2isqrt (example 3.6)

Percentage Accurate: 68.9% → 99.7%

Time: 13.5s

Alternatives: 15

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.9% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{-0.375}{{x}^{2}}\right)}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - e^{-0.5 \cdot \mathsf{log1p}\left(x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ x 1.0)))) 5e-9)
   (/ (+ (/ 0.5 x) (+ (/ 0.3125 (pow x 3.0)) (/ -0.375 (pow x 2.0)))) (sqrt x))
   (- (pow x -0.5) (exp (* -0.5 (log1p x))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 5e-9) {
		tmp = ((0.5 / x) + ((0.3125 / pow(x, 3.0)) + (-0.375 / pow(x, 2.0)))) / sqrt(x);
	} else {
		tmp = pow(x, -0.5) - exp((-0.5 * log1p(x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((x + 1.0)))) <= 5e-9) {
		tmp = ((0.5 / x) + ((0.3125 / Math.pow(x, 3.0)) + (-0.375 / Math.pow(x, 2.0)))) / Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) - Math.exp((-0.5 * Math.log1p(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((x + 1.0)))) <= 5e-9:
		tmp = ((0.5 / x) + ((0.3125 / math.pow(x, 3.0)) + (-0.375 / math.pow(x, 2.0)))) / math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) - math.exp((-0.5 * math.log1p(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(x + 1.0)))) <= 5e-9)
		tmp = Float64(Float64(Float64(0.5 / x) + Float64(Float64(0.3125 / (x ^ 3.0)) + Float64(-0.375 / (x ^ 2.0)))) / sqrt(x));
	else
		tmp = Float64((x ^ -0.5) - exp(Float64(-0.5 * log1p(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[(N[(0.5 / x), $MachinePrecision] + N[(N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Exp[N[(-0.5 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.0000000000000001e-9

   1. Initial program 40.5%

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

      1. frac-sub 40.6%

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

      2. *-un-lft-identity 40.6%

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

      3. *-rgt-identity 40.6%

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

      4. +-commutative 40.6%

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

      5. sqrt-unprod 40.6%

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

      6. +-commutative 40.6%

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

   4. Applied egg-rr 40.6%

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

      1. add-cube-cbrt 40.6%

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

      2. sqrt-prod 40.6%

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

      3. times-frac 40.6%

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

      4. pow2 40.6%

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

   6. Applied egg-rr 40.6%

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

      1. associate-*l/ 40.6%

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

      2. /-rgt-identity 40.6%

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

      3. times-frac 40.6%

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

      4. unpow2 40.6%

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

      5. rem-3cbrt-lft 40.6%

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

      6. *-lft-identity 40.6%

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

      7. div-sub 40.6%

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

      8. *-inverses 40.6%

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

   8. Simplified 40.6%

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

   9. Taylor expanded in x around inf 99.6%

      \[\leadsto \frac{\color{blue}{\left(0.3125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right) - 0.375 \cdot \frac{1}{{x}^{2}}}}{\sqrt{x}} \]
   10. Step-by-step derivation

       1. sub-neg 99.6%

          \[\leadsto \frac{\color{blue}{\left(0.3125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right) + \left(-0.375 \cdot \frac{1}{{x}^{2}}\right)}}{\sqrt{x}} \]

       2. +-commutative 99.6%

          \[\leadsto \frac{\color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right)} + \left(-0.375 \cdot \frac{1}{{x}^{2}}\right)}{\sqrt{x}} \]

       3. associate-+l+ 99.6%

          \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{1}{x} + \left(0.3125 \cdot \frac{1}{{x}^{3}} + \left(-0.375 \cdot \frac{1}{{x}^{2}}\right)\right)}}{\sqrt{x}} \]

       4. associate-*r/ 99.6%

          \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot 1}{x}} + \left(0.3125 \cdot \frac{1}{{x}^{3}} + \left(-0.375 \cdot \frac{1}{{x}^{2}}\right)\right)}{\sqrt{x}} \]

