2isqrt (example 3.6)

Percentage Accurate: 68.4% → 99.7%

Time: 14.4s

Alternatives: 11

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{-0.375}{x \cdot x}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0 - {\left(1 + x\right)}^{-0.3333333333333333}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-9)
   (* (+ (/ 0.5 x) (+ (/ 0.3125 (pow x 3.0)) (/ -0.375 (* x x)))) (pow x -0.5))
   (fma
    (- 0.0 (pow (+ 1.0 x) -0.3333333333333333))
    (cbrt (pow (+ 1.0 x) -0.5))
    (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-9) {
		tmp = ((0.5 / x) + ((0.3125 / pow(x, 3.0)) + (-0.375 / (x * x)))) * pow(x, -0.5);
	} else {
		tmp = fma((0.0 - pow((1.0 + x), -0.3333333333333333)), cbrt(pow((1.0 + x), -0.5)), pow(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-9)
		tmp = Float64(Float64(Float64(0.5 / x) + Float64(Float64(0.3125 / (x ^ 3.0)) + Float64(-0.375 / Float64(x * x)))) * (x ^ -0.5));
	else
		tmp = fma(Float64(0.0 - (Float64(1.0 + x) ^ -0.3333333333333333)), cbrt((Float64(1.0 + x) ^ -0.5)), (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], 2e-9], N[(N[(N[(0.5 / x), $MachinePrecision] + N[(N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - N[Power[N[(1.0 + x), $MachinePrecision], -0.3333333333333333], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, -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)))) < 2.00000000000000012e-9

   1. Initial program 38.2%

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

      1. frac-sub 38.3%

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

      2. div-inv 38.3%

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

      3. *-un-lft-identity 38.3%

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

      4. +-commutative 38.3%

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

      5. *-rgt-identity 38.3%

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

      6. metadata-eval 38.3%

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

      7. frac-times 38.3%

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

      8. un-div-inv 38.3%

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

      9. pow1/2 38.3%

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

      10. pow-flip 38.3%

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

      11. metadata-eval 38.3%

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

      12. +-commutative 38.3%

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

   3. Applied egg-rr 38.3%

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

      1. associate-*r/ 38.3%

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

      2. *-rgt-identity 38.3%

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

      3. times-frac 38.3%

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

      4. div-sub 38.3%

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

      5. *-inverses 38.3%

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

      6. unpow1 38.3%

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

      7. sqr-pow 38.3%

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

      8. metadata-eval 38.3%

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

      9. exp-to-pow 6.1%

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

      10. metadata-eval 6.1%

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

      11. exp-to-pow 6.0%

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

      12. hypot-1-def 6.0%

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

      13. exp-to-pow 38.3%

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

      14. unpow1/2 38.3%

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

      15. /-rgt-identity 38.3%

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

   5. Simplified 38.3%

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

      1. clear-num 38.3%

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

      2. inv-pow 38.3%

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

      3. hypot-1-def 38.2%

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

      4. add-sqr-sqrt 38.2%

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

      5. sqrt-undiv 38.3%

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

   7. Applied egg-rr 38.3%

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

      1. unpow-1 38.3%

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

   9. Simplified 38.3%

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

   10. Taylor expanded in x around inf 99.7%

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

       1. sub-neg 99.7%

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

       2. +-commutative 99.7%

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

       3. associate-+l+ 99.7%

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

       4. associate-*r/ 99.7%

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

       5. metadata-eval 99.7%

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

       6. associate-*r/ 99.7%

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

       7. metadata-eval 99.7%

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

       8. associate-*r/ 99.7%

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

       9. metadata-eval 99.7%

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

       10. distribute-neg-frac 99.7%

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

       11. metadata-eval 99.7%

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

       12. unpow2 99.7%

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

   12. Simplified 99.7%

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

   if 2.00000000000000012e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. +-commutative 99.4%

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

      3. add-cube-cbrt 99.4%

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

      4. distribute-lft-neg-in 99.4%

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

      5. fma-def 99.4%

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

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\sqrt[3]{\frac{1}{1 + x}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. pow1/3 99.9%

         \[\leadsto \mathsf{fma}\left(-\color{blue}{{\left(\frac{1}{1 + x}\right)}^{0.3333333333333333}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right) \]

