2isqrt (example 3.6)

Percentage Accurate: 68.4% → 99.8%

Time: 15.4s

Alternatives: 12

Speedup: 1.9×

Specification

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: 68.4% 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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(1 + x\right) + \sqrt{x \cdot \left(1 + x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} \cdot {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+105)
   (/ (pow x -0.5) (+ (+ 1.0 x) (sqrt (* x (+ 1.0 x)))))
   (* (/ 0.5 x) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 2e+105) {
		tmp = pow(x, -0.5) / ((1.0 + x) + sqrt((x * (1.0 + x))));
	} else {
		tmp = (0.5 / x) * pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2d+105) then
        tmp = (x ** (-0.5d0)) / ((1.0d0 + x) + sqrt((x * (1.0d0 + x))))
    else
        tmp = (0.5d0 / x) * ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e+105) {
		tmp = Math.pow(x, -0.5) / ((1.0 + x) + Math.sqrt((x * (1.0 + x))));
	} else {
		tmp = (0.5 / x) * Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e+105:
		tmp = math.pow(x, -0.5) / ((1.0 + x) + math.sqrt((x * (1.0 + x))))
	else:
		tmp = (0.5 / x) * math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+105)
		tmp = Float64((x ^ -0.5) / Float64(Float64(1.0 + x) + sqrt(Float64(x * Float64(1.0 + x)))));
	else
		tmp = Float64(Float64(0.5 / x) * (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e+105)
		tmp = (x ^ -0.5) / ((1.0 + x) + sqrt((x * (1.0 + x))));
	else
		tmp = (0.5 / x) * ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e+105], N[(N[Power[x, -0.5], $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] + N[Sqrt[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] * N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e105

   1. Initial program 71.4%

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

      1. frac-sub 71.5%

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

      2. *-un-lft-identity 71.5%

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

      3. +-commutative 71.5%

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

      4. *-rgt-identity 71.5%

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

      5. sqrt-unprod 71.5%

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

      6. +-commutative 71.5%

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

   4. Applied egg-rr 71.5%

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

      1. flip-- 72.1%

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

      2. add-sqr-sqrt 71.9%

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

      3. add-sqr-sqrt 72.5%

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

      4. +-commutative 72.5%

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

      5. +-commutative 72.5%

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

   6. Applied egg-rr 72.5%

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

      1. div-inv 72.5%

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

      2. +-commutative 72.5%

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

      3. sqrt-prod 72.4%

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

      4. times-frac 72.4%

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

      5. associate--l+ 72.4%

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

   8. Applied egg-rr 72.4%

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

      1. associate-+r- 72.4%

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

      2. +-commutative 72.4%

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

      3. associate-+r- 99.2%

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

      4. +-inverses 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-*l/ 99.3%

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

      7. *-lft-identity 99.3%

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

      8. associate-/l/ 99.3%

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

      9. distribute-lft-in 99.3%

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

      10. rem-square-sqrt 99.5%

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

      11. +-commutative 99.5%

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

      12. +-commutative 99.5%

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

   10. Simplified 99.5%

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

       1. div-inv 99.4%

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

       2. +-commutative 99.4%

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

       3. +-commutative 99.4%

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

       4. sqrt-unprod 99.5%

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

       5. +-commutative 99.5%

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

       6. pow1/2 99.5%

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

       7. pow-flip 99.8%

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

       8. metadata-eval 99.8%

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

   12. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. *-lft-identity 99.8%

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

       3. +-commutative 99.8%

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

       4. +-commutative 99.8%

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

       5. *-commutative 99.8%

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

       6. +-commutative 99.8%

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

   14. Simplified 99.8%

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

   if 1.9999999999999999e105 < x

   1. Initial program 50.3%

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

      1. frac-sub 50.3%

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

      2. associate-/r* 50.3%

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

      3. *-un-lft-identity 50.3%

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

      4. +-commutative 50.3%

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

      5. *-rgt-identity 50.3%

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

      6. +-commutative 50.3%

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

   4. Applied egg-rr 50.3%

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

      1. div-sub 50.3%

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

      2. sqrt-undiv 50.3%

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

      3. +-commutative 50.3%

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

      4. *-inverses 50.3%

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

   6. Applied egg-rr 50.3%

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

   7. Taylor expanded in x around inf 50.3%

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

      1. associate-*r/ 50.3%

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

      2. metadata-eval 50.3%

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

   9. Simplified 50.3%

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

       1. div-inv 50.3%

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

       2. add-exp-log 50.3%

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

       3. log1p-udef 50.3%

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

       4. expm1-udef 99.6%

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

       5. expm1-log1p-u 99.6%

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

       6. pow1/2 99.6%

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

       7. +-commutative 99.6%

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

       8. pow-flip 99.7%

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

       9. +-commutative 99.7%

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

       10. metadata-eval 99.7%

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

   11. Applied egg-rr 99.7%

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

       1. +-commutative 99.7%

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

   13. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(1 + x\right) + \sqrt{x \cdot \left(1 + x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} \cdot {\left(1 + x\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