       5. metadata-eval 99.6%

          \[\leadsto \frac{\frac{\color{blue}{0.5}}{x} + \left(0.3125 \cdot \frac{1}{{x}^{3}} + \left(-0.375 \cdot \frac{1}{{x}^{2}}\right)\right)}{\sqrt{x}} \]

       6. associate-*r/ 99.6%

          \[\leadsto \frac{\frac{0.5}{x} + \left(\color{blue}{\frac{0.3125 \cdot 1}{{x}^{3}}} + \left(-0.375 \cdot \frac{1}{{x}^{2}}\right)\right)}{\sqrt{x}} \]

       7. metadata-eval 99.6%

          \[\leadsto \frac{\frac{0.5}{x} + \left(\frac{\color{blue}{0.3125}}{{x}^{3}} + \left(-0.375 \cdot \frac{1}{{x}^{2}}\right)\right)}{\sqrt{x}} \]

       8. associate-*r/ 99.6%

          \[\leadsto \frac{\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \left(-\color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}\right)\right)}{\sqrt{x}} \]

       9. metadata-eval 99.6%

          \[\leadsto \frac{\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \left(-\frac{\color{blue}{0.375}}{{x}^{2}}\right)\right)}{\sqrt{x}} \]

       10. distribute-neg-frac 99.6%

           \[\leadsto \frac{\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \color{blue}{\frac{-0.375}{{x}^{2}}}\right)}{\sqrt{x}} \]

       11. metadata-eval 99.6%

           \[\leadsto \frac{\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{\color{blue}{-0.375}}{{x}^{2}}\right)}{\sqrt{x}} \]

   11. Simplified 99.6%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{-0.375}{{x}^{2}}\right)}}{\sqrt{x}} \]

   if 5.0000000000000001e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

      1. pow-to-exp 100.0%

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

      2. log1p-udef 100.0%

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

      3. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

      \[\leadsto {x}^{-0.5} - \color{blue}{e^{-0.5 \cdot \mathsf{log1p}\left(x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{-0.375}{{x}^{2}}\right)}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - e^{-0.5 \cdot \mathsf{log1p}\left(x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{t_0 + \sqrt{x}}}{\left(x + 0.5\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - e^{-0.5 \cdot \mathsf{log1p}\left(x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 5e-9)
     (/ (/ 1.0 (+ t_0 (sqrt x))) (- (+ x 0.5) (/ 0.125 x)))
     (- (pow x -0.5) (exp (* -0.5 (log1p x)))))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-9) {
		tmp = (1.0 / (t_0 + sqrt(x))) / ((x + 0.5) - (0.125 / x));
	} else {
		tmp = pow(x, -0.5) - exp((-0.5 * log1p(x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 5e-9) {
		tmp = (1.0 / (t_0 + Math.sqrt(x))) / ((x + 0.5) - (0.125 / x));
	} else {
		tmp = Math.pow(x, -0.5) - Math.exp((-0.5 * Math.log1p(x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 5e-9:
		tmp = (1.0 / (t_0 + math.sqrt(x))) / ((x + 0.5) - (0.125 / x))
	else:
		tmp = math.pow(x, -0.5) - math.exp((-0.5 * math.log1p(x)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 5e-9)
		tmp = Float64(Float64(1.0 / Float64(t_0 + sqrt(x))) / Float64(Float64(x + 0.5) - Float64(0.125 / x)));
	else
		tmp = Float64((x ^ -0.5) - exp(Float64(-0.5 * log1p(x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[(1.0 / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 0.5), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Exp[N[(-0.5 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.0000000000000001e-9

