2isqrt (example 3.6)

Percentage Accurate: 38.8% → 99.9%

Time: 10.7s

Alternatives: 10

Speedup: 2.0×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.8% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+17}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(x, x, x\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+17)
   (/ (/ 1.0 (fma x x x)) (+ (pow x -0.5) (pow (+ x 1.0) -0.5)))
   (* (pow x -1.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1e+17) {
		tmp = (1.0 / fma(x, x, x)) / (pow(x, -0.5) + pow((x + 1.0), -0.5));
	} else {
		tmp = pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+17)
		tmp = Float64(Float64(1.0 / fma(x, x, x)) / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)));
	else
		tmp = Float64((x ^ -1.5) * 0.5);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+17], N[(N[(1.0 / N[(x * x + x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e17

   1. Initial program 62.1%

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

      1. flip-- 62.7%

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

      2. div-inv 62.7%

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

      3. frac-times 63.4%

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

      4. metadata-eval 63.4%

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

      5. add-sqr-sqrt 63.1%

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

      6. frac-times 63.2%

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

      7. metadata-eval 63.2%

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

      8. add-sqr-sqrt 65.7%

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

      9. +-commutative 65.7%

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

      10. inv-pow 65.7%

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

      11. sqrt-pow2 65.7%

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

      12. metadata-eval 65.7%

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

      13. pow1/2 65.7%

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

      14. pow-flip 65.7%

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

      15. +-commutative 65.7%

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

      16. metadata-eval 65.7%

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

   4. Applied egg-rr 65.7%

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

      1. associate-*r/ 65.7%

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

      2. *-rgt-identity 65.7%

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

   6. Simplified 65.7%

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

      1. frac-sub 94.8%

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

      2. *-un-lft-identity 94.8%

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

      3. distribute-rgt-in 94.6%

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

      4. *-un-lft-identity 94.6%

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

      5. unpow2 94.6%

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

   8. Applied egg-rr 94.6%

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

      1. *-rgt-identity 94.6%

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

      2. associate--l+ 99.2%

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

      3. +-inverses 99.2%

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

      4. metadata-eval 99.2%

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

      5. +-commutative 99.2%

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

      6. unpow2 99.2%

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

      7. fma-define 99.4%

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

   10. Simplified 99.4%

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

   if 1e17 < x

   1. Initial program 38.9%

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

      1. flip-- 38.9%

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

      2. div-inv 38.9%

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

      3. frac-times 23.2%

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

      4. metadata-eval 23.2%

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

      5. add-sqr-sqrt 20.1%

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

      6. frac-times 25.5%

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

      7. metadata-eval 25.5%

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

      8. add-sqr-sqrt 38.9%

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

      9. +-commutative 38.9%

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

      10. inv-pow 38.9%

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

      11. sqrt-pow2 38.9%

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

      12. metadata-eval 38.9%

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

      13. pow1/2 38.9%

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

      14. pow-flip 38.9%

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

      15. +-commutative 38.9%

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

      16. metadata-eval 38.9%

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

   4. Applied egg-rr 38.9%

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

      1. associate-*r/ 38.9%

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

      2. *-rgt-identity 38.9%

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

   6. Simplified 38.9%

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

   7. Taylor expanded in x around inf 69.5%

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

      1. *-commutative 69.5%

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

      2. exp-to-pow 66.6%

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

      3. *-commutative 66.6%

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

      4. exp-neg 67.4%

         \[\leadsto \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \cdot 0.5 \]

      5. distribute-lft-neg-in 67.4%

         \[\leadsto \sqrt{e^{\color{blue}{\left(-3\right) \cdot \log x}}} \cdot 0.5 \]

      6. metadata-eval 67.4%

         \[\leadsto \sqrt{e^{\color{blue}{-3} \cdot \log x}} \cdot 0.5 \]

      7. *-commutative 67.4%

         \[\leadsto \sqrt{e^{\color{blue}{\log x \cdot -3}}} \cdot 0.5 \]

      8. exp-to-pow 70.4%

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

   9. Simplified 70.4%

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

       1. *-un-lft-identity 70.4%

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

       2. sqrt-pow1 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \cdot 0.5 \]

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \cdot 0.5 \]
   12. Step-by-step derivation