   1. frac-sub 64.0%

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

   2. *-un-lft-identity 64.0%

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

   3. +-commutative 64.0%

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

   4. *-rgt-identity 64.0%

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

   5. sqrt-unprod 64.0%

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

   6. +-commutative 64.0%

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

4. Applied egg-rr 64.0%

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

   1. flip-- 64.5%

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

   2. add-sqr-sqrt 64.2%

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

   3. add-sqr-sqrt 64.7%

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

   4. +-commutative 64.7%

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

   5. +-commutative 64.7%

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

6. Applied egg-rr 64.7%

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

   1. div-inv 64.7%

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

   2. +-commutative 64.7%

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

   3. sqrt-prod 64.7%

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

   4. times-frac 64.7%

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

   5. associate--l+ 64.7%

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

8. Applied egg-rr 64.7%

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

   1. associate-+r- 64.7%

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

   2. +-commutative 64.7%

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

   3. associate-+r- 99.3%

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

   4. +-inverses 99.3%

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

   5. metadata-eval 99.3%

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

   6. associate-*l/ 99.3%

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

   7. *-lft-identity 99.3%

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

   8. associate-/l/ 99.3%

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

   9. distribute-lft-in 99.3%

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

   10. rem-square-sqrt 99.5%

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

   11. +-commutative 99.5%

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

   12. +-commutative 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

    \[\leadsto \frac{\frac{1}{\sqrt{1 + x} \cdot \sqrt{x} + \left(1 + x\right)}}{\sqrt{x}} \]
12. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3550:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3550.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (/ (/ 1.0 (+ 1.5 (- (* x 2.0) (/ 0.125 x)))) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 3550.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = (1.0 / (1.5 + ((x * 2.0) - (0.125 / x)))) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3550.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = (1.0d0 / (1.5d0 + ((x * 2.0d0) - (0.125d0 / x)))) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3550.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = (1.0 / (1.5 + ((x * 2.0) - (0.125 / x)))) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3550.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = (1.0 / (1.5 + ((x * 2.0) - (0.125 / x)))) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3550.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(Float64(1.0 / Float64(1.5 + Float64(Float64(x * 2.0) - Float64(0.125 / x)))) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3550.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = (1.0 / (1.5 + ((x * 2.0) - (0.125 / x)))) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3550.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.5 + N[(N[(x * 2.0), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3550

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul0-lft 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+r+ 100.0%

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

      9. sub-neg 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   if 3550 < x

   1. Initial program 35.4%

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

      1. frac-sub 35.5%

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

      2. *-un-lft-identity 35.5%

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

      3. +-commutative 35.5%

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

      4. *-rgt-identity 35.5%

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

      5. sqrt-unprod 35.5%

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

      6. +-commutative 35.5%

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

   4. Applied egg-rr 35.5%

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

      1. flip-- 36.3%

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

      2. add-sqr-sqrt 35.9%

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

      3. add-sqr-sqrt 36.7%

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

      4. +-commutative 36.7%

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

      5. +-commutative 36.7%

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

   6. Applied egg-rr 36.7%

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

      1. div-inv 36.7%

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

      2. +-commutative 36.7%

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

      3. sqrt-prod 36.7%

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

      4. times-frac 36.7%

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

      5. associate--l+ 36.7%

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

   8. Applied egg-rr 36.7%

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

      1. associate-+r- 36.7%

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

      2. +-commutative 36.7%

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

      3. associate-+r- 99.1%

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

      4. +-inverses 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-*l/ 99.2%

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

      7. *-lft-identity 99.2%

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

      8. associate-/l/ 99.2%

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

      9. distribute-lft-in 99.2%

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

      10. rem-square-sqrt 99.4%

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

      11. +-commutative 99.4%

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

      12. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 99.5%

       \[\leadsto \frac{\frac{1}{\color{blue}{\left(1.5 + 2 \cdot x\right) - 0.125 \cdot \frac{1}{x}}}}{\sqrt{x}} \]
   12. Step-by-step derivation

