2isqrt (example 3.6)

Percentage Accurate: 68.5% → 99.9%

Time: 9.1s

Alternatives: 11

Speedup: 2.0×

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: 68.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;\frac{{x}^{-1.5}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{x + x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ x 1.0)))) 0.0)
   (/ (pow x -1.5) 2.0)
   (/ (pow x -0.5) (+ (+ x 1.0) (sqrt (+ x (* x x)))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 0.0) {
		tmp = pow(x, -1.5) / 2.0;
	} else {
		tmp = pow(x, -0.5) / ((x + 1.0) + sqrt((x + (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((x + 1.0d0)))) <= 0.0d0) then
        tmp = (x ** (-1.5d0)) / 2.0d0
    else
        tmp = (x ** (-0.5d0)) / ((x + 1.0d0) + sqrt((x + (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((x + 1.0)))) <= 0.0) {
		tmp = Math.pow(x, -1.5) / 2.0;
	} else {
		tmp = Math.pow(x, -0.5) / ((x + 1.0) + Math.sqrt((x + (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((x + 1.0)))) <= 0.0:
		tmp = math.pow(x, -1.5) / 2.0
	else:
		tmp = math.pow(x, -0.5) / ((x + 1.0) + math.sqrt((x + (x * x))))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[x, -1.5], $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[Sqrt[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 0.0

   1. Initial program 37.7%

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

      1. frac-sub 37.7%

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

      2. div-inv 37.7%

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

      3. *-un-lft-identity 37.7%

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

      4. +-commutative 37.7%

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

      5. *-rgt-identity 37.7%

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

      6. metadata-eval 37.7%

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

      7. frac-times 37.7%

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

      8. un-div-inv 37.7%

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

      9. pow1/2 37.7%

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

      10. pow-flip 37.7%

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

      11. metadata-eval 37.7%

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

      12. +-commutative 37.7%

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

   3. Applied egg-rr 37.7%

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

      1. *-commutative 37.7%

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

      2. clear-num 37.7%

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

      3. flip-- 37.7%

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

      4. frac-times 37.7%

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

      5. *-un-lft-identity 37.7%

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

      6. add-sqr-sqrt 37.4%

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

      7. add-sqr-sqrt 37.7%

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

   5. Applied egg-rr 37.7%

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

      1. associate--l+ 98.2%

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

      2. +-inverses 98.2%

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

      3. metadata-eval 98.2%

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

      4. associate-*l/ 98.1%

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

      5. associate-/l* 99.4%

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

      6. *-lft-identity 99.4%

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

      7. distribute-lft-in 99.4%

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

      8. rem-square-sqrt 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 99.7%

      \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{2 \cdot x}} \]
   9. Step-by-step derivation

      1. *-commutative 99.7%

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

   10. Simplified 99.7%

       \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{x \cdot 2}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 99.7%

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

       2. expm1-udef 37.7%

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

       3. associate-/r* 37.7%

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

       4. pow1 37.7%

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

       5. pow-div 37.7%

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

       6. metadata-eval 37.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{{x}^{\color{blue}{-1.5}}}{2}\right)} - 1 \]

   12. Applied egg-rr 37.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-1.5}}{2}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-1.5}}{2}\right)\right)} \]

       2. expm1-log1p 100.0%

          \[\leadsto \color{blue}{\frac{{x}^{-1.5}}{2}} \]

   14. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{{x}^{-1.5}}{2}} \]

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 98.4%

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

      1. frac-sub 98.4%

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

      2. div-inv 98.4%

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

      3. *-un-lft-identity 98.4%

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

      4. +-commutative 98.4%

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

      5. *-rgt-identity 98.4%

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

      6. metadata-eval 98.4%

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

      7. frac-times 98.4%

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

      8. un-div-inv 98.4%

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

      9. pow1/2 98.4%

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

      10. pow-flip 98.7%

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

      11. metadata-eval 98.7%

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

      12. +-commutative 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. clear-num 98.5%

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

      3. flip-- 98.7%

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

      4. frac-times 98.6%

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

      5. *-un-lft-identity 98.6%

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

      6. add-sqr-sqrt 98.6%

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

      7. add-sqr-sqrt 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate--l+ 99.7%