       1. *-lft-identity 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \color{blue}{{x}^{-1.5}} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+17}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(x, x, x\right)}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t\_0} \leq 0:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\sqrt{x} + t\_0\right) \cdot \left(x + 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 0.0)
     (* (pow x -1.5) 0.5)
     (/ 1.0 (* (+ (sqrt x) t_0) (+ x 0.5))))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 0.0) {
		tmp = pow(x, -1.5) * 0.5;
	} else {
		tmp = 1.0 / ((sqrt(x) + t_0) * (x + 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 0.0d0) then
        tmp = (x ** (-1.5d0)) * 0.5d0
    else
        tmp = 1.0d0 / ((sqrt(x) + t_0) * (x + 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 0.0) {
		tmp = Math.pow(x, -1.5) * 0.5;
	} else {
		tmp = 1.0 / ((Math.sqrt(x) + t_0) * (x + 0.5));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 0.0:
		tmp = math.pow(x, -1.5) * 0.5
	else:
		tmp = 1.0 / ((math.sqrt(x) + t_0) * (x + 0.5))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision], N[(1.0 / N[(N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 38.7%

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

      1. flip-- 38.7%

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

      2. div-inv 38.7%

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

      3. frac-times 23.1%

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

      4. metadata-eval 23.1%

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

      5. add-sqr-sqrt 20.0%

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

      6. frac-times 25.4%

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

      7. metadata-eval 25.4%

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

      8. add-sqr-sqrt 38.7%

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

      9. +-commutative 38.7%

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

      10. inv-pow 38.7%

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

      11. sqrt-pow2 38.7%

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

      12. metadata-eval 38.7%

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

      13. pow1/2 38.7%

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

      14. pow-flip 38.7%

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

      15. +-commutative 38.7%

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

      16. metadata-eval 38.7%

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

   4. Applied egg-rr 38.7%

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

      1. associate-*r/ 38.7%

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

      2. *-rgt-identity 38.7%

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

   6. Simplified 38.7%

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

   7. Taylor expanded in x around inf 69.6%

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

      1. *-commutative 69.6%

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

      2. exp-to-pow 66.7%

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

      3. *-commutative 66.7%

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

      4. exp-neg 67.6%

         \[\leadsto \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \cdot 0.5 \]

      5. distribute-lft-neg-in 67.6%

         \[\leadsto \sqrt{e^{\color{blue}{\left(-3\right) \cdot \log x}}} \cdot 0.5 \]

      6. metadata-eval 67.6%

         \[\leadsto \sqrt{e^{\color{blue}{-3} \cdot \log x}} \cdot 0.5 \]

      7. *-commutative 67.6%

         \[\leadsto \sqrt{e^{\color{blue}{\log x \cdot -3}}} \cdot 0.5 \]

      8. exp-to-pow 70.5%

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

   9. Simplified 70.5%

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

       1. *-un-lft-identity 70.5%

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

       2. sqrt-pow1 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \cdot 0.5 \]

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \cdot 0.5 \]
   12. Step-by-step derivation