   1. Initial program 40.5%

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

      1. frac-sub 40.6%

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

      2. *-un-lft-identity 40.6%

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

      3. *-rgt-identity 40.6%

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

      4. +-commutative 40.6%

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

      5. sqrt-unprod 40.6%

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

      6. +-commutative 40.6%

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

   4. Applied egg-rr 40.6%

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

      1. flip-- 41.0%

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

      2. div-inv 41.0%

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

      3. add-sqr-sqrt 40.7%

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

      4. add-sqr-sqrt 43.0%

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

   6. Applied egg-rr 43.0%

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

      1. associate-*r/ 43.0%

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

      2. *-rgt-identity 43.0%

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

      3. associate--l+ 81.0%

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

      4. +-inverses 81.0%

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

      5. metadata-eval 81.0%

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

      6. +-commutative 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in x around inf 99.6%

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

       1. +-commutative 99.6%

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

       2. associate-*r/ 99.6%

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

       3. metadata-eval 99.6%

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

   11. Simplified 99.6%

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

   if 5.0000000000000001e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

      1. pow-to-exp 100.0%

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

      2. log1p-udef 100.0%

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

      3. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

      \[\leadsto {x}^{-0.5} - \color{blue}{e^{-0.5 \cdot \mathsf{log1p}\left(x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\left(x + 0.5\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - e^{-0.5 \cdot \mathsf{log1p}\left(x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{t_0 + \sqrt{x}}}{\left(x + 0.5\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 5e-9)
     (/ (/ 1.0 (+ t_0 (sqrt x))) (- (+ x 0.5) (/ 0.125 x)))
     (- (pow x -0.5) (pow (+ x 1.0) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-9) {
		tmp = (1.0 / (t_0 + sqrt(x))) / ((x + 0.5) - (0.125 / x));
	} else {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 5d-9) then
        tmp = (1.0d0 / (t_0 + sqrt(x))) / ((x + 0.5d0) - (0.125d0 / x))
    else
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 5e-9) {
		tmp = (1.0 / (t_0 + Math.sqrt(x))) / ((x + 0.5) - (0.125 / x));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 5e-9:
		tmp = (1.0 / (t_0 + math.sqrt(x))) / ((x + 0.5) - (0.125 / x))
	else:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 5e-9)
		tmp = Float64(Float64(1.0 / Float64(t_0 + sqrt(x))) / Float64(Float64(x + 0.5) - Float64(0.125 / x)));
	else
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-9)
		tmp = (1.0 / (t_0 + sqrt(x))) / ((x + 0.5) - (0.125 / x));
	else
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[(1.0 / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 0.5), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.0000000000000001e-9

   1. Initial program 40.5%

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

      1. frac-sub 40.6%

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

      2. *-un-lft-identity 40.6%

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

      3. *-rgt-identity 40.6%

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

      4. +-commutative 40.6%

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

      5. sqrt-unprod 40.6%

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

      6. +-commutative 40.6%

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

   4. Applied egg-rr 40.6%

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

      1. flip-- 41.0%

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

      2. div-inv 41.0%

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

      3. add-sqr-sqrt 40.7%

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

      4. add-sqr-sqrt 43.0%

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

   6. Applied egg-rr 43.0%

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

      1. associate-*r/ 43.0%

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

      2. *-rgt-identity 43.0%

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

      3. associate--l+ 81.0%

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

      4. +-inverses 81.0%

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

      5. metadata-eval 81.0%

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

      6. +-commutative 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in x around inf 99.6%

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

       1. +-commutative 99.6%

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

       2. associate-*r/ 99.6%

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

       3. metadata-eval 99.6%

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

   11. Simplified 99.6%

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

   if 5.0000000000000001e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\left(x + 0.5\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{t_0 + \sqrt{x}}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 5e-9)
     (/ (/ 1.0 (+ t_0 (sqrt x))) (+ x 0.5))
     (- (pow x -0.5) (pow (+ x 1.0) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-9) {
		tmp = (1.0 / (t_0 + sqrt(x))) / (x + 0.5);
	} else {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 5d-9) then
        tmp = (1.0d0 / (t_0 + sqrt(x))) / (x + 0.5d0)
    else
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 5e-9) {
		tmp = (1.0 / (t_0 + Math.sqrt(x))) / (x + 0.5);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 5e-9:
		tmp = (1.0 / (t_0 + math.sqrt(x))) / (x + 0.5)
	else:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 5e-9)
		tmp = Float64(Float64(1.0 / Float64(t_0 + sqrt(x))) / Float64(x + 0.5));
	else
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-9)
		tmp = (1.0 / (t_0 + sqrt(x))) / (x + 0.5);
	else
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[(1.0 / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.0000000000000001e-9

