / (pow(x, -0.5) + pow((x + 1.0), -0.5))) * (1.0 / (x * (x + 1.0)));
	} else {
		tmp = 1.0 / (2.0 * fabs(pow(x, 1.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d+59) then
        tmp = (1.0d0 / ((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0)))) * (1.0d0 / (x * (x + 1.0d0)))
    else
        tmp = 1.0d0 / (2.0d0 * abs((x ** 1.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e+59) {
		tmp = (1.0 / (Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5))) * (1.0 / (x * (x + 1.0)));
	} else {
		tmp = 1.0 / (2.0 * Math.abs(Math.pow(x, 1.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e+59:
		tmp = (1.0 / (math.pow(x, -0.5) + math.pow((x + 1.0), -0.5))) * (1.0 / (x * (x + 1.0)))
	else:
		tmp = 1.0 / (2.0 * math.fabs(math.pow(x, 1.5)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+59)
		tmp = Float64(Float64(1.0 / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5))) * Float64(1.0 / Float64(x * Float64(x + 1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 * abs((x ^ 1.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e+59)
		tmp = (1.0 / ((x ^ -0.5) + ((x + 1.0) ^ -0.5))) * (1.0 / (x * (x + 1.0)));
	else
		tmp = 1.0 / (2.0 * abs((x ^ 1.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e+59], N[(N[(1.0 / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 * N[Abs[N[Power[x, 1.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999972e58

   1. Initial program 19.8%

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

      1. flip-- 19.3%

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

      2. frac-times 20.7%

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

      3. metadata-eval 20.7%

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

      4. add-sqr-sqrt 21.1%

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

      5. frac-times 20.4%

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

      6. metadata-eval 20.4%

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

      7. add-sqr-sqrt 20.7%

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

      8. +-commutative 20.7%

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

      9. pow1/2 20.7%

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

      10. pow-flip 20.7%

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

      11. metadata-eval 20.7%

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

      12. inv-pow 20.7%

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

      13. sqrt-pow2 20.7%

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

      14. +-commutative 20.7%

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

      15. metadata-eval 20.7%

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

   4. Applied egg-rr 20.7%

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

      1. div-inv 20.7%

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

      2. frac-sub 34.2%

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

      3. *-un-lft-identity 34.2%

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

      4. *-rgt-identity 34.2%

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

      5. associate-+r- 99.3%

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

      6. +-inverses 99.3%

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

      7. metadata-eval 99.3%

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

   6. Applied egg-rr 99.3%

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

   if 9.99999999999999972e58 < x

   1. Initial program 42.3%

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

      1. flip-- 42.3%

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

      2. clear-num 42.3%

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

      3. pow1/2 42.3%

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

      4. pow-flip 42.3%

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

      5. metadata-eval 42.3%

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

      6. inv-pow 42.3%

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

      7. sqrt-pow2 42.3%

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

      8. +-commutative 42.3%

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

      9. metadata-eval 42.3%

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

      10. frac-times 19.1%

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

      11. metadata-eval 19.1%

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

      12. add-sqr-sqrt 16.3%

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

      13. frac-times 28.1%

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

      14. metadata-eval 28.1%

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

      15. add-sqr-sqrt 42.3%

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

   4. Applied egg-rr 42.3%

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

   5. Taylor expanded in x around inf 59.6%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
   6. Step-by-step derivation

      1. sqr-pow 59.6%

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

      2. rem-sqrt-square 98.3%

         \[\leadsto \frac{1}{2 \cdot \color{blue}{\left|{x}^{\left(\frac{3}{2}\right)}\right|}} \]

      3. metadata-eval 98.3%

         \[\leadsto \frac{1}{2 \cdot \left|{x}^{\color{blue}{1.5}}\right|} \]