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

      2. +-inverses 99.7%

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

      3. metadata-eval 99.7%

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

      4. associate-*l/ 99.7%

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

      5. associate-/l* 99.9%

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

      6. *-lft-identity 99.9%

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

      7. distribute-lft-in 99.9%

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

      8. rem-square-sqrt 99.9%

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

   7. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-def 99.9%

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

      3. *-commutative 99.9%

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

      4. sqrt-unprod 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. fma-udef 99.9%

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

       2. *-lft-identity 99.9%

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

       3. +-commutative 99.9%

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

       4. distribute-rgt-in 99.9%

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

       5. *-lft-identity 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;\frac{{x}^{-1.5}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{x + x \cdot x}}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(\left(0.5 + \left(x + \frac{0.0625}{x \cdot x}\right)\right) - \frac{0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ x 1.0)))) 5e-6)
   (/
    (pow x -0.5)
    (+ (+ x 1.0) (- (+ 0.5 (+ x (/ 0.0625 (* x x)))) (/ 0.125 x))))
   (- (pow x -0.5) (pow (+ x 1.0) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 5e-6) {
		tmp = pow(x, -0.5) / ((x + 1.0) + ((0.5 + (x + (0.0625 / (x * x)))) - (0.125 / x)));
	} else {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((x + 1.0d0)))) <= 5d-6) then
        tmp = (x ** (-0.5d0)) / ((x + 1.0d0) + ((0.5d0 + (x + (0.0625d0 / (x * x)))) - (0.125d0 / x)))
    else
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((x + 1.0)))) <= 5e-6) {
		tmp = Math.pow(x, -0.5) / ((x + 1.0) + ((0.5 + (x + (0.0625 / (x * x)))) - (0.125 / x)));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((x + 1.0)))) <= 5e-6:
		tmp = math.pow(x, -0.5) / ((x + 1.0) + ((0.5 + (x + (0.0625 / (x * x)))) - (0.125 / x)))
	else:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(x + 1.0)))) <= 5e-6)
		tmp = Float64((x ^ -0.5) / Float64(Float64(x + 1.0) + Float64(Float64(0.5 + Float64(x + Float64(0.0625 / Float64(x * x)))) - Float64(0.125 / x))));
	else
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 5e-6)
		tmp = (x ^ -0.5) / ((x + 1.0) + ((0.5 + (x + (0.0625 / (x * x)))) - (0.125 / x)));
	else
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[Power[x, -0.5], $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(N[(0.5 + N[(x + N[(0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.00000000000000041e-6

   1. Initial program 38.5%

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

      1. frac-sub 38.5%

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

      2. div-inv 38.5%

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

      3. *-un-lft-identity 38.5%

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

      4. +-commutative 38.5%

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

      5. *-rgt-identity 38.5%

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

      6. metadata-eval 38.5%

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

      7. frac-times 38.5%

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

      8. un-div-inv 38.5%

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

      9. pow1/2 38.5%

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

      10. pow-flip 38.5%

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

      11. metadata-eval 38.5%

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

      12. +-commutative 38.5%

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

   3. Applied egg-rr 38.5%

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

      1. *-commutative 38.5%

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

      2. clear-num 38.5%

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

      3. flip-- 38.5%

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

      4. frac-times 38.5%

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

      5. *-un-lft-identity 38.5%

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

      6. add-sqr-sqrt 38.3%

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

      7. add-sqr-sqrt 39.4%

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

   5. Applied egg-rr 39.4%

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

      1. associate--l+ 98.2%

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

      2. +-inverses 98.2%

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

      3. metadata-eval 98.2%

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

      4. associate-*l/ 98.2%

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

      5. associate-/l* 99.4%

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

      6. *-lft-identity 99.4%

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

      7. distribute-lft-in 99.4%

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

      8. rem-square-sqrt 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 99.7%

      \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \color{blue}{\left(\left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + x\right)\right) - 0.125 \cdot \frac{1}{x}\right)}} \]
   9. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \left(\left(0.5 + \color{blue}{\left(x + 0.0625 \cdot \frac{1}{{x}^{2}}\right)}\right) - 0.125 \cdot \frac{1}{x}\right)} \]

