2isqrt (example 3.6)

Percentage Accurate: 69.2% → 99.8%

Time: 10.5s

Alternatives: 17

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: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+129}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(1 + x\right) + \sqrt{x + x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+129)
   (/ (pow x -0.5) (+ (+ 1.0 x) (sqrt (+ x (* x x)))))
   (* (pow x -0.5) (/ 0.5 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5e+129) {
		tmp = pow(x, -0.5) / ((1.0 + x) + sqrt((x + (x * x))));
	} else {
		tmp = pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5d+129) then
        tmp = (x ** (-0.5d0)) / ((1.0d0 + x) + sqrt((x + (x * x))))
    else
        tmp = (x ** (-0.5d0)) * (0.5d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e+129) {
		tmp = Math.pow(x, -0.5) / ((1.0 + x) + Math.sqrt((x + (x * x))));
	} else {
		tmp = Math.pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e+129:
		tmp = math.pow(x, -0.5) / ((1.0 + x) + math.sqrt((x + (x * x))))
	else:
		tmp = math.pow(x, -0.5) * (0.5 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+129)
		tmp = Float64((x ^ -0.5) / Float64(Float64(1.0 + x) + sqrt(Float64(x + Float64(x * x)))));
	else
		tmp = Float64((x ^ -0.5) * Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e+129)
		tmp = (x ^ -0.5) / ((1.0 + x) + sqrt((x + (x * x))));
	else
		tmp = (x ^ -0.5) * (0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e+129], N[(N[Power[x, -0.5], $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] + N[Sqrt[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000003e129

   1. Initial program 76.1%

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

      1. frac-sub 76.1%

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

      2. *-un-lft-identity 76.1%

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

      3. +-commutative 76.1%

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

      4. *-rgt-identity 76.1%

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

      5. sqrt-unprod 76.1%

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

      6. +-commutative 76.1%

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

   3. Applied egg-rr 76.1%

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

      1. flip-- 76.4%

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

      2. add-sqr-sqrt 76.9%

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

      3. +-commutative 76.9%

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

      4. add-sqr-sqrt 77.2%

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

      5. associate--l+ 77.2%

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

      6. +-commutative 77.2%

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

   5. Applied egg-rr 77.2%

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

      1. div-inv 77.2%

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

      2. sqrt-prod 77.2%

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

      3. +-commutative 77.2%

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

      4. times-frac 77.2%

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

   7. Applied egg-rr 77.2%

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

      1. associate-/l/ 77.2%

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

      2. associate-*r/ 77.2%

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

      3. metadata-eval 77.2%

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

      4. times-frac 77.2%

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

      5. *-rgt-identity 77.2%

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

      6. *-rgt-identity 77.2%

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

      7. associate-+r- 77.2%

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

      8. +-commutative 77.2%

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

      9. associate--l+ 99.5%

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

      10. distribute-lft-in 99.5%

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

      11. rem-square-sqrt 99.6%

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

      12. +-commutative 99.6%

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

      13. +-commutative 99.6%

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

   9. Simplified 99.6%

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

       1. expm1-log1p-u 94.5%

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

       2. expm1-udef 70.5%

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

       3. +-inverses 70.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1 + \color{blue}{0}}{\sqrt{x}}}{\left(1 + x\right) + \sqrt{1 + x} \cdot \sqrt{x}}\right)} - 1 \]

       4. metadata-eval 70.5%

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

       5. pow1/2 70.5%

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

       6. pow-flip 70.5%

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

       7. metadata-eval 70.5%

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

       8. *-commutative 70.5%

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

       9. sqrt-unprod 70.5%

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

   11. Applied egg-rr 70.5%

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

       1. expm1-def 94.6%

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

       2. expm1-log1p 99.9%

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

       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%

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

   13. Simplified 99.9%

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

   if 5.0000000000000003e129 < x

   1. Initial program 56.8%

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

      1. frac-sub 56.8%

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

      2. *-un-lft-identity 56.8%

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

      3. +-commutative 56.8%

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

      4. *-rgt-identity 56.8%

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

      5. sqrt-unprod 56.8%

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

      6. +-commutative 56.8%

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

   3. Applied egg-rr 56.8%

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

      1. add-cube-cbrt 56.8%

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

      2. sqrt-prod 56.8%

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

      3. times-frac 56.8%

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

      4. pow2 56.8%

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

      5. +-commutative 56.8%

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

      6. +-commutative 56.8%

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

      7. +-commutative 56.8%

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

   5. Applied egg-rr 56.8%

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

      1. associate-*l/ 56.8%

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

      2. associate-*r/ 56.8%

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

      3. unpow2 56.8%

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

      4. rem-3cbrt-lft 56.8%

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

      5. div-sub 56.8%

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

      6. *-inverses 56.8%

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

   7. Simplified 56.8%

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

   8. Taylor expanded in x around inf 99.8%

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

      1. div-inv 99.8%

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

      2. metadata-eval 99.8%

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

      3. add-sqr-sqrt 99.5%

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

      4. frac-times 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-inverses 99.5%