       1. *-lft-identity 100.0%

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

   13. Simplified 100.0%

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

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

   1. Initial program 65.0%

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

      1. frac-sub 66.1%

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

      2. *-rgt-identity 66.1%

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

      3. *-un-lft-identity 66.1%

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

      4. +-commutative 66.1%

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

      5. sqrt-unprod 66.1%

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

      6. +-commutative 66.1%

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

   4. Applied egg-rr 66.1%

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

   5. Taylor expanded in x around inf 52.5%

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

      1. distribute-lft-in 52.5%

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

      2. *-rgt-identity 52.5%

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

      3. *-commutative 52.5%

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

      4. associate-*r* 52.5%

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

      5. rgt-mult-inverse 52.5%

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

      6. metadata-eval 52.5%

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

   7. Simplified 52.5%

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

      1. flip-- 73.1%

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

      2. add-sqr-sqrt 85.9%

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

      3. add-sqr-sqrt 99.4%

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

      4. add-sqr-sqrt 99.3%

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

      5. hypot-1-def 99.2%

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

   9. Applied egg-rr 78.4%

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

       1. associate--l+ 99.2%

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

       2. +-inverses 99.2%

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

       3. metadata-eval 99.2%

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

       4. +-commutative 99.2%

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

       5. hypot-undefine 99.3%

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

       6. metadata-eval 99.3%

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

       7. rem-square-sqrt 99.4%

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

   11. Simplified 78.5%

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

       1. *-un-lft-identity 78.5%

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

       2. associate-/l/ 78.4%

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

       3. *-commutative 78.4%

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

   13. Applied egg-rr 78.4%

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

       1. *-lft-identity 78.4%

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

   15. Simplified 78.4%

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

4. Final simplification 98.3%

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

Alternative 3: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{x}}\\ \mathbf{if}\;t\_0 + \frac{-1}{\sqrt{x + 1}} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \sqrt{\frac{1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt x))))
   (if (<= (+ t_0 (/ -1.0 (sqrt (+ x 1.0)))) 2e-14)
     (* (pow x -1.5) 0.5)
     (- t_0 (sqrt (/ 1.0 (+ x 1.0)))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(x);
	double tmp;
	if ((t_0 + (-1.0 / sqrt((x + 1.0)))) <= 2e-14) {
		tmp = pow(x, -1.5) * 0.5;
	} else {
		tmp = t_0 - sqrt((1.0 / (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / sqrt(x)
    if ((t_0 + ((-1.0d0) / sqrt((x + 1.0d0)))) <= 2d-14) then
        tmp = (x ** (-1.5d0)) * 0.5d0
    else
        tmp = t_0 - sqrt((1.0d0 / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.sqrt(x);
	double tmp;
	if ((t_0 + (-1.0 / Math.sqrt((x + 1.0)))) <= 2e-14) {
		tmp = Math.pow(x, -1.5) * 0.5;
	} else {
		tmp = t_0 - Math.sqrt((1.0 / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / math.sqrt(x)
	tmp = 0
	if (t_0 + (-1.0 / math.sqrt((x + 1.0)))) <= 2e-14:
		tmp = math.pow(x, -1.5) * 0.5
	else:
		tmp = t_0 - math.sqrt((1.0 / (x + 1.0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(x))
	tmp = 0.0
	if (Float64(t_0 + Float64(-1.0 / sqrt(Float64(x + 1.0)))) <= 2e-14)
		tmp = Float64((x ^ -1.5) * 0.5);
	else
		tmp = Float64(t_0 - sqrt(Float64(1.0 / Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / sqrt(x);
	tmp = 0.0;
	if ((t_0 + (-1.0 / sqrt((x + 1.0)))) <= 2e-14)
		tmp = (x ^ -1.5) * 0.5;
	else
		tmp = t_0 - sqrt((1.0 / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-14], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision], N[(t$95$0 - N[Sqrt[N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 2e-14

   1. Initial program 38.9%

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

      1. flip-- 38.9%

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

      2. div-inv 38.9%

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

      3. frac-times 23.8%

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

      4. metadata-eval 23.8%

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

      5. add-sqr-sqrt 20.8%

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

      6. frac-times 26.1%

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

      7. metadata-eval 26.1%

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

      8. add-sqr-sqrt 39.0%

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

      9. +-commutative 39.0%

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

      10. inv-pow 39.0%

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

      11. sqrt-pow2 39.0%

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

      12. metadata-eval 39.0%

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

      13. pow1/2 39.0%

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

      14. pow-flip 39.0%

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

      15. +-commutative 39.0%

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

      16. metadata-eval 39.0%

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

   4. Applied egg-rr 39.0%

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

      1. associate-*r/ 39.0%

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

      2. *-rgt-identity 39.0%

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

   6. Simplified 39.0%

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

   7. Taylor expanded in x around inf 69.8%

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

      1. *-commutative 69.8%

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

      2. exp-to-pow 67.0%

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

      3. *-commutative 67.0%

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

      4. exp-neg 67.8%

         \[\leadsto \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \cdot 0.5 \]

      5. distribute-lft-neg-in 67.8%

         \[\leadsto \sqrt{e^{\color{blue}{\left(-3\right) \cdot \log x}}} \cdot 0.5 \]

      6. metadata-eval 67.8%

         \[\leadsto \sqrt{e^{\color{blue}{-3} \cdot \log x}} \cdot 0.5 \]

      7. *-commutative 67.8%

         \[\leadsto \sqrt{e^{\color{blue}{\log x \cdot -3}}} \cdot 0.5 \]

      8. exp-to-pow 70.6%

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

   9. Simplified 70.6%

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

       1. *-un-lft-identity 70.6%

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

       2. sqrt-pow1 99.2%

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

       3. metadata-eval 99.2%

          \[\leadsto \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \cdot 0.5 \]

   11. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \cdot 0.5 \]
   12. Step-by-step derivation