      2. associate-*r/ 99.7%

         \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \left(\left(0.5 + \left(x + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}}\right)\right) - 0.125 \cdot \frac{1}{x}\right)} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \left(\left(0.5 + \left(x + \frac{\color{blue}{0.0625}}{{x}^{2}}\right)\right) - 0.125 \cdot \frac{1}{x}\right)} \]

      4. unpow2 99.7%

         \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \left(\left(0.5 + \left(x + \frac{0.0625}{\color{blue}{x \cdot x}}\right)\right) - 0.125 \cdot \frac{1}{x}\right)} \]

      5. associate-*r/ 99.7%

         \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \left(\left(0.5 + \left(x + \frac{0.0625}{x \cdot x}\right)\right) - \color{blue}{\frac{0.125 \cdot 1}{x}}\right)} \]

      6. metadata-eval 99.7%

         \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \left(\left(0.5 + \left(x + \frac{0.0625}{x \cdot x}\right)\right) - \frac{\color{blue}{0.125}}{x}\right)} \]

   10. Simplified 99.7%

       \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \color{blue}{\left(\left(0.5 + \left(x + \frac{0.0625}{x \cdot x}\right)\right) - \frac{0.125}{x}\right)}} \]

   if 5.00000000000000041e-6 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. 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. inv-pow 99.6%

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

      9. sqrt-pow2 99.9%

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

      10. metadata-eval 99.9%

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

      11. pow1/2 99.9%

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

      12. pow-flip 99.9%

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

      13. +-commutative 99.9%

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

      14. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. distribute-lft1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. mul0-lft 99.9%

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

      5. +-rgt-identity 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(\left(0.5 + \left(x + \frac{0.0625}{x \cdot x}\right)\right) - \frac{0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.7%