      2. inv-pow 99.9%

         \[\leadsto \mathsf{fma}\left(-{\color{blue}{\left({\left(1 + x\right)}^{-1}\right)}}^{0.3333333333333333}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right) \]

      3. pow-pow 99.9%

         \[\leadsto \mathsf{fma}\left(-\color{blue}{{\left(1 + x\right)}^{\left(-1 \cdot 0.3333333333333333\right)}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right) \]

      4. metadata-eval 99.9%

         \[\leadsto \mathsf{fma}\left(-{\left(1 + x\right)}^{\color{blue}{-0.3333333333333333}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right) \]

   5. Applied egg-rr 99.9%

      \[\leadsto \mathsf{fma}\left(-\color{blue}{{\left(1 + x\right)}^{-0.3333333333333333}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{-0.375}{x \cdot x}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0 - {\left(1 + x\right)}^{-0.3333333333333333}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-9)
   (* (+ (/ 0.5 x) (+ (/ 0.3125 (pow x 3.0)) (/ -0.375 (* x x)))) (pow x -0.5))
   (* (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-9) {
		tmp = ((0.5 / x) + ((0.3125 / pow(x, 3.0)) + (-0.375 / (x * x)))) * pow(x, -0.5);
	} 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-9) then
        tmp = ((0.5d0 / x) + ((0.3125d0 / (x ** 3.0d0)) + ((-0.375d0) / (x * x)))) * (x ** (-0.5d0))
    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-9) {
		tmp = ((0.5 / x) + ((0.3125 / Math.pow(x, 3.0)) + (-0.375 / (x * x)))) * Math.pow(x, -0.5);
	} 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-9:
		tmp = ((0.5 / x) + ((0.3125 / math.pow(x, 3.0)) + (-0.375 / (x * x)))) * math.pow(x, -0.5)
	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-9)
		tmp = Float64(Float64(Float64(0.5 / x) + Float64(Float64(0.3125 / (x ^ 3.0)) + Float64(-0.375 / Float64(x * x)))) * (x ^ -0.5));
	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-9)
		tmp = ((0.5 / x) + ((0.3125 / (x ^ 3.0)) + (-0.375 / (x * x)))) * (x ^ -0.5);
	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-9], N[(N[(N[(0.5 / x), $MachinePrecision] + N[(N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.5], $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)))) < 2.00000000000000012e-9

   1. Initial program 38.2%

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

      1. frac-sub 38.3%

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

      2. div-inv 38.3%

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

      3. *-un-lft-identity 38.3%

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

      4. +-commutative 38.3%

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

      5. *-rgt-identity 38.3%

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

      6. metadata-eval 38.3%

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

      7. frac-times 38.3%

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

      8. un-div-inv 38.3%

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

      9. pow1/2 38.3%

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

      10. pow-flip 38.3%

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

      11. metadata-eval 38.3%

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

      12. +-commutative 38.3%

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

   3. Applied egg-rr 38.3%

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

      1. associate-*r/ 38.3%

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

      2. *-rgt-identity 38.3%

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

      3. times-frac 38.3%

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

      4. div-sub 38.3%

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

      5. *-inverses 38.3%

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

      6. unpow1 38.3%

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

      7. sqr-pow 38.3%

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

      8. metadata-eval 38.3%

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

      9. exp-to-pow 6.1%

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

      10. metadata-eval 6.1%

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

      11. exp-to-pow 6.0%

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

      12. hypot-1-def 6.0%

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

      13. exp-to-pow 38.3%

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

      14. unpow1/2 38.3%

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

      15. /-rgt-identity 38.3%

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

   5. Simplified 38.3%

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

      1. clear-num 38.3%

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

      2. inv-pow 38.3%

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

      3. hypot-1-def 38.2%

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

      4. add-sqr-sqrt 38.2%

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

      5. sqrt-undiv 38.3%

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

   7. Applied egg-rr 38.3%

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

      1. unpow-1 38.3%

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

   9. Simplified 38.3%

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

   10. Taylor expanded in x around inf 99.7%

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

       1. sub-neg 99.7%

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

       2. +-commutative 99.7%

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

       3. associate-+l+ 99.7%

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

       4. associate-*r/ 99.7%

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

       5. metadata-eval 99.7%

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

       6. associate-*r/ 99.7%

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

       7. metadata-eval 99.7%

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

       8. associate-*r/ 99.7%

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

       9. metadata-eval 99.7%

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

       10. distribute-neg-frac 99.7%

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

       11. metadata-eval 99.7%

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

       12. unpow2 99.7%

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

   12. Simplified 99.7%

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

   if 2.00000000000000012e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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 99.9%