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

      7. metadata-eval 99.5%

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

      8. +-inverses 99.5%

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

      9. *-un-lft-identity 99.5%

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

      10. times-frac 99.4%

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

      11. metadata-eval 99.4%

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

      12. un-div-inv 99.3%

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

      13. metadata-eval 99.3%

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

      14. +-inverses 99.3%

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

      15. pow3 99.3%

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

   10. Applied egg-rr 99.4%

       \[\leadsto \color{blue}{0.5 \cdot {\left({x}^{-0.5}\right)}^{3}} \]
   11. Step-by-step derivation

       1. unpow3 99.5%

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

       2. pow-sqr 99.8%

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

       3. metadata-eval 99.8%

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

       4. unpow-1 99.8%

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

       5. associate-*r* 99.8%

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

       6. associate-*r/ 99.8%

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

       7. metadata-eval 99.8%

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

   12. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+129}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(1 + x\right) + \sqrt{x + x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. frac-sub 70.6%

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

   2. *-un-lft-identity 70.6%

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

   3. +-commutative 70.6%

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

   4. *-rgt-identity 70.6%

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

   5. sqrt-unprod 70.6%

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

   6. +-commutative 70.6%

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

3. Applied egg-rr 70.6%

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

   1. flip-- 70.8%

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

   2. add-sqr-sqrt 71.2%

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

   3. +-commutative 71.2%

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

   4. add-sqr-sqrt 71.4%

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

   5. associate--l+ 71.4%

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

   6. +-commutative 71.4%

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

5. Applied egg-rr 71.4%

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

   1. div-inv 71.4%

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

   2. sqrt-prod 71.4%

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

   3. +-commutative 71.4%

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

   4. times-frac 71.4%

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

7. Applied egg-rr 71.4%

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

   1. associate-/l/ 71.4%

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

   2. associate-*r/ 71.4%

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

   3. metadata-eval 71.4%

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

   4. times-frac 71.4%

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

   5. *-rgt-identity 71.4%

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

   6. *-rgt-identity 71.4%

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

   7. associate-+r- 71.4%

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

   8. +-commutative 71.4%

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

   9. associate--l+ 99.5%

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

   10. distribute-lft-in 99.5%

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

   11. rem-square-sqrt 99.6%

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

   12. +-commutative 99.6%

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

   13. +-commutative 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(x - x\right)}{\sqrt{x}}}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (/ (/ (+ 1.0 (- x x)) (sqrt x)) (+ 1.5 (- (* x 2.0) (/ 0.125 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 6000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = ((1.0 + (x - x)) / sqrt(x)) / (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 <= 6000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = ((1.0d0 + (x - x)) / sqrt(x)) / (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 <= 6000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = ((1.0 + (x - x)) / Math.sqrt(x)) / (1.5 + ((x * 2.0) - (0.125 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = ((1.0 + (x - x)) / math.sqrt(x)) / (1.5 + ((x * 2.0) - (0.125 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x - x)) / sqrt(x)) / 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 <= 6000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = ((1.0 + (x - x)) / sqrt(x)) / (1.5 + ((x * 2.0) - (0.125 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $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 < 6e3

   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%

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

   if 6e3 < x

   1. Initial program 37.8%

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

      1. frac-sub 37.9%

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

      2. *-un-lft-identity 37.9%

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

      3. +-commutative 37.9%

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

      4. *-rgt-identity 37.9%

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

      5. sqrt-unprod 37.9%

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

      6. +-commutative 37.9%

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

   3. Applied egg-rr 37.9%

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

      1. flip-- 38.2%

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

      2. add-sqr-sqrt 38.9%

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

      3. +-commutative 38.9%

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

      4. add-sqr-sqrt 39.4%

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

      5. associate--l+ 39.4%

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

      6. +-commutative 39.4%

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

   5. Applied egg-rr 39.4%

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

      1. div-inv 39.4%

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

      2. sqrt-prod 39.4%

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

      3. +-commutative 39.4%

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

      4. times-frac 39.4%

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

   7. Applied egg-rr 39.4%

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

      1. associate-/l/ 39.4%

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

      2. associate-*r/ 39.4%

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

      3. metadata-eval 39.4%

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

      4. times-frac 39.4%

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

      5. *-rgt-identity 39.4%

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

      6. *-rgt-identity 39.4%

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

      7. associate-+r- 39.4%

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

      8. +-commutative 39.4%

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

      9. associate--l+ 99.3%

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

      10. distribute-lft-in 99.3%

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

      11. rem-square-sqrt 99.5%

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

      12. +-commutative 99.5%

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

      13. +-commutative 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in x around inf 99.7%