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

   1. frac-sub 64.7%

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

   2. div-inv 64.7%

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

   3. *-un-lft-identity 64.7%

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

   4. +-commutative 64.7%

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

   5. *-rgt-identity 64.7%

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

   6. metadata-eval 64.7%

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

   7. frac-times 64.7%

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

   8. un-div-inv 64.7%

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

   9. pow1/2 64.7%

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

   10. pow-flip 64.9%

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

   11. metadata-eval 64.9%

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

   12. +-commutative 64.9%

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

3. Applied egg-rr 64.9%

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

   1. *-commutative 64.9%

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

   2. clear-num 64.8%

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

   3. flip-- 64.9%

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

   4. frac-times 64.8%

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

   5. *-un-lft-identity 64.8%

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

   6. add-sqr-sqrt 64.7%

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

   7. add-sqr-sqrt 65.3%

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

5. Applied egg-rr 65.3%

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

   1. associate--l+ 98.9%

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

   2. +-inverses 98.9%

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

   3. metadata-eval 98.9%

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

   4. associate-*l/ 98.8%

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

   5. associate-/l* 99.6%

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

   6. *-lft-identity 99.6%

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

   7. distribute-lft-in 99.6%

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

   8. rem-square-sqrt 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.7%

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

   1. sub-neg 64.7%

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

   2. flip-+ 64.7%

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

   3. frac-times 54.8%

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

   4. metadata-eval 54.8%

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

   5. add-sqr-sqrt 53.2%

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

   6. distribute-neg-frac 53.2%

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

   7. metadata-eval 53.2%

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

   8. +-commutative 53.2%

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

   9. distribute-neg-frac 53.2%

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

   10. metadata-eval 53.2%

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

   11. +-commutative 53.2%

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

   12. pow1/2 53.2%

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

   13. pow-flip 53.2%

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

   14. metadata-eval 53.2%

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

3. Applied egg-rr 53.2%

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

   1. frac-times 59.1%

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

   2. metadata-eval 59.1%

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

   3. add-sqr-sqrt 64.6%

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

   4. frac-sub 65.1%

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

   5. *-un-lft-identity 65.1%

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

5. Applied egg-rr 65.1%

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

   1. *-rgt-identity 65.1%

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

   2. associate--l+ 90.9%

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

   3. +-inverses 90.9%

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

   4. metadata-eval 90.9%

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

   5. associate-/r* 91.0%

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

7. Simplified 91.0%

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

   1. associate-/l/ 99.4%

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

   2. div-inv 99.4%

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

   3. sub-neg 99.4%

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

   4. distribute-neg-frac 99.4%

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

   5. metadata-eval 99.4%

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

   6. pow1/2 99.4%

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

   7. pow-flip 99.4%

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

   8. metadata-eval 99.4%

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

9. Applied egg-rr 99.4%

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

    1. associate-*l/ 99.4%

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

    2. *-un-lft-identity 99.4%

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

    3. *-commutative 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 5: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.44:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(\left(0.5 + \left(x + \frac{0.0625}{x \cdot x}\right)\right) - \frac{0.125}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.44)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/
    (pow x -0.5)
    (+ (+ x 1.0) (- (+ 0.5 (+ x (/ 0.0625 (* x x)))) (/ 0.125 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.44) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = pow(x, -0.5) / ((x + 1.0) + ((0.5 + (x + (0.0625 / (x * x)))) - (0.125 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.44d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = (x ** (-0.5d0)) / ((x + 1.0d0) + ((0.5d0 + (x + (0.0625d0 / (x * x)))) - (0.125d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.44) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = Math.pow(x, -0.5) / ((x + 1.0) + ((0.5 + (x + (0.0625 / (x * x)))) - (0.125 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.44:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = math.pow(x, -0.5) / ((x + 1.0) + ((0.5 + (x + (0.0625 / (x * x)))) - (0.125 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.44)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64((x ^ -0.5) / Float64(Float64(x + 1.0) + Float64(Float64(0.5 + Float64(x + Float64(0.0625 / Float64(x * x)))) - Float64(0.125 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.44)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = (x ^ -0.5) / ((x + 1.0) + ((0.5 + (x + (0.0625 / (x * x)))) - (0.125 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.44], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(N[(0.5 + N[(x + N[(0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.440000000000000002

   1. Initial program 99.7%

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

      1. inv-pow 99.7%

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

      2. pow1/2 99.7%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.6%

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

      5. pow-exp 92.6%

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

      6. metadata-eval 92.6%

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

   3. Applied egg-rr 92.6%

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

   4. Taylor expanded in x around 0 99.9%

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

   if 0.440000000000000002 < x

   1. Initial program 39.2%

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

      1. frac-sub 39.3%

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

      2. div-inv 39.2%

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

      3. *-un-lft-identity 39.2%

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

      4. +-commutative 39.2%

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

      5. *-rgt-identity 39.2%

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

      6. metadata-eval 39.2%

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

      7. frac-times 39.3%

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

      8. un-div-inv 39.3%

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

      9. pow1/2 39.3%

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

      10. pow-flip 39.3%

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

      11. metadata-eval 39.3%

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

      12. +-commutative 39.3%

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

   3. Applied egg-rr 39.3%

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

      1. *-commutative 39.3%

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

      2. clear-num 39.2%

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

      3. flip-- 39.4%

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

      4. frac-times 39.3%

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

      5. *-un-lft-identity 39.3%

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

      6. add-sqr-sqrt 39.1%

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

      7. add-sqr-sqrt 40.2%

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

   5. Applied egg-rr 40.2%

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

      1. associate--l+ 98.2%

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

      2. +-inverses 98.2%

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

      3. metadata-eval 98.2%

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

      4. associate-*l/ 98.2%

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

      5. associate-/l* 99.4%

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

      6. *-lft-identity 99.4%

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

      7. distribute-lft-in 99.4%

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

      8. rem-square-sqrt 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 99.2%

      \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \color{blue}{\left(\left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + x\right)\right) - 0.125 \cdot \frac{1}{x}\right)}} \]
   9. Step-by-step derivation

      1. +-commutative 99.2%

         \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \left(\left(0.5 + \color{blue}{\left(x + 0.0625 \cdot \frac{1}{{x}^{2}}\right)}\right) - 0.125 \cdot \frac{1}{x}\right)} \]