       1. associate--l+ 99.5%

          \[\leadsto \frac{\frac{1}{\color{blue}{1.5 + \left(2 \cdot x - 0.125 \cdot \frac{1}{x}\right)}}}{\sqrt{x}} \]

       2. *-commutative 99.5%

          \[\leadsto \frac{\frac{1}{1.5 + \left(\color{blue}{x \cdot 2} - 0.125 \cdot \frac{1}{x}\right)}}{\sqrt{x}} \]

       3. associate-*r/ 99.5%

          \[\leadsto \frac{\frac{1}{1.5 + \left(x \cdot 2 - \color{blue}{\frac{0.125 \cdot 1}{x}}\right)}}{\sqrt{x}} \]

       4. metadata-eval 99.5%

          \[\leadsto \frac{\frac{1}{1.5 + \left(x \cdot 2 - \frac{\color{blue}{0.125}}{x}\right)}}{\sqrt{x}} \]

   13. Simplified 99.5%

       \[\leadsto \frac{\frac{1}{\color{blue}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}}}{\sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3550:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.51:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1.5 + x \cdot 2}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.51)
   (+ (/ 1.0 (sqrt x)) (- -1.0 (* x (+ -0.5 (* x 0.375)))))
   (/ (/ 1.0 (+ 1.5 (* x 2.0))) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.51) {
		tmp = (1.0 / sqrt(x)) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = (1.0 / (1.5 + (x * 2.0))) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.51d0) then
        tmp = (1.0d0 / sqrt(x)) + ((-1.0d0) - (x * ((-0.5d0) + (x * 0.375d0))))
    else
        tmp = (1.0d0 / (1.5d0 + (x * 2.0d0))) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.51) {
		tmp = (1.0 / Math.sqrt(x)) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = (1.0 / (1.5 + (x * 2.0))) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.51:
		tmp = (1.0 / math.sqrt(x)) + (-1.0 - (x * (-0.5 + (x * 0.375))))
	else:
		tmp = (1.0 / (1.5 + (x * 2.0))) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.51)
		tmp = Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 - Float64(x * Float64(-0.5 + Float64(x * 0.375)))));
	else
		tmp = Float64(Float64(1.0 / Float64(1.5 + Float64(x * 2.0))) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.51)
		tmp = (1.0 / sqrt(x)) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	else
		tmp = (1.0 / (1.5 + (x * 2.0))) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.51], N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 - N[(x * N[(-0.5 + N[(x * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.5 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.51000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 98.0%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(1 + \left(-0.5 \cdot x + 0.375 \cdot {x}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 98.0%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(\color{blue}{x \cdot -0.5} + 0.375 \cdot {x}^{2}\right)\right) \]

      2. *-commutative 98.0%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{{x}^{2} \cdot 0.375}\right)\right) \]

      3. unpow2 98.0%

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

      4. associate-*l* 98.0%

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

      5. distribute-lft-out 98.0%

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

   5. Simplified 98.0%

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

   if 0.51000000000000001 < x

   1. Initial program 35.4%

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

      1. frac-sub 35.5%

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

      2. *-un-lft-identity 35.5%

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

      3. +-commutative 35.5%

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

      4. *-rgt-identity 35.5%

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

      5. sqrt-unprod 35.5%

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

      6. +-commutative 35.5%

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

   4. Applied egg-rr 35.5%

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

      1. flip-- 36.3%

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

      2. add-sqr-sqrt 35.9%

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

      3. add-sqr-sqrt 36.7%

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

      4. +-commutative 36.7%

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

      5. +-commutative 36.7%

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

   6. Applied egg-rr 36.7%

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

      1. div-inv 36.7%

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

      2. +-commutative 36.7%

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

      3. sqrt-prod 36.7%

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

      4. times-frac 36.7%

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

      5. associate--l+ 36.7%

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

   8. Applied egg-rr 36.7%

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

      1. associate-+r- 36.7%

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

      2. +-commutative 36.7%

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

      3. associate-+r- 99.1%

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

      4. +-inverses 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-*l/ 99.2%

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

      7. *-lft-identity 99.2%

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

      8. associate-/l/ 99.2%

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

      9. distribute-lft-in 99.2%

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

      10. rem-square-sqrt 99.4%

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

      11. +-commutative 99.4%

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

      12. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 99.2%

       \[\leadsto \frac{\frac{1}{\color{blue}{1.5 + 2 \cdot x}}}{\sqrt{x}} \]
   12. Step-by-step derivation

