2isqrt (example 3.6)

Percentage Accurate: 69.6% → 99.8%

Time: 8.8s

Alternatives: 12

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + \left(\mathsf{fma}\left(2, x, \frac{0.0625}{x \cdot x}\right) - \frac{0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 4e-6)
   (/ (pow x -0.5) (+ 1.5 (- (fma 2.0 x (/ 0.0625 (* x x))) (/ 0.125 x))))
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 4e-6) {
		tmp = pow(x, -0.5) / (1.5 + (fma(2.0, x, (0.0625 / (x * x))) - (0.125 / x)));
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 4e-6)
		tmp = Float64((x ^ -0.5) / Float64(1.5 + Float64(fma(2.0, x, Float64(0.0625 / Float64(x * x))) - Float64(0.125 / x))));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-6], N[(N[Power[x, -0.5], $MachinePrecision] / N[(1.5 + N[(N[(2.0 * x + N[(0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $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)))) < 3.99999999999999982e-6

   1. Initial program 41.6%

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

      1. frac-sub 41.6%

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

      2. div-inv 41.6%

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

      3. *-un-lft-identity 41.6%

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

      4. +-commutative 41.6%

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

      5. *-rgt-identity 41.6%

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

      6. metadata-eval 41.6%

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

      7. frac-times 41.6%

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

      8. un-div-inv 41.6%

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

      9. pow1/2 41.6%

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

      10. pow-flip 41.6%

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

      11. metadata-eval 41.6%

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

      12. +-commutative 41.6%

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

   3. Applied egg-rr 41.6%

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

      1. associate-*r/ 41.6%

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

      2. *-rgt-identity 41.6%

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

      3. times-frac 41.6%

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

      4. div-sub 41.6%

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

      5. *-inverses 41.6%

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

      6. /-rgt-identity 41.6%

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

   5. Simplified 41.6%

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

      1. *-un-lft-identity 41.6%

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

      2. sqrt-undiv 41.9%

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

      3. +-commutative 41.9%

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

   7. Applied egg-rr 41.9%

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

      1. *-lft-identity 41.9%

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

   9. Simplified 41.9%

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

       1. *-commutative 41.9%

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

       2. metadata-eval 41.9%

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

       3. sqrt-pow2 41.9%

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

       4. inv-pow 41.9%

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

       5. flip-- 41.9%

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

       6. associate-*r/ 41.9%

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

       7. inv-pow 41.9%

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

       8. sqrt-pow2 41.9%

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

       9. metadata-eval 41.9%

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

       10. metadata-eval 41.9%

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

       11. add-sqr-sqrt 42.0%

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

       12. +-commutative 42.0%

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

       13. +-commutative 42.0%

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

   11. Applied egg-rr 42.0%

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

       1. associate-/l* 42.0%

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

   13. Simplified 42.0%

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

   14. Taylor expanded in x around inf 99.7%

       \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\left(1.5 + \left(2 \cdot x + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right) - 0.125 \cdot \frac{1}{x}}} \]
   15. Step-by-step derivation

       1. associate--l+ 99.7%

          \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{1.5 + \left(\left(2 \cdot x + 0.0625 \cdot \frac{1}{{x}^{2}}\right) - 0.125 \cdot \frac{1}{x}\right)}} \]

       2. fma-def 99.7%

          \[\leadsto \frac{{x}^{-0.5}}{1.5 + \left(\color{blue}{\mathsf{fma}\left(2, x, 0.0625 \cdot \frac{1}{{x}^{2}}\right)} - 0.125 \cdot \frac{1}{x}\right)} \]

       3. associate-*r/ 99.7%

          \[\leadsto \frac{{x}^{-0.5}}{1.5 + \left(\mathsf{fma}\left(2, x, \color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}}\right) - 0.125 \cdot \frac{1}{x}\right)} \]

       4. metadata-eval 99.7%

          \[\leadsto \frac{{x}^{-0.5}}{1.5 + \left(\mathsf{fma}\left(2, x, \frac{\color{blue}{0.0625}}{{x}^{2}}\right) - 0.125 \cdot \frac{1}{x}\right)} \]