      2. associate-*r/ 99.2%

         \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \left(\left(0.5 + \left(x + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}}\right)\right) - 0.125 \cdot \frac{1}{x}\right)} \]

      3. metadata-eval 99.2%

         \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \left(\left(0.5 + \left(x + \frac{\color{blue}{0.0625}}{{x}^{2}}\right)\right) - 0.125 \cdot \frac{1}{x}\right)} \]

      4. unpow2 99.2%

         \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \left(\left(0.5 + \left(x + \frac{0.0625}{\color{blue}{x \cdot x}}\right)\right) - 0.125 \cdot \frac{1}{x}\right)} \]

      5. associate-*r/ 99.2%

         \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \left(\left(0.5 + \left(x + \frac{0.0625}{x \cdot x}\right)\right) - \color{blue}{\frac{0.125 \cdot 1}{x}}\right)} \]

      6. metadata-eval 99.2%

         \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \left(\left(0.5 + \left(x + \frac{0.0625}{x \cdot x}\right)\right) - \frac{\color{blue}{0.125}}{x}\right)} \]

   10. Simplified 99.2%

       \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \color{blue}{\left(\left(0.5 + \left(x + \frac{0.0625}{x \cdot x}\right)\right) - \frac{0.125}{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.44:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(\left(0.5 + \left(x + \frac{0.0625}{x \cdot x}\right)\right) - \frac{0.125}{x}\right)}\\ \end{array} \]

Alternative 6: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.43:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(\left(x + 0.5\right) - \frac{0.125}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.43)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ (pow x -0.5) (+ (+ x 1.0) (- (+ x 0.5) (/ 0.125 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.43) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = pow(x, -0.5) / ((x + 1.0) + ((x + 0.5) - (0.125 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.43d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = (x ** (-0.5d0)) / ((x + 1.0d0) + ((x + 0.5d0) - (0.125d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.43) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = Math.pow(x, -0.5) / ((x + 1.0) + ((x + 0.5) - (0.125 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.43:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = math.pow(x, -0.5) / ((x + 1.0) + ((x + 0.5) - (0.125 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.43)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64((x ^ -0.5) / Float64(Float64(x + 1.0) + Float64(Float64(x + 0.5) - Float64(0.125 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.43)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = (x ^ -0.5) / ((x + 1.0) + ((x + 0.5) - (0.125 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.43], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(N[(x + 0.5), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.429999999999999993

   1. Initial program 99.7%

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

      1. inv-pow 99.7%

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

      2. pow1/2 99.7%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.6%

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

      5. pow-exp 92.6%

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

      6. metadata-eval 92.6%

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

   3. Applied egg-rr 92.6%

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

   4. Taylor expanded in x around 0 99.9%

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

   if 0.429999999999999993 < x

   1. Initial program 39.2%

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

      1. frac-sub 39.3%

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

      2. div-inv 39.2%

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

      3. *-un-lft-identity 39.2%

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

      4. +-commutative 39.2%

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

      5. *-rgt-identity 39.2%

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

      6. metadata-eval 39.2%

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

      7. frac-times 39.3%

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

      8. un-div-inv 39.3%

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

      9. pow1/2 39.3%

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

      10. pow-flip 39.3%

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

      11. metadata-eval 39.3%

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

      12. +-commutative 39.3%

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

   3. Applied egg-rr 39.3%

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

      1. *-commutative 39.3%

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

      2. clear-num 39.2%

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

      3. flip-- 39.4%

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

      4. frac-times 39.3%

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

      5. *-un-lft-identity 39.3%

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

      6. add-sqr-sqrt 39.1%

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

      7. add-sqr-sqrt 40.2%

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

   5. Applied egg-rr 40.2%

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

      1. associate--l+ 98.2%

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

      2. +-inverses 98.2%

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

      3. metadata-eval 98.2%

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

      4. associate-*l/ 98.2%

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

      5. associate-/l* 99.4%

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

      6. *-lft-identity 99.4%

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

      7. distribute-lft-in 99.4%

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

      8. rem-square-sqrt 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

   10. Simplified 98.9%

       \[\leadsto \frac{{x}^{-0.5}}{\left(1 + x\right) + \color{blue}{\left(\left(x + 0.5\right) - \frac{0.125}{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.43:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(\left(x + 0.5\right) - \frac{0.125}{x}\right)}\\ \end{array} \]