       1. *-commutative 99.2%

          \[\leadsto \frac{\frac{1}{1.5 + \color{blue}{x \cdot 2}}}{\sqrt{x}} \]

   13. Simplified 99.2%

       \[\leadsto \frac{\frac{1}{\color{blue}{1.5 + x \cdot 2}}}{\sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.51:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1.5 + x \cdot 2}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.51:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.51)
   (+ (/ 1.0 (sqrt x)) (- -1.0 (* x (+ -0.5 (* x 0.375)))))
   (/ (/ 1.0 (+ 1.5 (- (* x 2.0) (/ 0.125 x)))) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.51) {
		tmp = (1.0 / sqrt(x)) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = (1.0 / (1.5 + ((x * 2.0) - (0.125 / x)))) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.51d0) then
        tmp = (1.0d0 / sqrt(x)) + ((-1.0d0) - (x * ((-0.5d0) + (x * 0.375d0))))
    else
        tmp = (1.0d0 / (1.5d0 + ((x * 2.0d0) - (0.125d0 / x)))) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.51) {
		tmp = (1.0 / Math.sqrt(x)) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = (1.0 / (1.5 + ((x * 2.0) - (0.125 / x)))) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.51:
		tmp = (1.0 / math.sqrt(x)) + (-1.0 - (x * (-0.5 + (x * 0.375))))
	else:
		tmp = (1.0 / (1.5 + ((x * 2.0) - (0.125 / x)))) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.51)
		tmp = Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 - Float64(x * Float64(-0.5 + Float64(x * 0.375)))));
	else
		tmp = Float64(Float64(1.0 / Float64(1.5 + Float64(Float64(x * 2.0) - Float64(0.125 / x)))) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.51)
		tmp = (1.0 / sqrt(x)) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	else
		tmp = (1.0 / (1.5 + ((x * 2.0) - (0.125 / x)))) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.51], N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 - N[(x * N[(-0.5 + N[(x * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.5 + N[(N[(x * 2.0), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.51000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 98.0%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(1 + \left(-0.5 \cdot x + 0.375 \cdot {x}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 98.0%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(\color{blue}{x \cdot -0.5} + 0.375 \cdot {x}^{2}\right)\right) \]

      2. *-commutative 98.0%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{{x}^{2} \cdot 0.375}\right)\right) \]

      3. unpow2 98.0%

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

      4. associate-*l* 98.0%

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

      5. distribute-lft-out 98.0%

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

   5. Simplified 98.0%

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

   if 0.51000000000000001 < x

   1. Initial program 35.4%

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

      1. frac-sub 35.5%

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

      2. *-un-lft-identity 35.5%

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

      3. +-commutative 35.5%

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

      4. *-rgt-identity 35.5%

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

      5. sqrt-unprod 35.5%

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

      6. +-commutative 35.5%

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

   4. Applied egg-rr 35.5%

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

      1. flip-- 36.3%

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

      2. add-sqr-sqrt 35.9%

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

      3. add-sqr-sqrt 36.7%

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

      4. +-commutative 36.7%

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

      5. +-commutative 36.7%

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

   6. Applied egg-rr 36.7%

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

      1. div-inv 36.7%

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

      2. +-commutative 36.7%

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

      3. sqrt-prod 36.7%

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

      4. times-frac 36.7%

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

      5. associate--l+ 36.7%

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

   8. Applied egg-rr 36.7%

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

      1. associate-+r- 36.7%

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

      2. +-commutative 36.7%

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

      3. associate-+r- 99.1%

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

      4. +-inverses 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-*l/ 99.2%

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

      7. *-lft-identity 99.2%

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

      8. associate-/l/ 99.2%

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

      9. distribute-lft-in 99.2%

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

      10. rem-square-sqrt 99.4%

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

      11. +-commutative 99.4%

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

      12. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 99.5%

       \[\leadsto \frac{\frac{1}{\color{blue}{\left(1.5 + 2 \cdot x\right) - 0.125 \cdot \frac{1}{x}}}}{\sqrt{x}} \]
   12. Step-by-step derivation

       1. associate--l+ 99.5%

          \[\leadsto \frac{\frac{1}{\color{blue}{1.5 + \left(2 \cdot x - 0.125 \cdot \frac{1}{x}\right)}}}{\sqrt{x}} \]

       2. *-commutative 99.5%

          \[\leadsto \frac{\frac{1}{1.5 + \left(\color{blue}{x \cdot 2} - 0.125 \cdot \frac{1}{x}\right)}}{\sqrt{x}} \]