       5. unpow2 99.7%

          \[\leadsto \frac{{x}^{-0.5}}{1.5 + \left(\mathsf{fma}\left(2, x, \frac{0.0625}{\color{blue}{x \cdot x}}\right) - 0.125 \cdot \frac{1}{x}\right)} \]

       6. associate-*r/ 99.7%

          \[\leadsto \frac{{x}^{-0.5}}{1.5 + \left(\mathsf{fma}\left(2, x, \frac{0.0625}{x \cdot x}\right) - \color{blue}{\frac{0.125 \cdot 1}{x}}\right)} \]

       7. metadata-eval 99.7%

          \[\leadsto \frac{{x}^{-0.5}}{1.5 + \left(\mathsf{fma}\left(2, x, \frac{0.0625}{x \cdot x}\right) - \frac{\color{blue}{0.125}}{x}\right)} \]

   16. Simplified 99.7%

       \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{1.5 + \left(\mathsf{fma}\left(2, x, \frac{0.0625}{x \cdot x}\right) - \frac{0.125}{x}\right)}} \]

   if 3.99999999999999982e-6 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + \left(\mathsf{fma}\left(2, x, \frac{0.0625}{x \cdot x}\right) - \frac{0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-8)
   (/ (pow x -0.5) (+ 1.5 (- (* x 2.0) (/ 0.125 x))))
   (* (pow x -0.5) (- 1.0 (sqrt (/ x (+ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-8) {
		tmp = pow(x, -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)));
	} else {
		tmp = pow(x, -0.5) * (1.0 - sqrt((x / (1.0 + x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 2d-8) then
        tmp = (x ** (-0.5d0)) / (1.5d0 + ((x * 2.0d0) - (0.125d0 / x)))
    else
        tmp = (x ** (-0.5d0)) * (1.0d0 - sqrt((x / (1.0d0 + x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 2e-8) {
		tmp = Math.pow(x, -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)));
	} else {
		tmp = Math.pow(x, -0.5) * (1.0 - Math.sqrt((x / (1.0 + x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 2e-8:
		tmp = math.pow(x, -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)))
	else:
		tmp = math.pow(x, -0.5) * (1.0 - math.sqrt((x / (1.0 + x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-8)
		tmp = Float64((x ^ -0.5) / Float64(1.5 + Float64(Float64(x * 2.0) - Float64(0.125 / x))));
	else
		tmp = Float64((x ^ -0.5) * Float64(1.0 - sqrt(Float64(x / Float64(1.0 + x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-8)
		tmp = (x ^ -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)));
	else
		tmp = (x ^ -0.5) * (1.0 - sqrt((x / (1.0 + x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[Power[x, -0.5], $MachinePrecision] / N[(1.5 + N[(N[(x * 2.0), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(1.0 - N[Sqrt[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 2e-8