Alternative 7: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.43:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.43)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ (pow x -0.5) (+ 1.5 (- (* x 2.0) (/ 0.125 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.43) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = pow(x, -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.43d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = (x ** (-0.5d0)) / (1.5d0 + ((x * 2.0d0) - (0.125d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.43) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = Math.pow(x, -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.43:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = math.pow(x, -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.43)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64((x ^ -0.5) / Float64(1.5 + Float64(Float64(x * 2.0) - Float64(0.125 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.43)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = (x ^ -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.43], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] / N[(1.5 + N[(N[(x * 2.0), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.429999999999999993

   1. Initial program 99.7%

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

      1. inv-pow 99.7%

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

      2. pow1/2 99.7%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.6%

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

      5. pow-exp 92.6%

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

      6. metadata-eval 92.6%

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

   3. Applied egg-rr 92.6%

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

   4. Taylor expanded in x around 0 99.9%

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

   if 0.429999999999999993 < x

   1. Initial program 39.2%

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

      1. frac-sub 39.3%

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

      2. div-inv 39.2%

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

      3. *-un-lft-identity 39.2%

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

      4. +-commutative 39.2%

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

      5. *-rgt-identity 39.2%

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

      6. metadata-eval 39.2%

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

      7. frac-times 39.3%

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

      8. un-div-inv 39.3%

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

      9. pow1/2 39.3%

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

      10. pow-flip 39.3%

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

      11. metadata-eval 39.3%

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

      12. +-commutative 39.3%

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

   3. Applied egg-rr 39.3%

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

      1. *-commutative 39.3%

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

      2. clear-num 39.2%

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

      3. flip-- 39.4%

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

      4. frac-times 39.3%

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

      5. *-un-lft-identity 39.3%

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

      6. add-sqr-sqrt 39.1%

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

      7. add-sqr-sqrt 40.2%

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

   5. Applied egg-rr 40.2%

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

      1. associate--l+ 98.2%

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

      2. +-inverses 98.2%

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

      3. metadata-eval 98.2%

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

      4. associate-*l/ 98.2%

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

      5. associate-/l* 99.4%

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

      6. *-lft-identity 99.4%

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

      7. distribute-lft-in 99.4%

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

      8. rem-square-sqrt 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 98.9%

      \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\left(1.5 + 2 \cdot x\right) - 0.125 \cdot \frac{1}{x}}} \]
   9. Step-by-step derivation

      1. associate--l+ 98.9%

         \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{1.5 + \left(2 \cdot x - 0.125 \cdot \frac{1}{x}\right)}} \]

      2. *-commutative 98.9%

         \[\leadsto \frac{{x}^{-0.5}}{1.5 + \left(\color{blue}{x \cdot 2} - 0.125 \cdot \frac{1}{x}\right)} \]

      3. associate-*r/ 98.9%

         \[\leadsto \frac{{x}^{-0.5}}{1.5 + \left(x \cdot 2 - \color{blue}{\frac{0.125 \cdot 1}{x}}\right)} \]

      4. metadata-eval 98.9%

         \[\leadsto \frac{{x}^{-0.5}}{1.5 + \left(x \cdot 2 - \frac{\color{blue}{0.125}}{x}\right)} \]

   10. Simplified 98.9%

       \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.43:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}\\ \end{array} \]