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

       1. associate--l+ 99.7%

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

       2. *-commutative 99.7%

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

       3. associate-*r/ 99.7%

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

       4. metadata-eval 99.7%

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

   12. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(x - x\right)}{\sqrt{x}}}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}\\ \end{array} \]

Alternative 4: 99.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.43:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(x - x\right)}{\sqrt{x}}}{\left(1 + x\right) + \left(\frac{0.0625}{x \cdot x} + \left(x + \left(0.5 + \frac{-0.125}{x}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.43)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/
    (/ (+ 1.0 (- x x)) (sqrt x))
    (+ (+ 1.0 x) (+ (/ 0.0625 (* x x)) (+ 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 = ((1.0 + (x - x)) / sqrt(x)) / ((1.0 + x) + ((0.0625 / (x * x)) + (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 = ((1.0d0 + (x - x)) / sqrt(x)) / ((1.0d0 + x) + ((0.0625d0 / (x * x)) + (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 = ((1.0 + (x - x)) / Math.sqrt(x)) / ((1.0 + x) + ((0.0625 / (x * x)) + (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 = ((1.0 + (x - x)) / math.sqrt(x)) / ((1.0 + x) + ((0.0625 / (x * x)) + (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(Float64(Float64(1.0 + Float64(x - x)) / sqrt(x)) / Float64(Float64(1.0 + x) + Float64(Float64(0.0625 / Float64(x * x)) + Float64(x + Float64(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 = ((1.0 + (x - x)) / sqrt(x)) / ((1.0 + x) + ((0.0625 / (x * x)) + (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[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] + N[(N[(0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.5 + N[(-0.125 / x), $MachinePrecision]), $MachinePrecision]), $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. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.5%

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

   if 0.429999999999999993 < x

   1. Initial program 38.7%

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

      1. frac-sub 38.8%

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

      2. *-un-lft-identity 38.8%

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

      3. +-commutative 38.8%

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

      4. *-rgt-identity 38.8%

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

      5. sqrt-unprod 38.8%

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

      6. +-commutative 38.8%

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

   3. Applied egg-rr 38.8%

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

      1. flip-- 39.1%

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

      2. add-sqr-sqrt 39.9%

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

      3. +-commutative 39.9%

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

      4. add-sqr-sqrt 40.3%

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

      5. associate--l+ 40.3%

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

      6. +-commutative 40.3%

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

   5. Applied egg-rr 40.3%

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

      1. div-inv 40.3%

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

      2. sqrt-prod 40.4%

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

      3. +-commutative 40.4%

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

      4. times-frac 40.3%

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

   7. Applied egg-rr 40.3%

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

      1. associate-/l/ 40.3%

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

      2. associate-*r/ 40.4%

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

      3. metadata-eval 40.4%

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

      4. times-frac 40.4%

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

      5. *-rgt-identity 40.4%

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

      6. *-rgt-identity 40.4%

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

      7. associate-+r- 40.3%

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

      8. +-commutative 40.3%

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

      9. associate--l+ 99.3%

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

      10. distribute-lft-in 99.3%

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

      11. rem-square-sqrt 99.5%

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

      12. +-commutative 99.5%

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

      13. +-commutative 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in x around inf 98.9%

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

   12. Simplified 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.43:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(x - x\right)}{\sqrt{x}}}{\left(1 + x\right) + \left(\frac{0.0625}{x \cdot x} + \left(x + \left(0.5 + \frac{-0.125}{x}\right)\right)\right)}\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.43)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ (/ (+ 1.0 (- x x)) (sqrt x)) (+ (+ 1.0 x) (+ 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 = ((1.0 + (x - x)) / sqrt(x)) / ((1.0 + x) + (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 = ((1.0d0 + (x - x)) / sqrt(x)) / ((1.0d0 + x) + (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 = ((1.0 + (x - x)) / Math.sqrt(x)) / ((1.0 + x) + (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 = ((1.0 + (x - x)) / math.sqrt(x)) / ((1.0 + x) + (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(Float64(Float64(1.0 + Float64(x - x)) / sqrt(x)) / Float64(Float64(1.0 + x) + Float64(x + Float64(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 = ((1.0 + (x - x)) / sqrt(x)) / ((1.0 + x) + (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[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] + N[(x + N[(0.5 + N[(-0.125 / x), $MachinePrecision]), $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. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.5%