   1. Initial program 41.3%

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

      1. frac-sub 41.3%

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

      2. div-inv 41.3%

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

      3. *-un-lft-identity 41.3%

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

      4. +-commutative 41.3%

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

      5. *-rgt-identity 41.3%

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

      6. metadata-eval 41.3%

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

      7. frac-times 41.3%

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

      8. un-div-inv 41.3%

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

      9. pow1/2 41.3%

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

      10. pow-flip 41.3%

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

      11. metadata-eval 41.3%

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

      12. +-commutative 41.3%

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

   3. Applied egg-rr 41.3%

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

      1. associate-*r/ 41.3%

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

      2. *-rgt-identity 41.3%

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

      3. times-frac 41.3%

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

      4. div-sub 41.3%

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

      5. *-inverses 41.3%

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

      6. /-rgt-identity 41.3%

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

   5. Simplified 41.3%

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

      1. *-un-lft-identity 41.3%

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

      2. sqrt-undiv 41.5%

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

      3. +-commutative 41.5%

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

   7. Applied egg-rr 41.5%

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

      1. *-lft-identity 41.5%

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

   9. Simplified 41.5%

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

       1. *-commutative 41.5%

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

       2. metadata-eval 41.5%

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

       3. sqrt-pow2 41.5%

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

       4. inv-pow 41.5%

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

       5. flip-- 41.5%

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

       6. associate-*r/ 41.5%

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

       7. inv-pow 41.5%

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

       8. sqrt-pow2 41.5%

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

       9. metadata-eval 41.5%

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

       10. metadata-eval 41.5%

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

       11. add-sqr-sqrt 41.6%

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

       12. +-commutative 41.6%

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

       13. +-commutative 41.6%

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

   11. Applied egg-rr 41.6%

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

       1. associate-/l* 41.6%

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

   13. Simplified 41.6%

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

   14. Taylor expanded in x around inf 99.7%

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

       1. associate--l+ 99.7%

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

       2. *-commutative 99.7%

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

       3. associate-*r/ 99.7%

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

       4. metadata-eval 99.7%

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

   16. Simplified 99.7%

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

   if 2e-8 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.5%

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

      1. frac-sub 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. +-commutative 99.4%

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

      5. *-rgt-identity 99.4%

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

      6. metadata-eval 99.4%

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

      7. frac-times 99.4%

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

      8. un-div-inv 99.4%

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

      9. pow1/2 99.4%

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

      10. pow-flip 99.8%

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

      11. metadata-eval 99.8%

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

      12. +-commutative 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. times-frac 99.9%

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

      4. div-sub 99.9%

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

      5. *-inverses 99.9%

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

      6. /-rgt-identity 99.9%

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

   5. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. sqrt-undiv 99.9%

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

      3. +-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-8)
   (/ (pow x -0.5) (+ 1.5 (- (* x 2.0) (/ 0.125 x))))
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-8) {
		tmp = pow(x, -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)));
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -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((1.0d0 + x)))) <= 2d-8) then
        tmp = (x ** (-0.5d0)) / (1.5d0 + ((x * 2.0d0) - (0.125d0 / x)))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-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((1.0 + x)))) <= 2e-8) {
		tmp = Math.pow(x, -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 2e-8:
		tmp = math.pow(x, -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)))
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-8)
		tmp = Float64((x ^ -0.5) / Float64(1.5 + Float64(Float64(x * 2.0) - Float64(0.125 / x))));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-8)
		tmp = (x ^ -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)));
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -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[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[Power[x, -0.5], $MachinePrecision] / N[(1.5 + N[(N[(x * 2.0), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $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)))) < 2e-8