       3. associate-*r/ 99.5%

          \[\leadsto \frac{\frac{1}{1.5 + \left(x \cdot 2 - \color{blue}{\frac{0.125 \cdot 1}{x}}\right)}}{\sqrt{x}} \]

       4. metadata-eval 99.5%

          \[\leadsto \frac{\frac{1}{1.5 + \left(x \cdot 2 - \frac{\color{blue}{0.125}}{x}\right)}}{\sqrt{x}} \]

   13. Simplified 99.5%

       \[\leadsto \frac{\frac{1}{\color{blue}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}}}{\sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.51:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.41:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1.5 + x \cdot 2}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.41)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ (/ 1.0 (+ 1.5 (* x 2.0))) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.41) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = (1.0 / (1.5 + (x * 2.0))) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.41d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = (1.0d0 / (1.5d0 + (x * 2.0d0))) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.41) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = (1.0 / (1.5 + (x * 2.0))) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.41:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = (1.0 / (1.5 + (x * 2.0))) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.41)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(Float64(1.0 / Float64(1.5 + Float64(x * 2.0))) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.41)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = (1.0 / (1.5 + (x * 2.0))) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.41], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(1.0 / N[(1.5 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.409999999999999976

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul0-lft 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+r+ 100.0%

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

      9. sub-neg 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 97.5%

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

   if 0.409999999999999976 < x

   1. Initial program 35.4%

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

      1. frac-sub 35.5%

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

      2. *-un-lft-identity 35.5%

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

      3. +-commutative 35.5%

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

      4. *-rgt-identity 35.5%

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

      5. sqrt-unprod 35.5%

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

      6. +-commutative 35.5%

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

   4. Applied egg-rr 35.5%

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

      1. flip-- 36.3%

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

      2. add-sqr-sqrt 35.9%

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

      3. add-sqr-sqrt 36.7%

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

      4. +-commutative 36.7%

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

      5. +-commutative 36.7%

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

   6. Applied egg-rr 36.7%

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

      1. div-inv 36.7%

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

      2. +-commutative 36.7%

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

      3. sqrt-prod 36.7%

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

      4. times-frac 36.7%

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

      5. associate--l+ 36.7%

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

   8. Applied egg-rr 36.7%

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

      1. associate-+r- 36.7%

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

      2. +-commutative 36.7%

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

      3. associate-+r- 99.1%

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

      4. +-inverses 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-*l/ 99.2%

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

      7. *-lft-identity 99.2%

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

      8. associate-/l/ 99.2%

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

      9. distribute-lft-in 99.2%

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

      10. rem-square-sqrt 99.4%

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

      11. +-commutative 99.4%

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

      12. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 99.2%

       \[\leadsto \frac{\frac{1}{\color{blue}{1.5 + 2 \cdot x}}}{\sqrt{x}} \]
   12. Step-by-step derivation

       1. *-commutative 99.2%

          \[\leadsto \frac{\frac{1}{1.5 + \color{blue}{x \cdot 2}}}{\sqrt{x}} \]

   13. Simplified 99.2%

       \[\leadsto \frac{\frac{1}{\color{blue}{1.5 + x \cdot 2}}}{\sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.41:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1.5 + x \cdot 2}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{1 + x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.7)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ (/ 0.5 x) (sqrt (+ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.7) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = (0.5 / x) / sqrt((1.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = (0.5d0 / x) / sqrt((1.0d0 + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.7) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = (0.5 / x) / Math.sqrt((1.0 + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.7:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = (0.5 / x) / math.sqrt((1.0 + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(Float64(1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = (0.5 / x) / sqrt((1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.7], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul0-lft 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+r+ 100.0%

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

      9. sub-neg 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 97.5%

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

   if 0.69999999999999996 < x

   1. Initial program 35.4%

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

      1. frac-sub 35.5%

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

      2. associate-/r* 35.5%

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

      3. *-un-lft-identity 35.5%

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

      4. +-commutative 35.5%

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

      5. *-rgt-identity 35.5%

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

      6. +-commutative 35.5%

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

   4. Applied egg-rr 35.5%

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

      1. div-sub 35.5%

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

      2. sqrt-undiv 35.5%

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

      3. +-commutative 35.5%

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

      4. *-inverses 35.5%

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

   6. Applied egg-rr 35.5%

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

   7. Taylor expanded in x around inf 98.1%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{1 + x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.48) (+ (pow x -0.5) -1.0) (/ (/ 0.5 x) (sqrt (+ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.48) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) / sqrt((1.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.48d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (0.5d0 / x) / sqrt((1.0d0 + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.48) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) / Math.sqrt((1.0 + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.48:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = (0.5 / x) / math.sqrt((1.0 + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.48)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(Float64(1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.48)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (0.5 / x) / sqrt((1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.48], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.47999999999999998