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

   if 0.429999999999999993 < x

   1. Initial program 38.7%

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

      1. frac-sub 38.8%

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

      2. *-un-lft-identity 38.8%

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

      3. +-commutative 38.8%

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

      4. *-rgt-identity 38.8%

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

      5. sqrt-unprod 38.8%

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

      6. +-commutative 38.8%

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

   3. Applied egg-rr 38.8%

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

      1. flip-- 39.1%

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

      2. add-sqr-sqrt 39.9%

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

      3. +-commutative 39.9%

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

      4. add-sqr-sqrt 40.3%

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

      5. associate--l+ 40.3%

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

      6. +-commutative 40.3%

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

   5. Applied egg-rr 40.3%

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

      1. div-inv 40.3%

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

      2. sqrt-prod 40.4%

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

      3. +-commutative 40.4%

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

      4. times-frac 40.3%

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

   7. Applied egg-rr 40.3%

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

      1. associate-/l/ 40.3%

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

      2. associate-*r/ 40.4%

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

      3. metadata-eval 40.4%

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

      4. times-frac 40.4%

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

      5. *-rgt-identity 40.4%

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

      6. *-rgt-identity 40.4%

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

      7. associate-+r- 40.3%

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

      8. +-commutative 40.3%

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

      9. associate--l+ 99.3%

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

      10. distribute-lft-in 99.3%

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

      11. rem-square-sqrt 99.5%

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

      12. +-commutative 99.5%

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

      13. +-commutative 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in x around inf 98.8%

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

       1. associate--l+ 98.8%

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

       2. *-commutative 98.8%

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

       3. cancel-sign-sub-inv 98.8%

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

       4. *-lft-identity 98.8%

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

       5. metadata-eval 98.8%

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

       6. rem-square-sqrt 0.0%

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

       7. unpow2 0.0%

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

       8. associate-*r* 0.0%

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

       9. distribute-neg-frac 0.0%

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

       10. metadata-eval 0.0%

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

       11. metadata-eval 0.0%

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

       12. rem-square-sqrt 0.0%

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

       13. unpow2 0.0%

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

       14. associate-/r* 0.0%

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

       15. *-commutative 0.0%

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

       16. +-commutative 0.0%

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

       17. +-commutative 0.0%

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

   12. Simplified 98.8%

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

4. Final simplification 99.2%

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

Alternative 6: 99.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.43)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ (/ (+ 1.0 (- x x)) (sqrt x)) (+ 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 = ((1.0 + (x - x)) / sqrt(x)) / (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 = ((1.0d0 + (x - x)) / sqrt(x)) / (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 = ((1.0 + (x - x)) / Math.sqrt(x)) / (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 = ((1.0 + (x - x)) / math.sqrt(x)) / (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(Float64(Float64(1.0 + Float64(x - x)) / sqrt(x)) / 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 = ((1.0 + (x - x)) / sqrt(x)) / (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[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $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. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.5%

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

   if 0.429999999999999993 < x

   1. Initial program 38.7%

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

      1. frac-sub 38.8%

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

      2. *-un-lft-identity 38.8%

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

      3. +-commutative 38.8%

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

      4. *-rgt-identity 38.8%

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

      5. sqrt-unprod 38.8%

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

      6. +-commutative 38.8%

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

   3. Applied egg-rr 38.8%

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

      1. flip-- 39.1%

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

      2. add-sqr-sqrt 39.9%

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

      3. +-commutative 39.9%

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

      4. add-sqr-sqrt 40.3%

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

      5. associate--l+ 40.3%

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

      6. +-commutative 40.3%

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

   5. Applied egg-rr 40.3%

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

      1. div-inv 40.3%

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

      2. sqrt-prod 40.4%

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

      3. +-commutative 40.4%

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

      4. times-frac 40.3%

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

   7. Applied egg-rr 40.3%

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

      1. associate-/l/ 40.3%

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

      2. associate-*r/ 40.4%

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

      3. metadata-eval 40.4%

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

      4. times-frac 40.4%

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

      5. *-rgt-identity 40.4%

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

      6. *-rgt-identity 40.4%

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

      7. associate-+r- 40.3%

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

      8. +-commutative 40.3%

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

      9. associate--l+ 99.3%

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

      10. distribute-lft-in 99.3%

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

      11. rem-square-sqrt 99.5%

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

      12. +-commutative 99.5%

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

      13. +-commutative 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in x around inf 98.8%