       1. *-lft-identity 99.2%

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

   13. Simplified 99.2%

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

   if 2e-14 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 79.3%

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

      1. add-sqr-sqrt 78.1%

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

      2. sqrt-unprod 79.3%

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

      3. frac-times 79.1%

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

      4. metadata-eval 79.1%

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

      5. add-sqr-sqrt 79.8%

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

      6. +-commutative 79.8%

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

   4. Applied egg-rr 79.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x}} - \sqrt{\frac{1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x \cdot \left(x + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 8e+15)
   (/ (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))) (sqrt (* x (+ x 1.0))))
   (* (pow x -1.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 8e+15) {
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) / sqrt((x * (x + 1.0)));
	} else {
		tmp = pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8d+15) then
        tmp = (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) / sqrt((x * (x + 1.0d0)))
    else
        tmp = (x ** (-1.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8e+15) {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) / Math.sqrt((x * (x + 1.0)));
	} else {
		tmp = Math.pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8e+15:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) / math.sqrt((x * (x + 1.0)))
	else:
		tmp = math.pow(x, -1.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8e+15)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) / sqrt(Float64(x * Float64(x + 1.0))));
	else
		tmp = Float64((x ^ -1.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8e+15)
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) / sqrt((x * (x + 1.0)));
	else
		tmp = (x ^ -1.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8e+15], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8e15

   1. Initial program 65.0%

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

      1. frac-sub 66.1%

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

      2. *-rgt-identity 66.1%

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

      3. *-un-lft-identity 66.1%

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

      4. +-commutative 66.1%

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

      5. sqrt-unprod 66.1%

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

      6. +-commutative 66.1%

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

   4. Applied egg-rr 66.1%

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

      1. flip-- 73.1%

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

      2. add-sqr-sqrt 85.9%

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

      3. add-sqr-sqrt 99.4%

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

      4. add-sqr-sqrt 99.3%

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

      5. hypot-1-def 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. associate--l+ 99.2%

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

      2. +-inverses 99.2%

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

      3. metadata-eval 99.2%

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

      4. +-commutative 99.2%

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

      5. hypot-undefine 99.3%

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

      6. metadata-eval 99.3%

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

      7. rem-square-sqrt 99.4%

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

   8. Simplified 99.4%

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

   if 8e15 < x

   1. Initial program 38.7%

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

      1. flip-- 38.7%

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

      2. div-inv 38.7%

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

      3. frac-times 23.1%

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

      4. metadata-eval 23.1%

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

      5. add-sqr-sqrt 20.0%

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

      6. frac-times 25.4%

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

      7. metadata-eval 25.4%

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

      8. add-sqr-sqrt 38.7%

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

      9. +-commutative 38.7%

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

      10. inv-pow 38.7%

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

      11. sqrt-pow2 38.7%

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

      12. metadata-eval 38.7%

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

      13. pow1/2 38.7%

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

      14. pow-flip 38.7%

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

      15. +-commutative 38.7%

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

      16. metadata-eval 38.7%

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

   4. Applied egg-rr 38.7%

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

      1. associate-*r/ 38.7%

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

      2. *-rgt-identity 38.7%

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

   6. Simplified 38.7%

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

   7. Taylor expanded in x around inf 69.6%

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

      1. *-commutative 69.6%

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

      2. exp-to-pow 66.7%

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

      3. *-commutative 66.7%

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

      4. exp-neg 67.6%

         \[\leadsto \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \cdot 0.5 \]

      5. distribute-lft-neg-in 67.6%

         \[\leadsto \sqrt{e^{\color{blue}{\left(-3\right) \cdot \log x}}} \cdot 0.5 \]

      6. metadata-eval 67.6%

         \[\leadsto \sqrt{e^{\color{blue}{-3} \cdot \log x}} \cdot 0.5 \]

      7. *-commutative 67.6%

         \[\leadsto \sqrt{e^{\color{blue}{\log x \cdot -3}}} \cdot 0.5 \]

      8. exp-to-pow 70.5%

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

   9. Simplified 70.5%

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

       1. *-un-lft-identity 70.5%

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

       2. sqrt-pow1 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \cdot 0.5 \]

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \cdot 0.5 \]
   12. Step-by-step derivation