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul0-lft 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+r+ 100.0%

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

      9. sub-neg 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 96.2%

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

   if 0.47999999999999998 < x

   1. Initial program 35.4%

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

      1. frac-sub 35.5%

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

      2. associate-/r* 35.5%

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

      3. *-un-lft-identity 35.5%

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

      4. +-commutative 35.5%

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

      5. *-rgt-identity 35.5%

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

      6. +-commutative 35.5%

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

   4. Applied egg-rr 35.5%

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

      1. div-sub 35.5%

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

      2. sqrt-undiv 35.5%

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

      3. +-commutative 35.5%

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

      4. *-inverses 35.5%

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

   6. Applied egg-rr 35.5%

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

   7. Taylor expanded in x around inf 98.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.68) (+ (pow x -0.5) -1.0) (/ (/ 0.5 x) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (0.5d0 / x) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.68:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = (0.5 / x) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (0.5 / x) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.68], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul0-lft 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+r+ 100.0%

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

      9. sub-neg 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 96.2%

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

   if 0.680000000000000049 < x

   1. Initial program 35.4%

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

      1. frac-sub 35.5%

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

      2. *-un-lft-identity 35.5%

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

      3. +-commutative 35.5%

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

      4. *-rgt-identity 35.5%

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

      5. sqrt-unprod 35.5%

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

      6. +-commutative 35.5%

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

   4. Applied egg-rr 35.5%

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

      1. flip-- 36.3%

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

      2. add-sqr-sqrt 35.9%

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

      3. add-sqr-sqrt 36.7%

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

      4. +-commutative 36.7%

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

      5. +-commutative 36.7%

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

   6. Applied egg-rr 36.7%

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

      1. div-inv 36.7%

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

      2. +-commutative 36.7%

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

      3. sqrt-prod 36.7%

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

      4. times-frac 36.7%

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

      5. associate--l+ 36.7%

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

   8. Applied egg-rr 36.7%

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

      1. associate-+r- 36.7%

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

      2. +-commutative 36.7%

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

      3. associate-+r- 99.1%

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

      4. +-inverses 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-*l/ 99.2%

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

      7. *-lft-identity 99.2%

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

      8. associate-/l/ 99.2%

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

      9. distribute-lft-in 99.2%

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

      10. rem-square-sqrt 99.4%

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

      11. +-commutative 99.4%

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

      12. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 98.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.9% 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 64.0%

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

   1. add-cube-cbrt 51.1%

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

   2. pow3 51.0%

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

   3. pow1/2 51.0%

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

   4. pow-flip 51.5%

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

   5. metadata-eval 51.5%

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

4. Applied egg-rr 51.5%

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

5. Taylor expanded in x around inf 44.5%

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

   1. expm1-log1p-u 41.4%

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

   2. expm1-udef 57.1%

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

   3. inv-pow 57.1%

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

   4. sqrt-pow1 57.1%

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

   5. metadata-eval 57.1%

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

7. Applied egg-rr 57.1%

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

   1. expm1-def 41.4%

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

   2. expm1-log1p 44.6%

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

9. Simplified 44.6%

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

10. Final simplification 44.6%

    \[\leadsto {x}^{-0.5} \]
11. Add Preprocessing

Alternative 11: 1.9% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in x around 0 44.1%

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

4. Taylor expanded in x around inf 2.0%

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

5. Final simplification 2.0%

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

Alternative 12: 5.7% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 64.0%

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

   1. frac-sub 64.0%

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

   2. *-un-lft-identity 64.0%

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

   3. +-commutative 64.0%

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

   4. *-rgt-identity 64.0%

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

   5. sqrt-unprod 64.0%

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

   6. +-commutative 64.0%

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

4. Applied egg-rr 64.0%

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

5. Taylor expanded in x around inf 22.9%

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

6. Taylor expanded in x around 0 5.8%

   \[\leadsto \color{blue}{2} \]

7. Final simplification 5.8%

   \[\leadsto 2 \]
8. Add Preprocessing

Developer target: 98.9% 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 2024020 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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