   7. Simplified 98.3%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|{x}^{1.5}\right|}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+59}:\\ \;\;\;\;\frac{1}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \cdot \frac{1}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left|{x}^{1.5}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+69}:\\ \;\;\;\;\frac{\frac{-1}{x \cdot \left(-1 - x\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left|{x}^{1.5}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+69)
   (/ (/ -1.0 (* x (- -1.0 x))) (+ (pow x -0.5) (pow (+ x 1.0) -0.5)))
   (/ 1.0 (* 2.0 (fabs (pow x 1.5))))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+69) {
		tmp = (-1.0 / (x * (-1.0 - x))) / (pow(x, -0.5) + pow((x + 1.0), -0.5));
	} else {
		tmp = 1.0 / (2.0 * fabs(pow(x, 1.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d+69) then
        tmp = ((-1.0d0) / (x * ((-1.0d0) - x))) / ((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0)))
    else
        tmp = 1.0d0 / (2.0d0 * abs((x ** 1.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e+69) {
		tmp = (-1.0 / (x * (-1.0 - x))) / (Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5));
	} else {
		tmp = 1.0 / (2.0 * Math.abs(Math.pow(x, 1.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e+69:
		tmp = (-1.0 / (x * (-1.0 - x))) / (math.pow(x, -0.5) + math.pow((x + 1.0), -0.5))
	else:
		tmp = 1.0 / (2.0 * math.fabs(math.pow(x, 1.5)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+69)
		tmp = Float64(Float64(-1.0 / Float64(x * Float64(-1.0 - x))) / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)));
	else
		tmp = Float64(1.0 / Float64(2.0 * abs((x ^ 1.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e+69)
		tmp = (-1.0 / (x * (-1.0 - x))) / ((x ^ -0.5) + ((x + 1.0) ^ -0.5));
	else
		tmp = 1.0 / (2.0 * abs((x ^ 1.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e+69], 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[(1.0 / N[(2.0 * N[Abs[N[Power[x, 1.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0000000000000001e69

   1. Initial program 17.0%

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

      1. flip-- 16.5%

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

      2. frac-times 18.0%

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

      3. metadata-eval 18.0%

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

      4. add-sqr-sqrt 18.2%

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

      5. frac-times 17.4%

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

      6. metadata-eval 17.4%

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

      7. add-sqr-sqrt 17.7%

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

      8. +-commutative 17.7%

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

      9. pow1/2 17.7%

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

      10. pow-flip 17.7%

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

      11. metadata-eval 17.7%

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

      12. inv-pow 17.7%

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

      13. sqrt-pow2 17.7%

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

      14. +-commutative 17.7%

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

      15. metadata-eval 17.7%

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

   4. Applied egg-rr 17.7%

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

      1. frac-2neg 17.7%

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

      2. metadata-eval 17.7%

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

      3. frac-sub 28.7%

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

   6. Applied egg-rr 28.7%

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

      1. /-rgt-identity 28.7%

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

      2. /-rgt-identity 28.7%

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

      3. *-commutative 28.7%

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

      4. neg-mul-1 28.7%

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

      5. *-rgt-identity 28.7%

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

      6. *-lft-identity 28.7%

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

      7. neg-sub0 28.7%

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

      8. associate--l- 28.7%

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

      9. associate-+l+ 99.2%

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

      10. *-rgt-identity 99.2%

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

      11. sub-neg 99.2%

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

      12. +-inverses 99.2%

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

      13. metadata-eval 99.2%

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

      14. metadata-eval 99.2%

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

      15. distribute-neg-in 99.2%

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

      16. metadata-eval 99.2%

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

      17. unsub-neg 99.2%

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

   8. Simplified 99.2%

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

   if 1.0000000000000001e69 < x

   1. Initial program 44.0%

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

      1. flip-- 44.0%

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

      2. clear-num 44.0%

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

      3. pow1/2 44.0%

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

      4. pow-flip 44.0%

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

      5. metadata-eval 44.0%

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

      6. inv-pow 44.0%

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

      7. sqrt-pow2 44.0%

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

      8. +-commutative 44.0%

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

      9. metadata-eval 44.0%

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

      10. frac-times 19.7%

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

      11. metadata-eval 19.7%

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

      12. add-sqr-sqrt 16.8%

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

      13. frac-times 29.2%

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

      14. metadata-eval 29.2%

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

      15. add-sqr-sqrt 44.0%

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

   4. Applied egg-rr 44.0%

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

   5. Taylor expanded in x around inf 57.9%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
   6. Step-by-step derivation

      1. sqr-pow 57.9%

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

      2. rem-sqrt-square 98.3%

         \[\leadsto \frac{1}{2 \cdot \color{blue}{\left|{x}^{\left(\frac{3}{2}\right)}\right|}} \]