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

       1. associate--l+ 98.8%

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

       2. *-commutative 98.8%

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

       3. associate-*r/ 98.8%

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

       4. metadata-eval 98.8%

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

   12. Simplified 98.8%

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

4. Final simplification 99.2%

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

Alternative 7: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.4:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(x - x\right)}{\sqrt{x}}}{1.5 + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.4)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ (/ (+ 1.0 (- x x)) (sqrt x)) (+ 1.5 (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.4) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = ((1.0 + (x - x)) / sqrt(x)) / (1.5 + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.4d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = ((1.0d0 + (x - x)) / sqrt(x)) / (1.5d0 + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.4) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = ((1.0 + (x - x)) / Math.sqrt(x)) / (1.5 + (x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.4:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = ((1.0 + (x - x)) / math.sqrt(x)) / (1.5 + (x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.4)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x - x)) / sqrt(x)) / Float64(1.5 + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.4)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = ((1.0 + (x - x)) / sqrt(x)) / (1.5 + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.4], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / N[(1.5 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.40000000000000002

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.5%

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

   if 0.40000000000000002 < x

   1. Initial program 38.7%

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

      1. frac-sub 38.8%

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

      2. *-un-lft-identity 38.8%

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

      3. +-commutative 38.8%

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

      4. *-rgt-identity 38.8%

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

      5. sqrt-unprod 38.8%

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

      6. +-commutative 38.8%

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

   3. Applied egg-rr 38.8%

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

      1. flip-- 39.1%

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

      2. add-sqr-sqrt 39.9%

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

      3. +-commutative 39.9%

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

      4. add-sqr-sqrt 40.3%

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

      5. associate--l+ 40.3%

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

      6. +-commutative 40.3%

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

   5. Applied egg-rr 40.3%

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

      1. div-inv 40.3%

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

      2. sqrt-prod 40.4%

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

      3. +-commutative 40.4%

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

      4. times-frac 40.3%

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

   7. Applied egg-rr 40.3%

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

      1. associate-/l/ 40.3%

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

      2. associate-*r/ 40.4%

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

      3. metadata-eval 40.4%

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

      4. times-frac 40.4%

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

      5. *-rgt-identity 40.4%

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

      6. *-rgt-identity 40.4%

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

      7. associate-+r- 40.3%

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

      8. +-commutative 40.3%

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

      9. associate--l+ 99.3%

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

      10. distribute-lft-in 99.3%

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

      11. rem-square-sqrt 99.5%

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

      12. +-commutative 99.5%

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

      13. +-commutative 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in x around inf 98.6%

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

       1. +-commutative 98.6%

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

       2. *-commutative 98.6%

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

   12. Simplified 98.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.4:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(x - x\right)}{\sqrt{x}}}{1.5 + x \cdot 2}\\ \end{array} \]

Alternative 8: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x} - \frac{0.375}{x \cdot x}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.1)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ (- (/ 0.5 x) (/ 0.375 (* x x))) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.1) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = ((0.5 / x) - (0.375 / (x * x))) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.1d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = ((0.5d0 / x) - (0.375d0 / (x * x))) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.1) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = ((0.5 / x) - (0.375 / (x * x))) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.1:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = ((0.5 / x) - (0.375 / (x * x))) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.1)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * x))) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.1)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = ((0.5 / x) - (0.375 / (x * x))) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.1], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.5%

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

   if 1.1000000000000001 < x

   1. Initial program 38.7%

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

      1. frac-sub 38.8%

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

      2. *-un-lft-identity 38.8%

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

      3. +-commutative 38.8%

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

      4. *-rgt-identity 38.8%

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

      5. sqrt-unprod 38.8%

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

      6. +-commutative 38.8%

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

   3. Applied egg-rr 38.8%

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

      1. add-cube-cbrt 38.7%

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

      2. sqrt-prod 38.7%

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

      3. times-frac 38.7%

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

      4. pow2 38.7%

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

      5. +-commutative 38.7%

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

      6. +-commutative 38.7%

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

      7. +-commutative 38.7%

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

   5. Applied egg-rr 38.7%

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

      1. associate-*l/ 38.7%

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

      2. associate-*r/ 38.7%

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

      3. unpow2 38.7%

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

      4. rem-3cbrt-lft 38.8%

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

      5. div-sub 38.8%

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

      6. *-inverses 38.8%

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

   7. Simplified 38.8%

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

   8. Taylor expanded in x around inf 98.5%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{1}{x} - 0.375 \cdot \frac{1}{{x}^{2}}}}{\sqrt{x}} \]
   9. Step-by-step derivation

      1. associate-*r/ 98.5%

         \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.375 \cdot \frac{1}{{x}^{2}}}{\sqrt{x}} \]