   1. Initial program 41.3%

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

      1. frac-sub 41.3%

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

      2. div-inv 41.3%

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

      3. *-un-lft-identity 41.3%

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

      4. +-commutative 41.3%

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

      5. *-rgt-identity 41.3%

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

      6. metadata-eval 41.3%

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

      7. frac-times 41.3%

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

      8. un-div-inv 41.3%

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

      9. pow1/2 41.3%

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

      10. pow-flip 41.3%

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

      11. metadata-eval 41.3%

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

      12. +-commutative 41.3%

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

   3. Applied egg-rr 41.3%

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

      1. associate-*r/ 41.3%

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

      2. *-rgt-identity 41.3%

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

      3. times-frac 41.3%

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

      4. div-sub 41.3%

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

      5. *-inverses 41.3%

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

      6. /-rgt-identity 41.3%

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

   5. Simplified 41.3%

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

      1. *-un-lft-identity 41.3%

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

      2. sqrt-undiv 41.5%

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

      3. +-commutative 41.5%

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

   7. Applied egg-rr 41.5%

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

      1. *-lft-identity 41.5%

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

   9. Simplified 41.5%

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

       1. *-commutative 41.5%

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

       2. metadata-eval 41.5%

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

       3. sqrt-pow2 41.5%

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

       4. inv-pow 41.5%

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

       5. flip-- 41.5%

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

       6. associate-*r/ 41.5%

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

       7. inv-pow 41.5%

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

       8. sqrt-pow2 41.5%

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

       9. metadata-eval 41.5%

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

       10. metadata-eval 41.5%

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

       11. add-sqr-sqrt 41.6%

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

       12. +-commutative 41.6%

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

       13. +-commutative 41.6%

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

   11. Applied egg-rr 41.6%

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

       1. associate-/l* 41.6%

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

   13. Simplified 41.6%

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

   14. Taylor expanded in x around inf 99.7%

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

       1. associate--l+ 99.7%

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

       2. *-commutative 99.7%

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

       3. associate-*r/ 99.7%

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

       4. metadata-eval 99.7%

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

   16. Simplified 99.7%

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

   if 2e-8 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. inv-pow 99.5%

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

      9. sqrt-pow2 99.9%

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

      10. metadata-eval 99.9%

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

      11. pow1/2 99.9%

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

      12. pow-flip 99.9%

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

      13. +-commutative 99.9%

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

      14. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. distribute-lft1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. mul0-lft 99.9%

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

      5. +-rgt-identity 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 4: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.58)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (/ (pow x -0.5) (+ 1.5 (- (* x 2.0) (/ 0.125 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.58) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = pow(x, -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.58d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = (x ** (-0.5d0)) / (1.5d0 + ((x * 2.0d0) - (0.125d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.58) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = Math.pow(x, -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.58:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = math.pow(x, -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.58)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64((x ^ -0.5) / Float64(1.5 + Float64(Float64(x * 2.0) - Float64(0.125 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.58)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = (x ^ -0.5) / (1.5 + ((x * 2.0) - (0.125 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.58], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] / N[(1.5 + N[(N[(x * 2.0), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.57999999999999996

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around 0 99.2%

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

      1. add-log-exp 4.2%

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

      2. *-un-lft-identity 4.2%

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

      3. log-prod 4.2%

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

      4. metadata-eval 4.2%

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

      5. add-log-exp 99.2%

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

      6. pow1/2 99.2%

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

      7. pow-flip 99.6%

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

      8. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \frac{1}{0.5 \cdot x + 1} \]
   5. Step-by-step derivation

      1. +-lft-identity 99.6%

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

   6. Simplified 99.6%

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

   if 0.57999999999999996 < x

   1. Initial program 41.6%

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

      1. frac-sub 41.6%

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

      2. div-inv 41.6%

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

      3. *-un-lft-identity 41.6%

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

      4. +-commutative 41.6%

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

      5. *-rgt-identity 41.6%

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

      6. metadata-eval 41.6%

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

      7. frac-times 41.6%

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

      8. un-div-inv 41.6%

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

      9. pow1/2 41.6%

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

      10. pow-flip 41.6%

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

      11. metadata-eval 41.6%

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

      12. +-commutative 41.6%

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

   3. Applied egg-rr 41.6%

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

      1. associate-*r/ 41.6%

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

      2. *-rgt-identity 41.6%

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

      3. times-frac 41.6%

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

      4. div-sub 41.6%

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

      5. *-inverses 41.6%

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

      6. /-rgt-identity 41.6%

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

   5. Simplified 41.6%

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

      1. *-un-lft-identity 41.6%

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

      2. sqrt-undiv 41.9%

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

      3. +-commutative 41.9%

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

   7. Applied egg-rr 41.9%

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

      1. *-lft-identity 41.9%

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

   9. Simplified 41.9%

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

       1. *-commutative 41.9%

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

       2. metadata-eval 41.9%

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

       3. sqrt-pow2 41.9%

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

       4. inv-pow 41.9%

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

       5. flip-- 41.9%

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

       6. associate-*r/ 41.9%

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

       7. inv-pow 41.9%

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

       8. sqrt-pow2 41.9%

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

       9. metadata-eval 41.9%

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

       10. metadata-eval 41.9%

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

       11. add-sqr-sqrt 42.0%

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

       12. +-commutative 42.0%

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

       13. +-commutative 42.0%

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

   11. Applied egg-rr 42.0%

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

       1. associate-/l* 42.0%

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

   13. Simplified 42.0%

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

   14. Taylor expanded in x around inf 99.6%

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

       1. associate--l+ 99.6%

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

       2. *-commutative 99.6%

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

       3. associate-*r/ 99.6%

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

       4. metadata-eval 99.6%

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

   16. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.62)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (/ (pow x -0.5) (+ 1.5 (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.62) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = pow(x, -0.5) / (1.5 + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.62d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = (x ** (-0.5d0)) / (1.5d0 + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.62) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = Math.pow(x, -0.5) / (1.5 + (x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.62:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = math.pow(x, -0.5) / (1.5 + (x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.62)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64((x ^ -0.5) / Float64(1.5 + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.62)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = (x ^ -0.5) / (1.5 + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.62], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] / N[(1.5 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.619999999999999996