      3. metadata-eval 98.3%

         \[\leadsto \frac{1}{2 \cdot \left|{x}^{\color{blue}{1.5}}\right|} \]

   7. Simplified 98.3%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|{x}^{1.5}\right|}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+69}:\\ \;\;\;\;\frac{\frac{-1}{x \cdot \left(-1 - x\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left|{x}^{1.5}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e+102) (* 0.5 (sqrt (/ 1.0 (pow x 3.0)))) (pow x -1.5)))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e+102) {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	} else {
		tmp = pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.5d+102) then
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    else
        tmp = x ** (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e+102) {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	} else {
		tmp = Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e+102:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	else:
		tmp = math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e+102)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	else
		tmp = x ^ -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e+102)
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	else
		tmp = x ^ -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e+102], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[x, -1.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.49999999999999981e102

   1. Initial program 12.2%

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

      1. flip-- 11.9%

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

      2. clear-num 11.9%

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

      3. pow1/2 11.9%

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

      4. pow-flip 11.9%

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

      5. metadata-eval 11.9%

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

      6. inv-pow 11.9%

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

      7. sqrt-pow2 11.9%

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

      8. +-commutative 11.9%

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

      9. metadata-eval 11.9%

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

      10. frac-times 12.9%

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

      11. metadata-eval 12.9%

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

      12. add-sqr-sqrt 13.1%

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

      13. frac-times 12.5%

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

      14. metadata-eval 12.5%

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

      15. add-sqr-sqrt 12.7%

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

   4. Applied egg-rr 12.7%

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

   5. Taylor expanded in x around inf 94.2%

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

   if 5.49999999999999981e102 < x

   1. Initial program 50.7%

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

      1. sub-neg 50.7%

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

      2. flip-+ 50.7%

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

      3. frac-times 22.3%

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

      4. metadata-eval 22.3%

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

      5. add-sqr-sqrt 18.8%

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

      6. distribute-neg-frac 18.8%

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

      7. metadata-eval 18.8%

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

      8. +-commutative 18.8%

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

      9. distribute-neg-frac 18.8%

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

      10. metadata-eval 18.8%

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

      11. +-commutative 18.8%

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

      12. pow1/2 18.8%

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

      13. pow-flip 18.8%

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

      14. metadata-eval 18.8%

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

   4. Applied egg-rr 18.8%

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

      1. associate-*r/ 25.4%

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

      2. associate-*l/ 25.4%

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

      3. metadata-eval 25.4%

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

      4. associate-/l/ 33.4%

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

      5. rem-square-sqrt 50.7%

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

      6. sub-neg 50.7%

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

      7. distribute-neg-frac 50.7%

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

      8. metadata-eval 50.7%

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

      9. sub-neg 50.7%

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

      10. distribute-neg-frac 50.7%

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

      11. metadata-eval 50.7%

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

   6. Simplified 50.7%

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

      1. frac-add 50.7%

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

      2. *-un-lft-identity 50.7%

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

      3. distribute-rgt-in 50.7%

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

      4. *-un-lft-identity 50.7%

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

      5. pow2 50.7%

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

   8. Applied egg-rr 50.7%

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

      1. *-commutative 50.7%

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

      2. neg-mul-1 50.7%

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

      3. sub-neg 50.7%

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

      4. associate--l+ 73.7%

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

   10. Simplified 73.7%

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

   11. Taylor expanded in x around inf 50.7%

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

       1. metadata-eval 50.7%

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

       2. cube-div 51.2%

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

   13. Simplified 51.2%

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

       1. expm1-log1p-u 51.2%

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

       2. expm1-udef 50.7%

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

       3. sqrt-pow1 50.7%

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

       4. inv-pow 50.7%

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

       5. pow-pow 50.7%

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

       6. metadata-eval 50.7%

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

       7. metadata-eval 50.7%

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

   15. Applied egg-rr 50.7%

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

       1. expm1-def 56.7%

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

       2. expm1-log1p 56.7%

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

   17. Simplified 56.7%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \sqrt{{\left(\frac{1}{x}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 7.5e+107) (* 0.5 (sqrt (pow (/ 1.0 x) 3.0))) (pow x -1.5)))