      2. metadata-eval 98.5%

         \[\leadsto \frac{\frac{\color{blue}{0.5}}{x} - 0.375 \cdot \frac{1}{{x}^{2}}}{\sqrt{x}} \]

      3. unpow2 98.5%

         \[\leadsto \frac{\frac{0.5}{x} - 0.375 \cdot \frac{1}{\color{blue}{x \cdot x}}}{\sqrt{x}} \]

      4. associate-*r/ 98.5%

         \[\leadsto \frac{\frac{0.5}{x} - \color{blue}{\frac{0.375 \cdot 1}{x \cdot x}}}{\sqrt{x}} \]

      5. metadata-eval 98.5%

         \[\leadsto \frac{\frac{0.5}{x} - \frac{\color{blue}{0.375}}{x \cdot x}}{\sqrt{x}} \]

   10. Simplified 98.5%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{x} - \frac{0.375}{x \cdot x}}}{\sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x} - \frac{0.375}{x \cdot x}}{\sqrt{x}}\\ \end{array} \]

Alternative 9: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = (0.5 / x) / Math.sqrt(x);
	}
	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 = (0.5 / x) / math.sqrt(x)
	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(Float64(0.5 / x) / sqrt(x));
	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 = (0.5 / x) / sqrt(x);
	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[(0.5 / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $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. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.5%

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

   if 1 < x

   1. Initial program 38.7%

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

      1. frac-sub 38.8%

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

      2. *-un-lft-identity 38.8%

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

      3. +-commutative 38.8%

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

      4. *-rgt-identity 38.8%

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

      5. sqrt-unprod 38.8%

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

      6. +-commutative 38.8%

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

   3. Applied egg-rr 38.8%

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

      1. add-cube-cbrt 38.7%

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

      2. sqrt-prod 38.7%

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

      3. times-frac 38.7%

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

      4. pow2 38.7%

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

      5. +-commutative 38.7%

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

      6. +-commutative 38.7%

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

      7. +-commutative 38.7%

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

   5. Applied egg-rr 38.7%

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

      1. associate-*l/ 38.7%

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

      2. associate-*r/ 38.7%

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

      3. unpow2 38.7%

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

      4. rem-3cbrt-lft 38.8%

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

      5. div-sub 38.8%

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

      6. *-inverses 38.8%

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

   7. Simplified 38.8%

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

   8. Taylor expanded in x around inf 97.4%

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

4. Final simplification 98.5%

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

Alternative 10: 97.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.66)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(0.5 / Float64(x * sqrt(x)));
	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 = 0.5 / (x * sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.66], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(0.5 / N[(x * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $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. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.3%

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

   if 0.660000000000000031 < x

   1. Initial program 38.7%

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

      1. frac-sub 38.8%

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

      2. *-un-lft-identity 38.8%

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

      3. +-commutative 38.8%

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

      4. *-rgt-identity 38.8%

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

      5. sqrt-unprod 38.8%

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

      6. +-commutative 38.8%

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

   3. Applied egg-rr 38.8%

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

      1. add-cube-cbrt 38.7%

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

      2. sqrt-prod 38.7%

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

      3. times-frac 38.7%

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

      4. pow2 38.7%

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

      5. +-commutative 38.7%

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

      6. +-commutative 38.7%

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

      7. +-commutative 38.7%

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

   5. Applied egg-rr 38.7%

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

      1. associate-*l/ 38.7%

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

      2. associate-*r/ 38.7%

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

      3. unpow2 38.7%

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

      4. rem-3cbrt-lft 38.8%

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

      5. div-sub 38.8%

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

      6. *-inverses 38.8%

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

   7. Simplified 38.8%

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

   8. Taylor expanded in x around inf 97.4%

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

      1. add-cube-cbrt 96.3%

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

      2. pow3 96.3%

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

      3. associate-/l/ 95.6%

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

      4. cbrt-div 95.6%

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

      5. add-sqr-sqrt 95.4%

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

      6. cube-unmult 95.4%

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

      7. pow3 95.4%

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

      8. add-cbrt-cube 96.5%

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

   10. Applied egg-rr 96.5%

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

       1. cube-div 95.7%

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

       2. rem-cube-cbrt 96.2%

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

       3. unpow3 96.3%

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

       4. rem-square-sqrt 96.6%

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

   12. Simplified 96.6%

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

4. Final simplification 98.0%

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

Alternative 11: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.660000000000000031