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around 0 99.2%

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

      1. add-log-exp 4.2%

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

      2. *-un-lft-identity 4.2%

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

      3. log-prod 4.2%

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

      4. metadata-eval 4.2%

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

      5. add-log-exp 99.2%

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

      6. pow1/2 99.2%

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

      7. pow-flip 99.6%

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

      8. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \frac{1}{0.5 \cdot x + 1} \]
   5. Step-by-step derivation

      1. +-lft-identity 99.6%

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

   6. Simplified 99.6%

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

   if 0.619999999999999996 < x

   1. Initial program 41.6%

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

      1. frac-sub 41.6%

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

      2. div-inv 41.6%

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

      3. *-un-lft-identity 41.6%

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

      4. +-commutative 41.6%

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

      5. *-rgt-identity 41.6%

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

      6. metadata-eval 41.6%

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

      7. frac-times 41.6%

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

      8. un-div-inv 41.6%

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

      9. pow1/2 41.6%

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

      10. pow-flip 41.6%

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

      11. metadata-eval 41.6%

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

      12. +-commutative 41.6%

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

   3. Applied egg-rr 41.6%

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

      1. associate-*r/ 41.6%

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

      2. *-rgt-identity 41.6%

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

      3. times-frac 41.6%

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

      4. div-sub 41.6%

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

      5. *-inverses 41.6%

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

      6. /-rgt-identity 41.6%

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

   5. Simplified 41.6%

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

      1. *-un-lft-identity 41.6%

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

      2. sqrt-undiv 41.9%

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

      3. +-commutative 41.9%

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

   7. Applied egg-rr 41.9%

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

      1. *-lft-identity 41.9%

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

   9. Simplified 41.9%

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

       1. *-commutative 41.9%

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

       2. metadata-eval 41.9%

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

       3. sqrt-pow2 41.9%

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

       4. inv-pow 41.9%

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

       5. flip-- 41.9%

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

       6. associate-*r/ 41.9%

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

       7. inv-pow 41.9%

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

       8. sqrt-pow2 41.9%

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

       9. metadata-eval 41.9%

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

       10. metadata-eval 41.9%

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

       11. add-sqr-sqrt 42.0%

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

       12. +-commutative 42.0%

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

       13. +-commutative 42.0%

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

   11. Applied egg-rr 42.0%

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

       1. associate-/l* 42.0%

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

   13. Simplified 42.0%

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

   14. Taylor expanded in x around inf 99.2%

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

       1. +-commutative 99.2%

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

       2. *-commutative 99.2%

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

   16. Simplified 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + x \cdot 2}\\ \end{array} \]

Alternative 6: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(0.5 * N[Power[x, -1.5], $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. Step-by-step derivation

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.6%

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

   if 1 < x

   1. Initial program 41.6%

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

      1. frac-sub 41.6%

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

      2. div-inv 41.6%

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

      3. *-un-lft-identity 41.6%

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

      4. +-commutative 41.6%

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

      5. *-rgt-identity 41.6%

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

      6. metadata-eval 41.6%

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

      7. frac-times 41.6%

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

      8. un-div-inv 41.6%

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

      9. pow1/2 41.6%

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

      10. pow-flip 41.6%

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

      11. metadata-eval 41.6%

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

      12. +-commutative 41.6%

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

   3. Applied egg-rr 41.6%

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

      1. associate-*r/ 41.6%

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

      2. *-rgt-identity 41.6%

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

      3. times-frac 41.6%

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

      4. div-sub 41.6%

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

      5. *-inverses 41.6%

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

      6. /-rgt-identity 41.6%

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

   5. Simplified 41.6%

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

   6. Taylor expanded in x around inf 97.9%

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

      1. expm1-log1p-u 97.9%

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

      2. expm1-udef 40.1%

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

   8. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 98.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)\right)} \]