C

double code(double x) {
	double tmp;
	if (x <= 7.5e+107) {
		tmp = 0.5 * sqrt(pow((1.0 / x), 3.0));
	} else {
		tmp = pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 7.5d+107) then
        tmp = 0.5d0 * sqrt(((1.0d0 / x) ** 3.0d0))
    else
        tmp = x ** (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 7.5e+107) {
		tmp = 0.5 * Math.sqrt(Math.pow((1.0 / x), 3.0));
	} else {
		tmp = Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 7.5e+107:
		tmp = 0.5 * math.sqrt(math.pow((1.0 / x), 3.0))
	else:
		tmp = math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.5e+107)
		tmp = Float64(0.5 * sqrt((Float64(1.0 / x) ^ 3.0)));
	else
		tmp = x ^ -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 7.5e+107)
		tmp = 0.5 * sqrt(((1.0 / x) ^ 3.0));
	else
		tmp = x ^ -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 7.5e+107], N[(0.5 * N[Sqrt[N[Power[N[(1.0 / x), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[x, -1.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.4999999999999996e107

   1. Initial program 11.7%

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

      1. flip-- 11.4%

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

      2. clear-num 11.4%

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

      3. pow1/2 11.4%

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

      4. pow-flip 11.4%

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

      5. metadata-eval 11.4%

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

      6. inv-pow 11.4%

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

      7. sqrt-pow2 11.4%

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

      8. +-commutative 11.4%

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

      9. metadata-eval 11.4%

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

      10. frac-times 12.3%

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

      11. metadata-eval 12.3%

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

      12. add-sqr-sqrt 12.5%

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

      13. frac-times 12.0%

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

      14. metadata-eval 12.0%

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

      15. add-sqr-sqrt 12.1%

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

   4. Applied egg-rr 12.1%

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

   5. Taylor expanded in x around inf 88.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   6. Step-by-step derivation

      1. metadata-eval 88.0%

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

      2. cube-div 91.5%

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

   7. Simplified 91.5%

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

   if 7.4999999999999996e107 < x

   1. Initial program 52.4%

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

      1. sub-neg 52.4%

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

      2. flip-+ 52.4%

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

      3. frac-times 23.0%

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

      4. metadata-eval 23.0%

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

      5. add-sqr-sqrt 19.3%

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

      6. distribute-neg-frac 19.3%

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

      7. metadata-eval 19.3%

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

      8. +-commutative 19.3%

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

      9. distribute-neg-frac 19.3%

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

      10. metadata-eval 19.3%

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

      11. +-commutative 19.3%

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

      12. pow1/2 19.3%

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

      13. pow-flip 19.3%

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

      14. metadata-eval 19.3%

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

   4. Applied egg-rr 19.3%

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

      1. associate-*r/ 26.1%

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

      2. associate-*l/ 26.1%

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

      3. metadata-eval 26.1%

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

      4. associate-/l/ 34.4%

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

      5. rem-square-sqrt 52.4%

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

      6. sub-neg 52.4%

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

      7. distribute-neg-frac 52.4%

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

      8. metadata-eval 52.4%

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

      9. sub-neg 52.4%

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

      10. distribute-neg-frac 52.4%

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

      11. metadata-eval 52.4%

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

   6. Simplified 52.4%

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

      1. frac-add 52.4%

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

      2. *-un-lft-identity 52.4%

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

      3. distribute-rgt-in 52.4%

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

      4. *-un-lft-identity 52.4%

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

      5. pow2 52.4%

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

   8. Applied egg-rr 52.4%

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

      1. *-commutative 52.4%

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

      2. neg-mul-1 52.4%

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

      3. sub-neg 52.4%

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

      4. associate--l+ 72.8%

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

   10. Simplified 72.8%

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

   11. Taylor expanded in x around inf 52.4%

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

       1. metadata-eval 52.4%

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

       2. cube-div 52.4%

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

   13. Simplified 52.4%

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

       1. expm1-log1p-u 52.4%

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

       2. expm1-udef 52.4%

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

       3. sqrt-pow1 52.4%

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

       4. inv-pow 52.4%

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

       5. pow-pow 52.4%

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

       6. metadata-eval 52.4%

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

       7. metadata-eval 52.4%

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

   15. Applied egg-rr 52.4%

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

       1. expm1-def 58.1%

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

       2. expm1-log1p 58.1%

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

   17. Simplified 58.1%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \sqrt{{\left(\frac{1}{x}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 \cdot \left|{x}^{1.5}\right|} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* 2.0 (fabs (pow x 1.5)))))