   1. Initial program 40.5%

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

      1. frac-sub 40.6%

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

      2. *-un-lft-identity 40.6%

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

      3. *-rgt-identity 40.6%

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

      4. +-commutative 40.6%

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

      5. sqrt-unprod 40.6%

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

      6. +-commutative 40.6%

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

   4. Applied egg-rr 40.6%

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

      1. flip-- 41.0%

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

      2. div-inv 41.0%

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

      3. add-sqr-sqrt 40.7%

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

      4. add-sqr-sqrt 43.0%

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

   6. Applied egg-rr 43.0%

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

      1. associate-*r/ 43.0%

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

      2. *-rgt-identity 43.0%

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

      3. associate--l+ 81.0%

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

      4. +-inverses 81.0%

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

      5. metadata-eval 81.0%

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

      6. +-commutative 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in x around inf 99.5%

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

       1. +-commutative 99.5%

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

   11. Simplified 99.5%

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

   if 5.0000000000000001e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{0.5}{x} - \frac{0.375}{{x}^{2}}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ x 1.0)))) 5e-9)
   (/ (- (/ 0.5 x) (/ 0.375 (pow x 2.0))) (sqrt x))
   (- (pow x -0.5) (pow (+ x 1.0) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 5e-9) {
		tmp = ((0.5 / x) - (0.375 / pow(x, 2.0))) / sqrt(x);
	} else {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((x + 1.0d0)))) <= 5d-9) then
        tmp = ((0.5d0 / x) - (0.375d0 / (x ** 2.0d0))) / sqrt(x)
    else
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((x + 1.0)))) <= 5e-9) {
		tmp = ((0.5 / x) - (0.375 / Math.pow(x, 2.0))) / Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((x + 1.0)))) <= 5e-9:
		tmp = ((0.5 / x) - (0.375 / math.pow(x, 2.0))) / math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(x + 1.0)))) <= 5e-9)
		tmp = Float64(Float64(Float64(0.5 / x) - Float64(0.375 / (x ^ 2.0))) / sqrt(x));
	else
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 5e-9)
		tmp = ((0.5 / x) - (0.375 / (x ^ 2.0))) / sqrt(x);
	else
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.0000000000000001e-9