      2. expm1-log1p 98.3%

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

   10. Simplified 98.3%

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

4. Final simplification 98.9%

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

Alternative 7: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.40000000000000002

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.6%

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

   if 0.40000000000000002 < x

   1. Initial program 41.6%

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

      1. frac-sub 41.6%

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

      2. div-inv 41.6%

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

      3. *-un-lft-identity 41.6%

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

      4. +-commutative 41.6%

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

      5. *-rgt-identity 41.6%

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

      6. metadata-eval 41.6%

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

      7. frac-times 41.6%

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

      8. un-div-inv 41.6%

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

      9. pow1/2 41.6%

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

      10. pow-flip 41.6%

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

      11. metadata-eval 41.6%

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

      12. +-commutative 41.6%

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

   3. Applied egg-rr 41.6%

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

      1. associate-*r/ 41.6%

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

      2. *-rgt-identity 41.6%

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

      3. times-frac 41.6%

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

      4. div-sub 41.6%

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

      5. *-inverses 41.6%

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

      6. /-rgt-identity 41.6%

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

   5. Simplified 41.6%

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

      1. *-un-lft-identity 41.6%

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

      2. sqrt-undiv 41.9%

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

      3. +-commutative 41.9%

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

   7. Applied egg-rr 41.9%

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

      1. *-lft-identity 41.9%

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

   9. Simplified 41.9%

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

       1. *-commutative 41.9%

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

       2. metadata-eval 41.9%

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

       3. sqrt-pow2 41.9%

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

       4. inv-pow 41.9%

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

       5. flip-- 41.9%

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

       6. associate-*r/ 41.9%

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

       7. inv-pow 41.9%

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

       8. sqrt-pow2 41.9%

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

       9. metadata-eval 41.9%

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

       10. metadata-eval 41.9%

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

       11. add-sqr-sqrt 42.0%

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

       12. +-commutative 42.0%

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

       13. +-commutative 42.0%

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

   11. Applied egg-rr 42.0%

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

       1. associate-/l* 42.0%

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

   13. Simplified 42.0%

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

   14. Taylor expanded in x around inf 99.2%

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

       1. +-commutative 99.2%

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

       2. *-commutative 99.2%

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

   16. Simplified 99.2%

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

4. Final simplification 99.4%

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

Alternative 8: 66.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 8.5e+122) (sqrt (/ 1.0 x)) (* (pow x -0.5) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = sqrt((1.0 / x));
	} else {
		tmp = pow(x, -0.5) * 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.5d+122) then
        tmp = sqrt((1.0d0 / x))
    else
        tmp = (x ** (-0.5d0)) * 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = Math.sqrt((1.0 / x));
	} else {
		tmp = Math.pow(x, -0.5) * 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.5e+122:
		tmp = math.sqrt((1.0 / x))
	else:
		tmp = math.pow(x, -0.5) * 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.5e+122)
		tmp = sqrt(Float64(1.0 / x));
	else
		tmp = Float64((x ^ -0.5) * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.5e+122)
		tmp = sqrt((1.0 / x));
	else
		tmp = (x ^ -0.5) * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.5e+122], N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.50000000000000003e122

   1. Initial program 72.7%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around 0 71.5%

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

      1. add-log-exp 4.2%

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

      2. *-un-lft-identity 4.2%

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

      3. log-prod 4.2%

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

      4. metadata-eval 4.2%

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

      5. add-log-exp 71.5%

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

      6. pow1/2 71.5%

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

      7. pow-flip 71.8%

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

      8. metadata-eval 71.8%

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

   4. Applied egg-rr 71.8%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \frac{1}{0.5 \cdot x + 1} \]
   5. Step-by-step derivation