       1. *-lft-identity 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \color{blue}{{x}^{-1.5}} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x \cdot \left(x + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 90000000:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 90000000.0)
   (- (pow x -0.5) (sqrt (/ 1.0 (+ x 1.0))))
   (* (pow x -1.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 90000000.0) {
		tmp = pow(x, -0.5) - sqrt((1.0 / (x + 1.0)));
	} else {
		tmp = pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 90000000.0d0) then
        tmp = (x ** (-0.5d0)) - sqrt((1.0d0 / (x + 1.0d0)))
    else
        tmp = (x ** (-1.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 90000000.0) {
		tmp = Math.pow(x, -0.5) - Math.sqrt((1.0 / (x + 1.0)));
	} else {
		tmp = Math.pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 90000000.0:
		tmp = math.pow(x, -0.5) - math.sqrt((1.0 / (x + 1.0)))
	else:
		tmp = math.pow(x, -1.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 90000000.0)
		tmp = Float64((x ^ -0.5) - sqrt(Float64(1.0 / Float64(x + 1.0))));
	else
		tmp = Float64((x ^ -1.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 90000000.0)
		tmp = (x ^ -0.5) - sqrt((1.0 / (x + 1.0)));
	else
		tmp = (x ^ -1.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 90000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Sqrt[N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9e7

   1. Initial program 79.3%

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

      1. sub-neg 79.3%

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

      2. inv-pow 79.3%

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

      3. sqrt-pow2 79.3%

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

      4. metadata-eval 79.3%

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

      5. distribute-neg-frac 79.3%

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

      6. metadata-eval 79.3%

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

      7. +-commutative 79.3%

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

   4. Applied egg-rr 79.3%

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

      1. *-rgt-identity 79.3%

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

      2. cancel-sign-sub 79.3%

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

      3. distribute-lft-neg-in 79.3%

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

      4. *-rgt-identity 79.3%

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

      5. distribute-neg-frac 79.3%

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

      6. metadata-eval 79.3%

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

      7. unpow1/2 79.3%

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

      8. exp-to-pow 78.0%

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

      9. log1p-undefine 78.0%

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

      10. *-commutative 78.0%

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

      11. exp-neg 78.2%

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

      12. *-commutative 78.2%

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

      13. distribute-rgt-neg-in 78.2%

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

      14. log1p-undefine 78.2%

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

      15. metadata-eval 78.2%

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

      16. exp-to-pow 79.4%

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

   6. Simplified 79.4%

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

      1. add-sqr-sqrt 78.4%

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

      2. sqrt-unprod 79.4%

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

      3. pow-prod-up 79.8%

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

      4. metadata-eval 79.8%

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

   8. Applied egg-rr 79.8%

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

      1. unpow-1 79.8%

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

   10. Simplified 79.8%

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

   if 9e7 < x

   1. Initial program 38.9%

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

      1. flip-- 38.9%

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

      2. div-inv 38.9%

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

      3. frac-times 23.8%

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

      4. metadata-eval 23.8%

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

      5. add-sqr-sqrt 20.8%

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

      6. frac-times 26.1%

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

      7. metadata-eval 26.1%

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

      8. add-sqr-sqrt 39.0%

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

      9. +-commutative 39.0%

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

      10. inv-pow 39.0%

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

      11. sqrt-pow2 39.0%

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

      12. metadata-eval 39.0%

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

      13. pow1/2 39.0%

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

      14. pow-flip 39.0%

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

      15. +-commutative 39.0%

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

      16. metadata-eval 39.0%

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

   4. Applied egg-rr 39.0%

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

      1. associate-*r/ 39.0%

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

      2. *-rgt-identity 39.0%

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

   6. Simplified 39.0%

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

   7. Taylor expanded in x around inf 69.8%

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

      1. *-commutative 69.8%

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

      2. exp-to-pow 67.0%

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

      3. *-commutative 67.0%

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

      4. exp-neg 67.8%

         \[\leadsto \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \cdot 0.5 \]

      5. distribute-lft-neg-in 67.8%

         \[\leadsto \sqrt{e^{\color{blue}{\left(-3\right) \cdot \log x}}} \cdot 0.5 \]

      6. metadata-eval 67.8%

         \[\leadsto \sqrt{e^{\color{blue}{-3} \cdot \log x}} \cdot 0.5 \]

      7. *-commutative 67.8%

         \[\leadsto \sqrt{e^{\color{blue}{\log x \cdot -3}}} \cdot 0.5 \]

      8. exp-to-pow 70.6%

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

   9. Simplified 70.6%

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

       1. *-un-lft-identity 70.6%

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

       2. sqrt-pow1 99.2%

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

       3. metadata-eval 99.2%

          \[\leadsto \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \cdot 0.5 \]

   11. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \cdot 0.5 \]
   12. Step-by-step derivation