Alternative 8: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.7%

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

      1. inv-pow 99.7%

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

      2. pow1/2 99.7%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.6%

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

      5. pow-exp 92.6%

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

      6. metadata-eval 92.6%

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

   3. Applied egg-rr 92.6%

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

   4. Taylor expanded in x around 0 99.9%

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

   if 1 < x

   1. Initial program 39.2%

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

      1. frac-sub 39.3%

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

      2. div-inv 39.2%

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

      3. *-un-lft-identity 39.2%

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

      4. +-commutative 39.2%

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

      5. *-rgt-identity 39.2%

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

      6. metadata-eval 39.2%

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

      7. frac-times 39.3%

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

      8. un-div-inv 39.3%

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

      9. pow1/2 39.3%

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

      10. pow-flip 39.3%

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

      11. metadata-eval 39.3%

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

      12. +-commutative 39.3%

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

   3. Applied egg-rr 39.3%

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

      1. *-commutative 39.3%

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

      2. clear-num 39.2%

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

      3. flip-- 39.4%

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

      4. frac-times 39.3%

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

      5. *-un-lft-identity 39.3%

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

      6. add-sqr-sqrt 39.1%

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

      7. add-sqr-sqrt 40.2%

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

   5. Applied egg-rr 40.2%

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

      1. associate--l+ 98.2%

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

      2. +-inverses 98.2%

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

      3. metadata-eval 98.2%

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

      4. associate-*l/ 98.2%

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

      5. associate-/l* 99.4%

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

      6. *-lft-identity 99.4%

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

      7. distribute-lft-in 99.4%

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

      8. rem-square-sqrt 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 97.5%

      \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{2 \cdot x}} \]
   9. Step-by-step derivation

      1. *-commutative 97.5%

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

   10. Simplified 97.5%

       \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{x \cdot 2}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 97.5%

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

       2. expm1-udef 37.5%

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

       3. associate-/r* 37.5%

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

       4. pow1 37.5%

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

       5. pow-div 37.5%

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

       6. metadata-eval 37.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{{x}^{\color{blue}{-1.5}}}{2}\right)} - 1 \]

   12. Applied egg-rr 37.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-1.5}}{2}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 97.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-1.5}}{2}\right)\right)} \]

       2. expm1-log1p 97.7%

          \[\leadsto \color{blue}{\frac{{x}^{-1.5}}{2}} \]

   14. Simplified 97.7%

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

4. Final simplification 98.7%

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

Alternative 9: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.41) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = pow(x, -0.5) / (1.5 + (x * 2.0));
	}
	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 = (x ** (-0.5d0)) / (1.5d0 + (x * 2.0d0))
    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 = Math.pow(x, -0.5) / (1.5 + (x * 2.0));
	}
	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 = math.pow(x, -0.5) / (1.5 + (x * 2.0))
	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((x ^ -0.5) / Float64(1.5 + Float64(x * 2.0)));
	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 = (x ^ -0.5) / (1.5 + (x * 2.0));
	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[Power[x, -0.5], $MachinePrecision] / N[(1.5 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.409999999999999976

   1. Initial program 99.7%

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

      1. inv-pow 99.7%

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

      2. pow1/2 99.7%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.6%

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

      5. pow-exp 92.6%

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

      6. metadata-eval 92.6%

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

   3. Applied egg-rr 92.6%

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

   4. Taylor expanded in x around 0 99.9%

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

   if 0.409999999999999976 < x

   1. Initial program 39.2%

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

      1. frac-sub 39.3%

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

      2. div-inv 39.2%

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

      3. *-un-lft-identity 39.2%

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

      4. +-commutative 39.2%

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

      5. *-rgt-identity 39.2%

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

      6. metadata-eval 39.2%

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

      7. frac-times 39.3%

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

      8. un-div-inv 39.3%

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

      9. pow1/2 39.3%

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

      10. pow-flip 39.3%

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

      11. metadata-eval 39.3%

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

      12. +-commutative 39.3%

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

   3. Applied egg-rr 39.3%

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

      1. *-commutative 39.3%

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

      2. clear-num 39.2%

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

      3. flip-- 39.4%

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

      4. frac-times 39.3%

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

      5. *-un-lft-identity 39.3%

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

      6. add-sqr-sqrt 39.1%

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

      7. add-sqr-sqrt 40.2%

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

   5. Applied egg-rr 40.2%

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

      1. associate--l+ 98.2%

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

      2. +-inverses 98.2%

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

      3. metadata-eval 98.2%

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

      4. associate-*l/ 98.2%

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

      5. associate-/l* 99.4%

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

      6. *-lft-identity 99.4%

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

      7. distribute-lft-in 99.4%

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

      8. rem-square-sqrt 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 98.5%

      \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{1.5 + 2 \cdot x}} \]
   9. Step-by-step derivation