      1. +-lft-identity 71.8%

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

   6. Simplified 71.8%

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

   7. Taylor expanded in x around inf 68.2%

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

   if 8.50000000000000003e122 < x

   1. Initial program 62.4%

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

      1. sub-neg 62.4%

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

      2. +-commutative 62.4%

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

      3. add-sqr-sqrt 35.6%

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

      4. distribute-rgt-neg-in 35.6%

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

      5. fma-def 4.5%

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

      6. inv-pow 4.5%

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

      7. sqrt-pow2 4.5%

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

      8. +-commutative 4.5%

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

      9. metadata-eval 4.5%

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

      10. inv-pow 4.5%

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

      11. sqrt-pow2 4.5%

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

      12. +-commutative 4.5%

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

      13. metadata-eval 4.5%

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

      14. pow1/2 4.5%

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

      15. pow-flip 4.4%

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

      16. metadata-eval 4.4%

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

   3. Applied egg-rr 4.4%

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

   4. Taylor expanded in x around inf 62.4%

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

      1. unpow1/2 62.4%

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

      2. distribute-rgt1-in 62.4%

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

      3. metadata-eval 62.4%

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

      4. unpow1/2 62.4%

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

      5. unpow-1 62.4%

         \[\leadsto 0 \cdot \sqrt{\color{blue}{{x}^{-1}}} \]

      6. metadata-eval 62.4%

         \[\leadsto 0 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      7. pow-sqr 62.4%

         \[\leadsto 0 \cdot \sqrt{\color{blue}{{x}^{-0.5} \cdot {x}^{-0.5}}} \]

      8. rem-sqrt-square 62.4%

         \[\leadsto 0 \cdot \color{blue}{\left|{x}^{-0.5}\right|} \]

      9. metadata-eval 62.4%

         \[\leadsto 0 \cdot \left|{x}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \]

      10. pow-sqr 62.4%

          \[\leadsto 0 \cdot \left|\color{blue}{{x}^{-0.25} \cdot {x}^{-0.25}}\right| \]

      11. fabs-sqr 62.4%

          \[\leadsto 0 \cdot \color{blue}{\left({x}^{-0.25} \cdot {x}^{-0.25}\right)} \]

      12. pow-sqr 62.4%

          \[\leadsto 0 \cdot \color{blue}{{x}^{\left(2 \cdot -0.25\right)}} \]

      13. metadata-eval 62.4%

          \[\leadsto 0 \cdot {x}^{\color{blue}{-0.5}} \]

   6. Simplified 62.4%

      \[\leadsto \color{blue}{0 \cdot {x}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0\\ \end{array} \]

Alternative 9: 96.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around 0 99.2%

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

      1. add-log-exp 4.2%

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

      2. *-un-lft-identity 4.2%

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

      3. log-prod 4.2%

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

      4. metadata-eval 4.2%

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

      5. add-log-exp 99.2%

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

      6. pow1/2 99.2%

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

      7. pow-flip 99.6%

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

      8. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \frac{1}{0.5 \cdot x + 1} \]
   5. Step-by-step derivation

      1. +-lft-identity 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 94.4%

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

   if 0.5 < x

   1. Initial program 41.6%

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

      1. frac-sub 41.6%

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

      2. div-inv 41.6%

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

      3. *-un-lft-identity 41.6%

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

      4. +-commutative 41.6%

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

      5. *-rgt-identity 41.6%

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

      6. metadata-eval 41.6%

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

      7. frac-times 41.6%

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

      8. un-div-inv 41.6%

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

      9. pow1/2 41.6%

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

      10. pow-flip 41.6%

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

      11. metadata-eval 41.6%

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

      12. +-commutative 41.6%

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

   3. Applied egg-rr 41.6%

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

      1. associate-*r/ 41.6%

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

      2. *-rgt-identity 41.6%

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

      3. times-frac 41.6%

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

      4. div-sub 41.6%

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

      5. *-inverses 41.6%

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

      6. /-rgt-identity 41.6%

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

   5. Simplified 41.6%

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

   6. Taylor expanded in x around inf 97.9%

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

      1. expm1-log1p-u 97.9%

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

      2. expm1-udef 40.1%

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

   8. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 98.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)\right)} \]