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.3%

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

   if 0.660000000000000031 < x

   1. Initial program 38.7%

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

      1. frac-sub 38.8%

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

      2. *-un-lft-identity 38.8%

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

      3. +-commutative 38.8%

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

      4. *-rgt-identity 38.8%

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

      5. sqrt-unprod 38.8%

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

      6. +-commutative 38.8%

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

   3. Applied egg-rr 38.8%

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

      1. add-cube-cbrt 38.7%

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

      2. sqrt-prod 38.7%

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

      3. times-frac 38.7%

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

      4. pow2 38.7%

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

      5. +-commutative 38.7%

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

      6. +-commutative 38.7%

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

      7. +-commutative 38.7%

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

   5. Applied egg-rr 38.7%

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

      1. associate-*l/ 38.7%

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

      2. associate-*r/ 38.7%

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

      3. unpow2 38.7%

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

      4. rem-3cbrt-lft 38.8%

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

      5. div-sub 38.8%

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

      6. *-inverses 38.8%

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

   7. Simplified 38.8%

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

   8. Taylor expanded in x around inf 97.4%

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

4. Final simplification 98.4%

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

Alternative 12: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{0.5}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.62) (+ (pow x -0.5) -1.0) (- (/ 1.0 x) (/ 0.5 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.62) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = (1.0 / x) - (0.5 / (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.62d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (1.0d0 / x) - (0.5d0 / (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.62) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = (1.0 / x) - (0.5 / (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.62:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = (1.0 / x) - (0.5 / (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.62)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.62)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (1.0 / x) - (0.5 / (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.62], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.619999999999999996

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.3%

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

   if 0.619999999999999996 < x

   1. Initial program 38.7%

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

      1. frac-sub 38.8%

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

      2. *-un-lft-identity 38.8%

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

      3. +-commutative 38.8%

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

      4. *-rgt-identity 38.8%

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

      5. sqrt-unprod 38.8%

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

      6. +-commutative 38.8%

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

   3. Applied egg-rr 38.8%

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

   4. Taylor expanded in x around inf 37.8%

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

      1. +-commutative 37.8%

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

   6. Simplified 37.8%

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

   7. Taylor expanded in x around 0 7.8%

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

   8. Taylor expanded in x around inf 7.8%

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

      1. associate-*r/ 7.8%

         \[\leadsto \frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} \]

      2. metadata-eval 7.8%

         \[\leadsto \frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}} \]

      3. unpow2 7.8%

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

   10. Simplified 7.8%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{0.5}{x \cdot x}\\ \end{array} \]

Alternative 13: 97.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.66) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 0.5 / pow(x, 1.5);
	}
	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 = 0.5d0 / (x ** 1.5d0)
    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 = 0.5 / Math.pow(x, 1.5);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.66)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(0.5 / (x ^ 1.5));
	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 = 0.5 / (x ^ 1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.66], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(0.5 / N[Power[x, 1.5], $MachinePrecision]), $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. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.3%

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

   if 0.660000000000000031 < x

   1. Initial program 38.7%

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

      1. frac-sub 38.8%

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

      2. *-un-lft-identity 38.8%

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

      3. +-commutative 38.8%

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

      4. *-rgt-identity 38.8%

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

      5. sqrt-unprod 38.8%

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

      6. +-commutative 38.8%

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

   3. Applied egg-rr 38.8%

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

      1. add-cube-cbrt 38.7%

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

      2. sqrt-prod 38.7%

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

      3. times-frac 38.7%

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

      4. pow2 38.7%

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

      5. +-commutative 38.7%

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

      6. +-commutative 38.7%

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

      7. +-commutative 38.7%

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

   5. Applied egg-rr 38.7%

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

      1. associate-*l/ 38.7%

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

      2. associate-*r/ 38.7%

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

      3. unpow2 38.7%

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

      4. rem-3cbrt-lft 38.8%

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

      5. div-sub 38.8%

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

      6. *-inverses 38.8%

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

   7. Simplified 38.8%

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

   8. Taylor expanded in x around inf 97.4%

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

      1. expm1-log1p-u 97.4%

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

      2. expm1-udef 36.8%

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

      3. associate-/l/ 36.8%

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

      4. pow1/2 36.8%

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

      5. pow-plus 36.8%

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

      6. metadata-eval 36.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{{x}^{\color{blue}{1.5}}}\right)} - 1 \]

   10. Applied egg-rr 36.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5}{{x}^{1.5}}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 96.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{{x}^{1.5}}\right)\right)} \]

       2. expm1-log1p 96.6%

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

   12. Simplified 96.6%

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

4. Final simplification 98.0%

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

Alternative 14: 50.4% 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 70.6%