   1. Initial program 40.5%

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

      1. frac-sub 40.6%

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

      2. *-un-lft-identity 40.6%

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

      3. *-rgt-identity 40.6%

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

      4. +-commutative 40.6%

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

      5. sqrt-unprod 40.6%

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

      6. +-commutative 40.6%

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

   4. Applied egg-rr 40.6%

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

      1. add-cube-cbrt 40.6%

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

      2. sqrt-prod 40.6%

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

      3. times-frac 40.6%

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

      4. pow2 40.6%

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

   6. Applied egg-rr 40.6%

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

      1. associate-*l/ 40.6%

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

      2. /-rgt-identity 40.6%

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

      3. times-frac 40.6%

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

      4. unpow2 40.6%

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

      5. rem-3cbrt-lft 40.6%

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

      6. *-lft-identity 40.6%

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

      7. div-sub 40.6%

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

      8. *-inverses 40.6%

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

   8. Simplified 40.6%

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

   9. Taylor expanded in x around inf 99.4%

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

       1. associate-*r/ 99.4%

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

       2. metadata-eval 99.4%

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

       3. associate-*r/ 99.4%

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

       4. metadata-eval 99.4%

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

   11. Simplified 99.4%

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

   if 5.0000000000000001e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{0.5}{x} - \frac{0.375}{{x}^{2}}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 10^{-12}:\\ \;\;\;\;\frac{\frac{1}{t_0 + \sqrt{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 1e-12)
     (/ (/ 1.0 (+ t_0 (sqrt x))) x)
     (- (pow x -0.5) (pow (+ x 1.0) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 1e-12) {
		tmp = (1.0 / (t_0 + sqrt(x))) / x;
	} else {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 1d-12) then
        tmp = (1.0d0 / (t_0 + sqrt(x))) / x
    else
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 1e-12) {
		tmp = (1.0 / (t_0 + Math.sqrt(x))) / x;
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 1e-12:
		tmp = (1.0 / (t_0 + math.sqrt(x))) / x
	else:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 1e-12)
		tmp = Float64(Float64(1.0 / Float64(t_0 + sqrt(x))) / x);
	else
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 1e-12)
		tmp = (1.0 / (t_0 + sqrt(x))) / x;
	else
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 1e-12], N[(N[(1.0 / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 9.9999999999999998e-13

   1. Initial program 40.1%

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

      1. frac-sub 40.1%

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

      2. *-un-lft-identity 40.1%

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

      3. *-rgt-identity 40.1%

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

      4. +-commutative 40.1%

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

      5. sqrt-unprod 40.1%

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

      6. +-commutative 40.1%

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

   4. Applied egg-rr 40.1%

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

      1. flip-- 40.0%

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

      2. div-inv 40.0%

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

      3. add-sqr-sqrt 40.1%

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

      4. add-sqr-sqrt 42.0%

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

   6. Applied egg-rr 42.0%

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

      1. associate-*r/ 42.0%

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

      2. *-rgt-identity 42.0%

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

      3. associate--l+ 80.8%

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

      4. +-inverses 80.8%

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

      5. metadata-eval 80.8%

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

      6. +-commutative 80.8%

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

   8. Simplified 80.8%

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

   9. Taylor expanded in x around inf 98.9%

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

   if 9.9999999999999998e-13 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. clear-num 99.2%

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

      3. associate-/r/ 99.2%

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

      4. prod-diff 99.2%

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

      5. *-un-lft-identity 99.2%

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

      6. fma-neg 99.2%

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

      7. *-un-lft-identity 99.2%

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

      8. pow1/2 99.2%

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

      9. pow-flip 99.5%

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

      10. metadata-eval 99.5%

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

      11. pow1/2 99.5%

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

      12. pow-flip 99.5%

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

      13. +-commutative 99.5%

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

      14. metadata-eval 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. associate-+l- 99.5%

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

      2. expm1-log1p 99.5%

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

      3. expm1-def 99.3%

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

      4. associate--l- 99.3%

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

      5. fma-udef 99.3%

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

      6. distribute-lft1-in 99.3%

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

      7. metadata-eval 99.3%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 99.3%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 99.3%

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

      10. expm1-def 99.5%

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

      11. expm1-log1p 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 10^{-12}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 10^{-12}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ x 1.0)))) 1e-12)
   (/ (/ 0.5 x) (sqrt x))
   (- (pow x -0.5) (pow (+ x 1.0) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 1e-12) {
		tmp = (0.5 / x) / sqrt(x);
	} else {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((x + 1.0d0)))) <= 1d-12) then
        tmp = (0.5d0 / x) / sqrt(x)
    else
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((x + 1.0)))) <= 1e-12) {
		tmp = (0.5 / x) / Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((x + 1.0)))) <= 1e-12:
		tmp = (0.5 / x) / math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(x + 1.0)))) <= 1e-12)
		tmp = Float64(Float64(0.5 / x) / sqrt(x));
	else
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 1e-12)
		tmp = (0.5 / x) / sqrt(x);
	else
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-12], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 9.9999999999999998e-13