      2. expm1-log1p 98.3%

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

   10. Simplified 98.3%

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

4. Final simplification 96.4%

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

Alternative 10: 96.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.6%

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

      1. frac-sub 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. +-commutative 99.6%

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

      5. *-rgt-identity 99.6%

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

      6. metadata-eval 99.6%

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

      7. frac-times 99.5%

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

      8. un-div-inv 99.5%

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

      9. pow1/2 99.5%

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

      10. pow-flip 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. times-frac 100.0%

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

      4. div-sub 100.0%

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

      5. *-inverses 100.0%

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

      6. /-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 94.7%

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

   if 0.5 < x

   1. Initial program 41.6%

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

      1. frac-sub 41.6%

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

      2. div-inv 41.6%

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

      3. *-un-lft-identity 41.6%

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

      4. +-commutative 41.6%

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

      5. *-rgt-identity 41.6%

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

      6. metadata-eval 41.6%

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

      7. frac-times 41.6%

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

      8. un-div-inv 41.6%

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

      9. pow1/2 41.6%

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

      10. pow-flip 41.6%

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

      11. metadata-eval 41.6%

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

      12. +-commutative 41.6%

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

   3. Applied egg-rr 41.6%

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

      1. associate-*r/ 41.6%

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

      2. *-rgt-identity 41.6%

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

      3. times-frac 41.6%

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

      4. div-sub 41.6%

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

      5. *-inverses 41.6%

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

      6. /-rgt-identity 41.6%

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

   5. Simplified 41.6%

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

   6. Taylor expanded in x around inf 97.9%

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

      1. expm1-log1p-u 97.9%

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

      2. expm1-udef 40.1%

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

   8. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 98.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)\right)} \]

      2. expm1-log1p 98.3%

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

   10. Simplified 98.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 11: 98.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.68) (+ (pow x -0.5) -1.0) (* 0.5 (pow x -1.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 0.5 * pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = 0.5d0 * (x ** (-1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = 0.5 * Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.68:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = 0.5 * math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(0.5 * (x ^ -1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 0.5 * (x ^ -1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.68], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.3%

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

   if 0.680000000000000049 < x

   1. Initial program 41.6%

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

      1. frac-sub 41.6%

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

      2. div-inv 41.6%

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

      3. *-un-lft-identity 41.6%

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

      4. +-commutative 41.6%

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

      5. *-rgt-identity 41.6%

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

      6. metadata-eval 41.6%

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

      7. frac-times 41.6%

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

      8. un-div-inv 41.6%

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

      9. pow1/2 41.6%

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

      10. pow-flip 41.6%

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

      11. metadata-eval 41.6%

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

      12. +-commutative 41.6%

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

   3. Applied egg-rr 41.6%

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

      1. associate-*r/ 41.6%

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

      2. *-rgt-identity 41.6%

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

      3. times-frac 41.6%

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

      4. div-sub 41.6%

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

      5. *-inverses 41.6%

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

      6. /-rgt-identity 41.6%

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

   5. Simplified 41.6%

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

   6. Taylor expanded in x around inf 97.9%

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

      1. expm1-log1p-u 97.9%

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

      2. expm1-udef 40.1%

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

   8. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 98.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)\right)} \]

      2. expm1-log1p 98.3%

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

   10. Simplified 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 12: 50.8% 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 69.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

2. Taylor expanded in x around 0 50.6%

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

   1. add-log-exp 4.8%

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

   2. *-un-lft-identity 4.8%

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

   3. log-prod 4.8%

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

   4. metadata-eval 4.8%

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

   5. add-log-exp 50.6%

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

   6. pow1/2 50.6%

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

   7. pow-flip 50.8%

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

   8. metadata-eval 50.8%

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

4. Applied egg-rr 50.8%

   \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \frac{1}{0.5 \cdot x + 1} \]
5. Step-by-step derivation

   1. +-lft-identity 50.8%

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

6. Simplified 50.8%

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

7. Taylor expanded in x around inf 48.3%

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

8. Final simplification 48.3%

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

Developer target: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023263 
(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)))))