       1. *-lft-identity 99.2%

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

   13. Simplified 99.2%

       \[\leadsto \color{blue}{{x}^{-1.5}} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 90000000:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 90000000:\\ \;\;\;\;\frac{1}{\sqrt{x}} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 90000000.0)
   (- (/ 1.0 (sqrt x)) (pow (+ x 1.0) -0.5))
   (* (pow x -1.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 90000000.0) {
		tmp = (1.0 / sqrt(x)) - pow((x + 1.0), -0.5);
	} else {
		tmp = pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 90000000.0d0) then
        tmp = (1.0d0 / sqrt(x)) - ((x + 1.0d0) ** (-0.5d0))
    else
        tmp = (x ** (-1.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 90000000.0) {
		tmp = (1.0 / Math.sqrt(x)) - Math.pow((x + 1.0), -0.5);
	} else {
		tmp = Math.pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 90000000.0:
		tmp = (1.0 / math.sqrt(x)) - math.pow((x + 1.0), -0.5)
	else:
		tmp = math.pow(x, -1.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 90000000.0)
		tmp = Float64(Float64(1.0 / sqrt(x)) - (Float64(x + 1.0) ^ -0.5));
	else
		tmp = Float64((x ^ -1.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 90000000.0)
		tmp = (1.0 / sqrt(x)) - ((x + 1.0) ^ -0.5);
	else
		tmp = (x ^ -1.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 90000000.0], N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9e7

   1. Initial program 79.3%

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

      1. frac-2neg 79.3%

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

      2. metadata-eval 79.3%

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

      3. div-inv 79.3%

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

      4. frac-2neg 79.3%

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

      5. metadata-eval 79.3%

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

      6. div-inv 79.3%

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

      7. distribute-neg-frac2 79.3%

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

      8. prod-diff 79.3%

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

      9. distribute-neg-frac 79.3%

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

      10. metadata-eval 79.3%

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

      11. +-commutative 79.3%

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

   4. Applied egg-rr 79.4%

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

   6. Simplified 79.4%

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

   if 9e7 < x

   1. Initial program 38.9%

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

      1. flip-- 38.9%

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

      2. div-inv 38.9%

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

      3. frac-times 23.8%

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

      4. metadata-eval 23.8%

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

      5. add-sqr-sqrt 20.8%

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

      6. frac-times 26.1%

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

      7. metadata-eval 26.1%

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

      8. add-sqr-sqrt 39.0%

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

      9. +-commutative 39.0%

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

      10. inv-pow 39.0%

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

      11. sqrt-pow2 39.0%

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

      12. metadata-eval 39.0%

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

      13. pow1/2 39.0%

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

      14. pow-flip 39.0%

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

      15. +-commutative 39.0%

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

      16. metadata-eval 39.0%

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

   4. Applied egg-rr 39.0%

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

      1. associate-*r/ 39.0%

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

      2. *-rgt-identity 39.0%

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

   6. Simplified 39.0%

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

   7. Taylor expanded in x around inf 69.8%

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

      1. *-commutative 69.8%

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

      2. exp-to-pow 67.0%

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

      3. *-commutative 67.0%

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

      4. exp-neg 67.8%

         \[\leadsto \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \cdot 0.5 \]

      5. distribute-lft-neg-in 67.8%

         \[\leadsto \sqrt{e^{\color{blue}{\left(-3\right) \cdot \log x}}} \cdot 0.5 \]

      6. metadata-eval 67.8%

         \[\leadsto \sqrt{e^{\color{blue}{-3} \cdot \log x}} \cdot 0.5 \]

      7. *-commutative 67.8%

         \[\leadsto \sqrt{e^{\color{blue}{\log x \cdot -3}}} \cdot 0.5 \]

      8. exp-to-pow 70.6%

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

   9. Simplified 70.6%

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

       1. *-un-lft-identity 70.6%

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

       2. sqrt-pow1 99.2%

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

       3. metadata-eval 99.2%

          \[\leadsto \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \cdot 0.5 \]

   11. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \cdot 0.5 \]
   12. Step-by-step derivation