   1. Initial program 40.1%

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

      1. frac-sub 40.1%

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

      2. *-un-lft-identity 40.1%

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

      3. *-rgt-identity 40.1%

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

      4. +-commutative 40.1%

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

      5. sqrt-unprod 40.1%

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

      6. +-commutative 40.1%

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

   4. Applied egg-rr 40.1%

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

      1. add-cube-cbrt 40.1%

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

      2. sqrt-prod 40.1%

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

      3. times-frac 40.1%

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

      4. pow2 40.1%

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

   6. Applied egg-rr 40.1%

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

      1. associate-*l/ 40.1%

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

      2. /-rgt-identity 40.1%

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

      3. times-frac 40.1%

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

      4. unpow2 40.1%

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

      5. rem-3cbrt-lft 40.1%

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

      6. *-lft-identity 40.1%

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

      7. div-sub 40.1%

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

      8. *-inverses 40.1%

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

   8. Simplified 40.1%

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

   9. Taylor expanded in x around inf 98.9%

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

   if 9.9999999999999998e-13 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. clear-num 99.2%

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

      3. associate-/r/ 99.2%

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

      4. prod-diff 99.2%

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

      5. *-un-lft-identity 99.2%

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

      6. fma-neg 99.2%

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

      7. *-un-lft-identity 99.2%

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

      8. pow1/2 99.2%

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

      9. pow-flip 99.5%

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

      10. metadata-eval 99.5%

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

      11. pow1/2 99.5%

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

      12. pow-flip 99.5%

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

      13. +-commutative 99.5%

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

      14. metadata-eval 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. associate-+l- 99.5%

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

      2. expm1-log1p 99.5%

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

      3. expm1-def 99.3%

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

      4. associate--l- 99.3%

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

      5. fma-udef 99.3%

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

      6. distribute-lft1-in 99.3%

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

      7. metadata-eval 99.3%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 99.3%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 99.3%

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

      10. expm1-def 99.5%

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

      11. expm1-log1p 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 10^{-12}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \frac{{x}^{-0.5}}{t_0 \cdot \left(t_0 + \sqrt{x}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0)))) (/ (pow x -0.5) (* t_0 (+ t_0 (sqrt x))))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	return pow(x, -0.5) / (t_0 * (t_0 + sqrt(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((x + 1.0d0))
    code = (x ** (-0.5d0)) / (t_0 * (t_0 + sqrt(x)))
end function

Java

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

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	return math.pow(x, -0.5) / (t_0 * (t_0 + math.sqrt(x)))

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	return Float64((x ^ -0.5) / Float64(t_0 * Float64(t_0 + sqrt(x))))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = (x ^ -0.5) / (t_0 * (t_0 + sqrt(x)));
end

Wolfram

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

Derivation

1. Initial program 71.5%

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

   1. frac-sub 71.5%

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

   2. *-un-lft-identity 71.5%

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

   3. *-rgt-identity 71.5%

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

   4. +-commutative 71.5%

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

   5. sqrt-unprod 71.5%

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

   6. +-commutative 71.5%

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

4. Applied egg-rr 71.5%

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

   1. flip-- 71.6%

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

   2. div-inv 71.7%

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

   3. add-sqr-sqrt 71.5%

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

   4. add-sqr-sqrt 72.6%

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

6. Applied egg-rr 72.6%

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

   1. associate-*r/ 72.6%

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

   2. *-rgt-identity 72.6%

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

   3. associate--l+ 90.8%

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

   4. +-inverses 90.8%

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

   5. metadata-eval 90.8%

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

   6. +-commutative 90.8%

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

8. Simplified 90.8%

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

   1. *-un-lft-identity 90.8%

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

   2. sqrt-prod 99.4%

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

   3. times-frac 99.3%

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

   4. pow1/2 99.3%

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

   5. pow-flip 99.6%

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

   6. metadata-eval 99.6%

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

10. Applied egg-rr 99.6%

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

    1. associate-/l/ 99.6%

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

    2. associate-*r/ 99.7%

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

    3. *-commutative 99.7%

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

    4. *-lft-identity 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 9: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (/ (/ 0.5 x) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} 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 <= 1.7d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = (0.5d0 / x) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = (0.5 / x) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = (0.5 / x) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = (0.5 / x) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.7], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.6%