      1. +-commutative 98.5%

         \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{2 \cdot x + 1.5}} \]

      2. *-commutative 98.5%

         \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{x \cdot 2} + 1.5} \]

   10. Simplified 98.5%

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

4. Final simplification 99.1%

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

Alternative 10: 98.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-1.5}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.66) (+ (pow x -0.5) -1.0) (/ (pow x -1.5) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.66) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = pow(x, -1.5) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.66d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (x ** (-1.5d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.66) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = Math.pow(x, -1.5) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.66:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = math.pow(x, -1.5) / 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.66)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64((x ^ -1.5) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.66)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (x ^ -1.5) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.66], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[x, -1.5], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.660000000000000031

   1. Initial program 99.7%

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

      1. inv-pow 99.7%

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

      2. pow1/2 99.7%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.6%

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

      5. pow-exp 92.6%

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

      6. metadata-eval 92.6%

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

   3. Applied egg-rr 92.6%

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

   4. Taylor expanded in x around 0 99.5%

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

   if 0.660000000000000031 < x

   1. Initial program 39.2%

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

      1. frac-sub 39.3%

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

      2. div-inv 39.2%

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

      3. *-un-lft-identity 39.2%

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

      4. +-commutative 39.2%

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

      5. *-rgt-identity 39.2%

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

      6. metadata-eval 39.2%

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

      7. frac-times 39.3%

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

      8. un-div-inv 39.3%

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

      9. pow1/2 39.3%

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

      10. pow-flip 39.3%

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

      11. metadata-eval 39.3%

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

      12. +-commutative 39.3%

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

   3. Applied egg-rr 39.3%

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

      1. *-commutative 39.3%

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

      2. clear-num 39.2%

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

      3. flip-- 39.4%

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

      4. frac-times 39.3%

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

      5. *-un-lft-identity 39.3%

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

      6. add-sqr-sqrt 39.1%

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

      7. add-sqr-sqrt 40.2%

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

   5. Applied egg-rr 40.2%

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

      1. associate--l+ 98.2%

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

      2. +-inverses 98.2%

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

      3. metadata-eval 98.2%

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

      4. associate-*l/ 98.2%

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

      5. associate-/l* 99.4%

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

      6. *-lft-identity 99.4%

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

      7. distribute-lft-in 99.4%

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

      8. rem-square-sqrt 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 97.5%

      \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{2 \cdot x}} \]
   9. Step-by-step derivation

      1. *-commutative 97.5%

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

   10. Simplified 97.5%

       \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{x \cdot 2}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 97.5%

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

       2. expm1-udef 37.5%

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

       3. associate-/r* 37.5%

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

       4. pow1 37.5%

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

       5. pow-div 37.5%

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

       6. metadata-eval 37.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{{x}^{\color{blue}{-1.5}}}{2}\right)} - 1 \]

   12. Applied egg-rr 37.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-1.5}}{2}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 97.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-1.5}}{2}\right)\right)} \]

       2. expm1-log1p 97.7%

          \[\leadsto \color{blue}{\frac{{x}^{-1.5}}{2}} \]

   14. Simplified 97.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-1.5}}{2}\\ \end{array} \]

Alternative 11: 50.3% 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 64.7%

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

   1. inv-pow 64.7%

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

   2. pow1/2 64.7%

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

   3. pow-pow 59.8%

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

   4. add-exp-log 43.2%

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

   5. pow-exp 43.2%

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

   6. metadata-eval 43.2%

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

3. Applied egg-rr 43.2%

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

4. Taylor expanded in x around inf 44.0%

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

   1. inv-pow 44.0%

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

   2. sqrt-pow1 44.1%

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

   3. metadata-eval 44.1%

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

   4. *-un-lft-identity 44.1%

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

6. Applied egg-rr 44.1%

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

   1. *-lft-identity 44.1%

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

8. Simplified 44.1%

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

9. Final simplification 44.1%

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

Developer target: 99.0% 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 2023240 
(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)))))