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

      11. metadata-eval 99.9%

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

      12. +-commutative 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      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. unpow1 99.9%

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

      7. sqr-pow 99.9%

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

      8. metadata-eval 99.9%

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

      9. exp-to-pow 99.9%

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

      10. metadata-eval 99.9%

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

      11. exp-to-pow 99.9%

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

      12. hypot-1-def 99.9%

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

      13. exp-to-pow 99.9%

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

      14. unpow1/2 99.9%

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

      15. /-rgt-identity 99.9%

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

   5. Simplified 99.9%

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

      1. expm1-log1p-u 92.2%

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

      2. expm1-udef 92.1%

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

      3. hypot-1-def 92.1%

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

      4. add-sqr-sqrt 92.1%

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

      5. sqrt-undiv 92.1%

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

   7. Applied egg-rr 92.1%

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

      1. expm1-def 92.2%

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

      2. expm1-log1p 99.9%

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

      3. *-commutative 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-9)
   (* (pow x -0.5) (- (/ 0.5 x) (/ 0.375 (* x 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-9) {
		tmp = pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * 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-9) then
        tmp = (x ** (-0.5d0)) * ((0.5d0 / x) - (0.375d0 / (x * 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-9) {
		tmp = Math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * 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-9:
		tmp = math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * 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-9)
		tmp = Float64((x ^ -0.5) * Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * 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-9)
		tmp = (x ^ -0.5) * ((0.5 / x) - (0.375 / (x * 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-9], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * 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)))) < 2.00000000000000012e-9

   1. Initial program 38.2%

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

      1. frac-sub 38.3%

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

      2. div-inv 38.3%

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

      3. *-un-lft-identity 38.3%

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

      4. +-commutative 38.3%

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

      5. *-rgt-identity 38.3%

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

      6. metadata-eval 38.3%

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

      7. frac-times 38.3%

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

      8. un-div-inv 38.3%

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

      9. pow1/2 38.3%

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

      10. pow-flip 38.3%

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

      11. metadata-eval 38.3%

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

      12. +-commutative 38.3%

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

   3. Applied egg-rr 38.3%

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

      1. associate-*r/ 38.3%

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

      2. *-rgt-identity 38.3%

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

      3. times-frac 38.3%

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

      4. div-sub 38.3%

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

      5. *-inverses 38.3%

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

      6. unpow1 38.3%

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

      7. sqr-pow 38.3%

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

      8. metadata-eval 38.3%

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

      9. exp-to-pow 6.1%

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

      10. metadata-eval 6.1%

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

      11. exp-to-pow 6.0%

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

      12. hypot-1-def 6.0%

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

      13. exp-to-pow 38.3%

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

      14. unpow1/2 38.3%

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

      15. /-rgt-identity 38.3%

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

   5. Simplified 38.3%

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

      1. clear-num 38.3%

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

      2. inv-pow 38.3%

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

      3. hypot-1-def 38.2%

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

      4. add-sqr-sqrt 38.2%

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

      5. sqrt-undiv 38.3%

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

   7. Applied egg-rr 38.3%

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

      1. unpow-1 38.3%

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

   9. Simplified 38.3%

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

   10. Taylor expanded in x around inf 99.5%

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

       1. associate-*r/ 99.5%

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

       2. metadata-eval 99.5%

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

       3. associate-*r/ 99.5%

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

       4. metadata-eval 99.5%

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

       5. unpow2 99.5%

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

   12. Simplified 99.5%

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

   if 2.00000000000000012e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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 99.9%

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

      11. metadata-eval 99.9%

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

      12. +-commutative 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      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. unpow1 99.9%