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

      1. expm1-log1p-u 92.7%

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

      2. expm1-udef 92.7%

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

      3. pow1/2 92.7%

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

      4. pow-flip 92.7%

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

      5. metadata-eval 92.7%

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

   4. Applied egg-rr 92.7%

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

      1. expm1-def 92.7%

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

      2. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 97.5%

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

      1. *-commutative 97.5%

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

   9. Simplified 97.5%

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

   if 1.69999999999999996 < x

   1. Initial program 40.5%

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

      1. frac-sub 40.6%

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

      2. *-un-lft-identity 40.6%

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

      3. *-rgt-identity 40.6%

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

      4. +-commutative 40.6%

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

      5. sqrt-unprod 40.6%

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

      6. +-commutative 40.6%

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

   4. Applied egg-rr 40.6%

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

      1. add-cube-cbrt 40.6%

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

      2. sqrt-prod 40.6%

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

      3. times-frac 40.6%

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

      4. pow2 40.6%

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

   6. Applied egg-rr 40.6%

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

      1. associate-*l/ 40.6%

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

      2. /-rgt-identity 40.6%

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

      3. times-frac 40.6%

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

      4. unpow2 40.6%

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

      5. rem-3cbrt-lft 40.6%

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

      6. *-lft-identity 40.6%

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

      7. div-sub 40.6%

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

      8. *-inverses 40.6%

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

   8. Simplified 40.6%

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

   9. Taylor expanded in x around inf 98.1%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $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 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 97.3%

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

      1. *-commutative 97.3%

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

   9. Simplified 97.3%

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

   if 1 < x

   1. Initial program 40.5%

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

      1. frac-sub 40.6%

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

      2. *-un-lft-identity 40.6%

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

      3. *-rgt-identity 40.6%

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

      4. +-commutative 40.6%

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

      5. sqrt-unprod 40.6%

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

      6. +-commutative 40.6%

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

   4. Applied egg-rr 40.6%

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

      1. add-cube-cbrt 40.6%

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

      2. sqrt-prod 40.6%

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

      3. times-frac 40.6%

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

      4. pow2 40.6%

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

   6. Applied egg-rr 40.6%

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

      1. associate-*l/ 40.6%

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

      2. /-rgt-identity 40.6%

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

      3. times-frac 40.6%

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

      4. unpow2 40.6%

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

      5. rem-3cbrt-lft 40.6%

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

      6. *-lft-identity 40.6%

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

      7. div-sub 40.6%

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

      8. *-inverses 40.6%

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

   8. Simplified 40.6%

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

   9. Taylor expanded in x around inf 98.1%

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

4. Final simplification 97.7%

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

Alternative 11: 98.3% 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 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 96.2%