       1. *-lft-identity 99.2%

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

   13. Simplified 99.2%

       \[\leadsto \color{blue}{{x}^{-1.5}} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 90000000:\\ \;\;\;\;\frac{1}{\sqrt{x}} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 90000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 90000000.0)
   (- (pow x -0.5) (pow (+ x 1.0) -0.5))
   (* (pow x -1.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 90000000.0) {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	} else {
		tmp = pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 90000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    else
        tmp = (x ** (-1.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 90000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	} else {
		tmp = Math.pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 90000000.0:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	else:
		tmp = math.pow(x, -1.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 90000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	else
		tmp = Float64((x ^ -1.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 90000000.0)
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	else
		tmp = (x ^ -1.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 90000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9e7

   1. Initial program 79.3%

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

      1. sub-neg 79.3%

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

      2. inv-pow 79.3%

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

      3. sqrt-pow2 79.3%

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

      4. metadata-eval 79.3%

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

      5. distribute-neg-frac 79.3%

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

      6. metadata-eval 79.3%

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

      7. +-commutative 79.3%

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

   4. Applied egg-rr 79.3%

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

      1. *-rgt-identity 79.3%

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

      2. cancel-sign-sub 79.3%

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

      3. distribute-lft-neg-in 79.3%

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

      4. *-rgt-identity 79.3%

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

      5. distribute-neg-frac 79.3%

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

      6. metadata-eval 79.3%

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

      7. unpow1/2 79.3%

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

      8. exp-to-pow 78.0%

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

      9. log1p-undefine 78.0%

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

      10. *-commutative 78.0%

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

      11. exp-neg 78.2%

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

      12. *-commutative 78.2%

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

      13. distribute-rgt-neg-in 78.2%

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

      14. log1p-undefine 78.2%

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

      15. metadata-eval 78.2%

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

      16. exp-to-pow 79.4%

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

   6. Simplified 79.4%

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

   if 9e7 < x

   1. Initial program 38.9%

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

      1. flip-- 38.9%

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

      2. div-inv 38.9%

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

      3. frac-times 23.8%

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

      4. metadata-eval 23.8%

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

      5. add-sqr-sqrt 20.8%

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

      6. frac-times 26.1%

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

      7. metadata-eval 26.1%

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

      8. add-sqr-sqrt 39.0%

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

      9. +-commutative 39.0%

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

      10. inv-pow 39.0%

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

      11. sqrt-pow2 39.0%

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

      12. metadata-eval 39.0%

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

      13. pow1/2 39.0%

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

      14. pow-flip 39.0%

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

      15. +-commutative 39.0%

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

      16. metadata-eval 39.0%

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

   4. Applied egg-rr 39.0%

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

      1. associate-*r/ 39.0%

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

      2. *-rgt-identity 39.0%

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

   6. Simplified 39.0%

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

   7. Taylor expanded in x around inf 69.8%

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

      1. *-commutative 69.8%

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

      2. exp-to-pow 67.0%

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

      3. *-commutative 67.0%

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

      4. exp-neg 67.8%

         \[\leadsto \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \cdot 0.5 \]

      5. distribute-lft-neg-in 67.8%

         \[\leadsto \sqrt{e^{\color{blue}{\left(-3\right) \cdot \log x}}} \cdot 0.5 \]

      6. metadata-eval 67.8%

         \[\leadsto \sqrt{e^{\color{blue}{-3} \cdot \log x}} \cdot 0.5 \]

      7. *-commutative 67.8%

         \[\leadsto \sqrt{e^{\color{blue}{\log x \cdot -3}}} \cdot 0.5 \]

      8. exp-to-pow 70.6%

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

   9. Simplified 70.6%

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

       1. *-un-lft-identity 70.6%

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

       2. sqrt-pow1 99.2%

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

       3. metadata-eval 99.2%

          \[\leadsto \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \cdot 0.5 \]

   11. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \cdot 0.5 \]
   12. Step-by-step derivation

       1. *-lft-identity 99.2%

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

   13. Simplified 99.2%

       \[\leadsto \color{blue}{{x}^{-1.5}} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 90000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