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

      7. sqr-pow 99.9%

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

      8. metadata-eval 99.9%

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

      9. exp-to-pow 99.9%

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

      10. metadata-eval 99.9%

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

      11. exp-to-pow 99.9%

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

      12. hypot-1-def 99.9%

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

      13. exp-to-pow 99.9%

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

      14. unpow1/2 99.9%

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

      15. /-rgt-identity 99.9%

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

   5. Simplified 99.9%

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

      1. expm1-log1p-u 92.2%

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

      2. expm1-udef 92.1%

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

      3. hypot-1-def 92.1%

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

      4. add-sqr-sqrt 92.1%

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

      5. sqrt-undiv 92.1%

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

   7. Applied egg-rr 92.1%

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

      1. expm1-def 92.2%

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

      2. expm1-log1p 99.9%

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

      3. *-commutative 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-9)
   (* (pow x -0.5) (- (/ 0.5 x) (/ 0.375 (* x 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-9) {
		tmp = pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * 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-9) then
        tmp = (x ** (-0.5d0)) * ((0.5d0 / x) - (0.375d0 / (x * 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-9) {
		tmp = Math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * 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-9:
		tmp = math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * 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-9)
		tmp = Float64((x ^ -0.5) * Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * 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-9)
		tmp = (x ^ -0.5) * ((0.5 / x) - (0.375 / (x * 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-9], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * 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)))) < 2.00000000000000012e-9

   1. Initial program 38.2%

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

      1. frac-sub 38.3%

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

      2. div-inv 38.3%

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

      3. *-un-lft-identity 38.3%

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

      4. +-commutative 38.3%

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

      5. *-rgt-identity 38.3%

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

      6. metadata-eval 38.3%

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

      7. frac-times 38.3%

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

      8. un-div-inv 38.3%

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

      9. pow1/2 38.3%

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

      10. pow-flip 38.3%

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

      11. metadata-eval 38.3%

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

      12. +-commutative 38.3%

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

   3. Applied egg-rr 38.3%

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

      1. associate-*r/ 38.3%

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

      2. *-rgt-identity 38.3%

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

      3. times-frac 38.3%

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

      4. div-sub 38.3%

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

      5. *-inverses 38.3%

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

      6. unpow1 38.3%

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

      7. sqr-pow 38.3%

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

      8. metadata-eval 38.3%

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

      9. exp-to-pow 6.1%

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

      10. metadata-eval 6.1%

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

      11. exp-to-pow 6.0%

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

      12. hypot-1-def 6.0%

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

      13. exp-to-pow 38.3%

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

      14. unpow1/2 38.3%

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

      15. /-rgt-identity 38.3%

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

   5. Simplified 38.3%

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

      1. clear-num 38.3%

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

      2. inv-pow 38.3%

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

      3. hypot-1-def 38.2%

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

      4. add-sqr-sqrt 38.2%

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

      5. sqrt-undiv 38.3%

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

   7. Applied egg-rr 38.3%

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

      1. unpow-1 38.3%

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

   9. Simplified 38.3%

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

   10. Taylor expanded in x around inf 99.5%

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

       1. associate-*r/ 99.5%

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

       2. metadata-eval 99.5%

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

       3. associate-*r/ 99.5%

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

       4. metadata-eval 99.5%

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

       5. unpow2 99.5%

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

   12. Simplified 99.5%

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

   if 2.00000000000000012e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. clear-num 99.4%

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

      3. associate-/r/ 99.4%

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

      4. prod-diff 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. fma-neg 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. inv-pow 99.4%

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

      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. neg-mul-1 99.9%

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

      3. rem-log-exp 99.9%

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

      4. log-rec 99.9%

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

      5. +-commutative 99.9%

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

      6. log-rec 99.9%

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

      7. rem-log-exp 99.9%

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

      8. sub-neg 99.9%

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

      9. +-inverses 99.9%

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

      10. +-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 99.7%

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

Alternative 5: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.42)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (* (pow x -0.5) (- (/ 0.5 x) (/ 0.375 (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.42) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.42d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = (x ** (-0.5d0)) * ((0.5d0 / x) - (0.375d0 / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.42) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = Math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.42:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.42)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64((x ^ -0.5) * Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.42)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = (x ^ -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.42], 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[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