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

   if 0.660000000000000031 < x

   1. Initial program 40.5%

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

      1. frac-sub 40.6%

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

      2. *-un-lft-identity 40.6%

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

      3. *-rgt-identity 40.6%

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

      4. +-commutative 40.6%

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

      5. sqrt-unprod 40.6%

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

      6. +-commutative 40.6%

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

   4. Applied egg-rr 40.6%

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

      1. add-cube-cbrt 40.6%

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

      2. sqrt-prod 40.6%

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

      3. times-frac 40.6%

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

      4. pow2 40.6%

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

   6. Applied egg-rr 40.6%

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

      1. associate-*l/ 40.6%

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

      2. /-rgt-identity 40.6%

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

      3. times-frac 40.6%

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

      4. unpow2 40.6%

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

      5. rem-3cbrt-lft 40.6%

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

      6. *-lft-identity 40.6%

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

      7. div-sub 40.6%

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

      8. *-inverses 40.6%

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

   8. Simplified 40.6%

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

   9. Taylor expanded in x around inf 98.1%

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

4. Final simplification 97.1%

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

Alternative 12: 50.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

   1. *-un-lft-identity 71.5%

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

   2. clear-num 71.5%

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

   3. associate-/r/ 71.5%

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

   4. prod-diff 71.5%

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

   5. *-un-lft-identity 71.5%

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

   6. fma-neg 71.5%

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

   7. *-un-lft-identity 71.5%

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

   8. pow1/2 71.5%

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

   9. pow-flip 67.2%

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

   10. metadata-eval 67.2%

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

   11. pow1/2 67.2%

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

   12. pow-flip 71.6%

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

   13. +-commutative 71.6%

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

   14. metadata-eval 71.6%

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

4. Applied egg-rr 71.6%

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

   1. associate-+l- 71.6%

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

   2. expm1-log1p 71.6%

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

   3. expm1-def 55.2%

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

   4. associate--l- 55.2%

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

   5. fma-udef 55.2%

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

   6. distribute-lft1-in 55.2%

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

   7. metadata-eval 55.2%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

   8. mul0-lft 55.2%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

   9. metadata-eval 55.2%

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

   10. expm1-def 71.6%

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

   11. expm1-log1p 71.6%

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

6. Simplified 71.6%

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

7. Taylor expanded in x around 0 51.6%

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

8. Final simplification 51.6%

   \[\leadsto {x}^{-0.5} + -1 \]
9. Add Preprocessing

Alternative 13: 50.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

   1. expm1-log1p-u 67.8%

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

   2. expm1-udef 50.9%

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

   3. pow1/2 50.9%

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

   4. pow-flip 50.9%

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

   5. metadata-eval 50.9%

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

4. Applied egg-rr 50.9%

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

   1. expm1-def 63.4%

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

   2. expm1-log1p 67.2%

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

6. Simplified 67.2%

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

7. Taylor expanded in x around inf 50.7%

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

8. Final simplification 50.7%

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

Alternative 14: 3.9% accurate, 69.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x 0.5))

C

double code(double x) {
	return 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 x * 0.5;
}

Python

def code(x):
	return x * 0.5

Julia

function code(x)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.5;
end

Wolfram

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

Derivation

1. Initial program 71.5%

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

   1. *-un-lft-identity 71.5%

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

   2. clear-num 71.5%

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

   3. associate-/r/ 71.5%

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

   4. prod-diff 71.5%

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

   5. *-un-lft-identity 71.5%

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

   6. fma-neg 71.5%

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

   7. *-un-lft-identity 71.5%

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

   8. pow1/2 71.5%

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

   9. pow-flip 67.2%

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

   10. metadata-eval 67.2%

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

   11. pow1/2 67.2%

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

   12. pow-flip 71.6%

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

   13. +-commutative 71.6%

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

   14. metadata-eval 71.6%

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

4. Applied egg-rr 71.6%

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

   1. associate-+l- 71.6%

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

   2. expm1-log1p 71.6%

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

   3. expm1-def 55.2%

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

   4. associate--l- 55.2%

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

   5. fma-udef 55.2%

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

   6. distribute-lft1-in 55.2%

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

   7. metadata-eval 55.2%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

   8. mul0-lft 55.2%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

   9. metadata-eval 55.2%

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

   10. expm1-def 71.6%

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

   11. expm1-log1p 71.6%

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

6. Simplified 71.6%

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

7. Taylor expanded in x around 0 52.5%

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

   1. *-commutative 52.5%

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

9. Simplified 52.5%

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

10. Taylor expanded in x around inf 4.1%

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

    1. *-commutative 4.1%

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

12. Simplified 4.1%

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

13. Final simplification 4.1%

    \[\leadsto x \cdot 0.5 \]
14. Add Preprocessing

Alternative 15: 1.9% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 71.5%

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

3. Taylor expanded in x around 0 51.4%

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

4. Taylor expanded in x around inf 1.9%

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

5. Final simplification 1.9%

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

Developer target: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024013 
(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)))))