C

double code(double x) {
	return 1.0 / (2.0 * fabs(pow(x, 1.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (2.0d0 * abs((x ** 1.5d0)))
end function

Java

public static double code(double x) {
	return 1.0 / (2.0 * Math.abs(Math.pow(x, 1.5)));
}

Python

def code(x):
	return 1.0 / (2.0 * math.fabs(math.pow(x, 1.5)))

Julia

function code(x)
	return Float64(1.0 / Float64(2.0 * abs((x ^ 1.5))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (2.0 * abs((x ^ 1.5)));
end

Wolfram

code[x_] := N[(1.0 / N[(2.0 * N[Abs[N[Power[x, 1.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. 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%

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

4. Applied egg-rr 38.8%

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

5. Taylor expanded in x around inf 64.3%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
6. Step-by-step derivation

   1. sqr-pow 64.2%

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

   2. rem-sqrt-square 96.8%

      \[\leadsto \frac{1}{2 \cdot \color{blue}{\left|{x}^{\left(\frac{3}{2}\right)}\right|}} \]

   3. metadata-eval 96.8%

      \[\leadsto \frac{1}{2 \cdot \left|{x}^{\color{blue}{1.5}}\right|} \]

7. Simplified 96.8%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|{x}^{1.5}\right|}} \]

8. Final simplification 96.8%

   \[\leadsto \frac{1}{2 \cdot \left|{x}^{1.5}\right|} \]
9. Add Preprocessing

Alternative 10: 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. Add Preprocessing
3. 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%

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

4. Applied egg-rr 17.0%

   \[\leadsto \color{blue}{\frac{\frac{1}{x} - \frac{-1}{\sqrt{1 + x}} \cdot \frac{-1}{\sqrt{1 + x}}}{{x}^{-0.5} - \frac{-1}{\sqrt{1 + x}}}} \]
5. 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%

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

6. Simplified 38.8%

   \[\leadsto \color{blue}{\frac{\frac{1}{x} + \frac{-1}{1 + x}}{{x}^{-0.5} + \frac{1}{\sqrt{1 + x}}}} \]
7. 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%

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

8. Applied egg-rr 41.0%

   \[\leadsto \frac{\color{blue}{\frac{\left(1 + x\right) + x \cdot -1}{x + {x}^{2}}}}{{x}^{-0.5} + \frac{1}{\sqrt{1 + x}}} \]
9. 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%

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

10. Simplified 81.7%

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

11. Taylor expanded in x around inf 40.7%

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

    1. metadata-eval 40.7%

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

    2. cube-div 41.1%

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

13. Simplified 41.1%

    \[\leadsto \color{blue}{\sqrt{{\left(\frac{1}{x}\right)}^{3}}} \]
14. 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%

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

15. Applied egg-rr 36.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({x}^{-1.5}\right)} - 1} \]
16. 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%

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

17. Simplified 44.9%

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

18. Final simplification 44.9%

    \[\leadsto {x}^{-1.5} \]
19. Add Preprocessing

Alternative 11: 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. Add Preprocessing
3. 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%

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

4. Applied egg-rr 17.0%

   \[\leadsto \color{blue}{\frac{\frac{1}{x} - \frac{-1}{\sqrt{1 + x}} \cdot \frac{-1}{\sqrt{1 + x}}}{{x}^{-0.5} - \frac{-1}{\sqrt{1 + x}}}} \]
5. 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%

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

6. Simplified 38.8%

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

7. Taylor expanded in x around 0 7.7%

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

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

9. Simplified 7.7%

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

10. Taylor expanded in x around inf 7.7%

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

11. Final simplification 7.7%

    \[\leadsto \frac{1}{x} \]
12. Add Preprocessing

Alternative 12: 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. Add Preprocessing

3. Taylor expanded in x around 0 2.6%

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

4. Taylor expanded in x around inf 2.6%

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

5. Final simplification 2.6%

   \[\leadsto -1 \]
6. Add Preprocessing

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)))))