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

   4. Simplified 99.0%

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

      1. add-log-exp 4.0%

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

      2. *-un-lft-identity 4.0%

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

      3. log-prod 4.0%

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

      4. metadata-eval 4.0%

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

      5. add-log-exp 99.0%

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

      6. pow1/2 99.0%

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

      7. pow-flip 99.5%

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

      8. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. +-lft-identity 99.5%

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

   8. Simplified 99.5%

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

   if 1.4199999999999999 < x

   1. Initial program 39.2%

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

      1. frac-sub 39.3%

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

      2. div-inv 39.3%

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

      3. *-un-lft-identity 39.3%

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

      4. +-commutative 39.3%

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

      5. *-rgt-identity 39.3%

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

      6. metadata-eval 39.3%

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

      7. frac-times 39.3%

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

      8. un-div-inv 39.3%

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

      9. pow1/2 39.3%

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

      10. pow-flip 39.3%

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

      11. metadata-eval 39.3%

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

      12. +-commutative 39.3%

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

   3. Applied egg-rr 39.3%

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

      1. associate-*r/ 39.3%

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

      2. *-rgt-identity 39.3%

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

      3. times-frac 39.3%

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

      4. div-sub 39.3%

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

      5. *-inverses 39.3%

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

      6. unpow1 39.3%

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

      7. sqr-pow 39.3%

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

      8. metadata-eval 39.3%

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

      9. exp-to-pow 7.7%

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

      10. metadata-eval 7.7%

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

      11. exp-to-pow 7.6%

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

      12. hypot-1-def 7.6%

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

      13. exp-to-pow 39.3%

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

      14. unpow1/2 39.3%

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

      15. /-rgt-identity 39.3%

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

   5. Simplified 39.3%

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

      1. clear-num 39.3%

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

      2. inv-pow 39.3%

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

      3. hypot-1-def 39.3%

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

      4. add-sqr-sqrt 39.3%

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

      5. sqrt-undiv 39.3%

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

   7. Applied egg-rr 39.3%

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

      1. unpow-1 39.3%

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

   9. Simplified 39.3%

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

   10. Taylor expanded in x around inf 98.3%

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

       1. associate-*r/ 98.3%

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

       2. metadata-eval 98.3%

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

       3. associate-*r/ 98.3%

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

       4. metadata-eval 98.3%

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

       5. unpow2 98.3%

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

   12. Simplified 98.3%

       \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)} \cdot {x}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \end{array} \]

Alternative 6: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (* 0.5 (pow x -1.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 0.5 * pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.7d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = 0.5d0 * (x ** (-1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 0.5 * Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = 0.5 * math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(0.5 * (x ^ -1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = 0.5 * (x ^ -1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.7], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

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

   4. Simplified 99.0%

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

      1. add-log-exp 4.0%

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

      2. *-un-lft-identity 4.0%

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

      3. log-prod 4.0%

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

      4. metadata-eval 4.0%

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

      5. add-log-exp 99.0%

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

      6. pow1/2 99.0%

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

      7. pow-flip 99.5%

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

      8. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. +-lft-identity 99.5%

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

   8. Simplified 99.5%

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

   if 1.69999999999999996 < x

   1. Initial program 39.2%

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

      1. flip-- 39.3%

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

      2. div-inv 39.3%

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

      3. frac-times 25.8%

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

      4. metadata-eval 25.8%

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

      5. add-sqr-sqrt 23.6%

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

      6. frac-times 21.4%

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

      7. metadata-eval 21.4%

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

      8. add-sqr-sqrt 39.5%

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

      9. +-commutative 39.5%

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

      10. pow1/2 39.5%

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

      11. pow-flip 39.5%

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

      12. metadata-eval 39.5%

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

      13. inv-pow 39.5%

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

      14. sqrt-pow2 39.5%

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

      15. +-commutative 39.5%

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

      16. metadata-eval 39.5%

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

   3. Applied egg-rr 39.5%

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

      1. frac-sub 40.4%

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

      2. *-un-lft-identity 40.4%

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

   5. Applied egg-rr 40.4%

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

      1. *-rgt-identity 40.4%

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

      2. associate--l+ 84.2%

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

      3. +-inverses 84.2%

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

      4. metadata-eval 84.2%

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

      5. distribute-lft-in 84.2%

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

      6. *-rgt-identity 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in x around inf 60.9%