   1. frac-sub 40.9%

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

   2. *-rgt-identity 40.9%

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

   3. *-un-lft-identity 40.9%

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

   4. +-commutative 40.9%

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

   5. sqrt-unprod 40.9%

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

   6. +-commutative 40.9%

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

4. Applied egg-rr 40.9%

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

5. Taylor expanded in x around inf 39.8%

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

   1. distribute-lft-in 39.8%

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

   2. *-rgt-identity 39.8%

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

   3. *-commutative 39.8%

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

   4. associate-*r* 39.8%

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

   5. rgt-mult-inverse 39.8%

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

   6. metadata-eval 39.8%

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

7. Simplified 39.8%

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

   1. flip-- 41.4%

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

   2. add-sqr-sqrt 42.6%

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

   3. add-sqr-sqrt 43.5%

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

   4. add-sqr-sqrt 43.5%

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

   5. hypot-1-def 43.5%

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

9. Applied egg-rr 41.8%

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

    1. associate--l+ 83.9%

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

    2. +-inverses 83.9%

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

    3. metadata-eval 83.9%

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

    4. +-commutative 83.9%

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

    5. hypot-undefine 83.9%

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

    6. metadata-eval 83.9%

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

    7. rem-square-sqrt 83.9%

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

11. Simplified 97.9%

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

12. Final simplification 97.9%

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

Alternative 9: 97.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

   1. flip-- 40.8%

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

   2. div-inv 40.8%

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

   3. frac-times 26.5%

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

   4. metadata-eval 26.5%

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

   5. add-sqr-sqrt 23.6%

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

   6. frac-times 28.6%

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

   7. metadata-eval 28.6%

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

   8. add-sqr-sqrt 41.1%

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

   9. +-commutative 41.1%

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

   10. inv-pow 41.1%

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

   11. sqrt-pow2 41.1%

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

   12. metadata-eval 41.1%

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

   13. pow1/2 41.1%

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

   14. pow-flip 41.1%

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

   15. +-commutative 41.1%

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

   16. metadata-eval 41.1%

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

4. Applied egg-rr 41.1%

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

   1. associate-*r/ 41.1%

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

   2. *-rgt-identity 41.1%

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

6. Simplified 41.1%

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

7. Taylor expanded in x around inf 68.4%

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

   1. *-commutative 68.4%

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

   2. exp-to-pow 65.7%

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

   3. *-commutative 65.7%

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

   4. exp-neg 66.5%

      \[\leadsto \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \cdot 0.5 \]

   5. distribute-lft-neg-in 66.5%

      \[\leadsto \sqrt{e^{\color{blue}{\left(-3\right) \cdot \log x}}} \cdot 0.5 \]

   6. metadata-eval 66.5%

      \[\leadsto \sqrt{e^{\color{blue}{-3} \cdot \log x}} \cdot 0.5 \]

   7. *-commutative 66.5%

      \[\leadsto \sqrt{e^{\color{blue}{\log x \cdot -3}}} \cdot 0.5 \]

   8. exp-to-pow 69.2%

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

9. Simplified 69.2%

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

    1. *-un-lft-identity 69.2%

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

    2. sqrt-pow1 96.4%

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

    3. metadata-eval 96.4%

       \[\leadsto \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \cdot 0.5 \]

11. Applied egg-rr 96.4%

    \[\leadsto \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \cdot 0.5 \]
12. Step-by-step derivation

    1. *-lft-identity 96.4%

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

13. Simplified 96.4%

    \[\leadsto \color{blue}{{x}^{-1.5}} \cdot 0.5 \]
14. Add Preprocessing

Alternative 10: 5.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 40.8%

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

   1. flip-- 40.8%

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

   2. div-inv 40.8%

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

   3. frac-times 26.5%

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

   4. metadata-eval 26.5%

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

   5. add-sqr-sqrt 23.6%

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

   6. frac-times 28.6%

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

   7. metadata-eval 28.6%

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

   8. add-sqr-sqrt 41.1%

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

   9. +-commutative 41.1%

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

   10. inv-pow 41.1%

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

   11. sqrt-pow2 41.1%

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

   12. metadata-eval 41.1%

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

   13. pow1/2 41.1%

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

   14. pow-flip 41.1%

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

   15. +-commutative 41.1%

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

   16. metadata-eval 41.1%

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

4. Applied egg-rr 41.1%

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

   1. associate-*r/ 41.1%

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

   2. *-rgt-identity 41.1%

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

6. Simplified 41.1%

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

7. Taylor expanded in x around 0 5.8%

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

   1. unpow1/2 5.8%

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

   2. rem-exp-log 5.8%

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

   3. exp-neg 5.8%

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

   4. exp-prod 5.8%

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

   5. distribute-lft-neg-out 5.8%

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

   6. distribute-rgt-neg-in 5.8%

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

   7. metadata-eval 5.8%

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

   8. exp-to-pow 5.8%

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

9. Simplified 5.8%

   \[\leadsto \color{blue}{{x}^{-0.5}} \]
10. Add Preprocessing

Developer Target 1: 98.2% 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 2024137 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

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