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

   1. sub-neg 70.6%

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

   2. +-commutative 70.6%

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

   3. add-cube-cbrt 57.2%

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

   4. distribute-lft-neg-in 57.2%

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

   5. fma-def 56.0%

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

3. Applied egg-rr 56.3%

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

4. Taylor expanded in x around inf 52.9%

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

   1. inv-pow 52.9%

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

   2. sqrt-pow1 52.9%

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

   3. metadata-eval 52.9%

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

   4. expm1-log1p-u 49.3%

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

   5. expm1-udef 63.9%

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

6. Applied egg-rr 63.9%

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

   1. expm1-def 49.3%

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

   2. expm1-log1p 52.9%

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

8. Simplified 52.9%

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

9. Final simplification 52.9%

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

Alternative 15: 7.5% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / (x + 0.5)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. frac-sub 70.6%

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

   2. *-un-lft-identity 70.6%

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

   3. +-commutative 70.6%

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

   4. *-rgt-identity 70.6%

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

   5. sqrt-unprod 70.6%

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

   6. +-commutative 70.6%

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

3. Applied egg-rr 70.6%

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

4. Taylor expanded in x around inf 21.6%

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

   1. +-commutative 21.6%

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

6. Simplified 21.6%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{x + 0.5}} \]

7. Taylor expanded in x around 0 7.3%

   \[\leadsto \frac{\color{blue}{1}}{x + 0.5} \]

8. Final simplification 7.3%

   \[\leadsto \frac{1}{x + 0.5} \]

Alternative 16: 7.4% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 x))

C

double code(double x) {
	return 1.0 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / x
end function

Java

public static double code(double x) {
	return 1.0 / x;
}

Python

def code(x):
	return 1.0 / x

Julia

function code(x)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / x;
end

Wolfram

code[x_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 70.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Step-by-step derivation

   1. frac-sub 70.6%

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   2. *-un-lft-identity 70.6%

      \[\leadsto \frac{\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   3. +-commutative 70.6%

      \[\leadsto \frac{\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   4. *-rgt-identity 70.6%

      \[\leadsto \frac{\sqrt{1 + x} - \color{blue}{\sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   5. sqrt-unprod 70.6%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{\sqrt{x \cdot \left(x + 1\right)}}} \]

   6. +-commutative 70.6%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot \color{blue}{\left(1 + x\right)}}} \]

3. Applied egg-rr 70.6%

   \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot \left(1 + x\right)}}} \]

4. Taylor expanded in x around inf 21.6%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{0.5 + x}} \]
5. Step-by-step derivation

   1. +-commutative 21.6%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{x + 0.5}} \]

6. Simplified 21.6%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{x + 0.5}} \]

7. Taylor expanded in x around 0 7.3%

   \[\leadsto \frac{\color{blue}{1}}{x + 0.5} \]

8. Taylor expanded in x around inf 7.3%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

9. Final simplification 7.3%

   \[\leadsto \frac{1}{x} \]

Alternative 17: 5.8% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 70.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Step-by-step derivation

   1. frac-sub 70.6%

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   2. *-un-lft-identity 70.6%

      \[\leadsto \frac{\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   3. +-commutative 70.6%

      \[\leadsto \frac{\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   4. *-rgt-identity 70.6%

      \[\leadsto \frac{\sqrt{1 + x} - \color{blue}{\sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   5. sqrt-unprod 70.6%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{\sqrt{x \cdot \left(x + 1\right)}}} \]

   6. +-commutative 70.6%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot \color{blue}{\left(1 + x\right)}}} \]

3. Applied egg-rr 70.6%

   \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot \left(1 + x\right)}}} \]

4. Taylor expanded in x around inf 21.6%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{0.5 + x}} \]
5. Step-by-step derivation

   1. +-commutative 21.6%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{x + 0.5}} \]

6. Simplified 21.6%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{x + 0.5}} \]

7. Taylor expanded in x around 0 5.8%

   \[\leadsto \color{blue}{2} \]

8. Final simplification 5.8%

   \[\leadsto 2 \]

Developer target: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0))))))

C

double code(double x) {
	return 1.0 / (((x + 1.0) * sqrt(x)) + (x * sqrt((x + 1.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (((x + 1.0d0) * sqrt(x)) + (x * sqrt((x + 1.0d0))))
end function

Java

public static double code(double x) {
	return 1.0 / (((x + 1.0) * Math.sqrt(x)) + (x * Math.sqrt((x + 1.0))));
}

Python

def code(x):
	return 1.0 / (((x + 1.0) * math.sqrt(x)) + (x * math.sqrt((x + 1.0))))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(x + 1.0) * sqrt(x)) + Float64(x * sqrt(Float64(x + 1.0)))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (((x + 1.0) * sqrt(x)) + (x * sqrt((x + 1.0))));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(x + 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x * N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023200 
(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)))))