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

      1. unpow-1 60.9%

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

      2. exp-to-pow 58.8%

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

      3. *-commutative 58.8%

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

      4. log-pow 59.2%

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

      5. associate-*r* 59.2%

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

      6. metadata-eval 59.2%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

      7. *-commutative 59.2%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

      8. exp-to-pow 61.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-3}}} \]

      9. metadata-eval 61.5%

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

      10. pow-sqr 61.6%

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

      11. rem-sqrt-square 97.5%

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

      12. rem-square-sqrt 97.0%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

      13. fabs-sqr 97.0%

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

      14. rem-square-sqrt 97.5%

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

   10. Simplified 97.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 7: 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 99.5%

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

      1. inv-pow 99.5%

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

      2. pow1/2 99.5%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.1%

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

      5. pow-exp 92.1%

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

      6. metadata-eval 92.1%

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

   3. Applied egg-rr 92.1%

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

   4. Taylor expanded in x around 0 99.5%

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

   if 1 < x

   1. Initial program 39.2%

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

      1. flip-- 39.3%

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

      2. div-inv 39.3%

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

      3. frac-times 25.8%

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

      4. metadata-eval 25.8%

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

      5. add-sqr-sqrt 23.6%

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

      6. frac-times 21.4%

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

      7. metadata-eval 21.4%

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

      8. add-sqr-sqrt 39.5%

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

      9. +-commutative 39.5%

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

      10. pow1/2 39.5%

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

      11. pow-flip 39.5%

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

      12. metadata-eval 39.5%

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

      13. inv-pow 39.5%

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

      14. sqrt-pow2 39.5%

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

      15. +-commutative 39.5%

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

      16. metadata-eval 39.5%

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

   3. Applied egg-rr 39.5%

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

      1. frac-sub 40.4%

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

      2. *-un-lft-identity 40.4%

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

   5. Applied egg-rr 40.4%

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

      1. *-rgt-identity 40.4%

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

      2. associate--l+ 84.2%

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

      3. +-inverses 84.2%

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

      4. metadata-eval 84.2%

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

      5. distribute-lft-in 84.2%

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

      6. *-rgt-identity 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in x around inf 60.9%

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

      1. unpow-1 60.9%

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

      2. exp-to-pow 58.8%

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

      3. *-commutative 58.8%

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

      4. log-pow 59.2%

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

      5. associate-*r* 59.2%

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

      6. metadata-eval 59.2%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

      7. *-commutative 59.2%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

      8. exp-to-pow 61.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-3}}} \]

      9. metadata-eval 61.5%

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

      10. pow-sqr 61.6%

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

      11. rem-sqrt-square 97.5%

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

      12. rem-square-sqrt 97.0%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

      13. fabs-sqr 97.0%

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

      14. rem-square-sqrt 97.5%

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

   10. Simplified 97.5%

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

4. Final simplification 98.6%

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

Alternative 8: 97.0% 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 99.5%

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

      1. inv-pow 99.5%

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

      2. pow1/2 99.5%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.1%

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

      5. pow-exp 92.1%

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

      6. metadata-eval 92.1%

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

   3. Applied egg-rr 92.1%

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

   4. Taylor expanded in x around inf 97.3%

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

      1. pow1/2 97.3%

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

      2. inv-pow 97.3%

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

      3. pow-pow 97.5%

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

      4. metadata-eval 97.5%

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

      5. expm1-log1p-u 89.9%

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

      6. expm1-udef 89.9%

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

   6. Applied egg-rr 89.9%

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

      1. expm1-def 89.9%

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

      2. expm1-log1p 97.5%

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

   8. Simplified 97.5%

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

   if 0.5 < x

   1. Initial program 39.8%

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

      1. flip-- 39.9%

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

      2. div-inv 39.9%

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

      3. frac-times 26.4%

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

      4. metadata-eval 26.4%

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

      5. add-sqr-sqrt 24.3%

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

      6. frac-times 22.1%

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

      7. metadata-eval 22.1%

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

      8. add-sqr-sqrt 40.0%

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

      9. +-commutative 40.0%

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

      10. pow1/2 40.0%

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

      11. pow-flip 40.0%

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

      12. metadata-eval 40.0%

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

      13. inv-pow 40.0%

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

      14. sqrt-pow2 40.0%

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

      15. +-commutative 40.0%

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

      16. metadata-eval 40.0%

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

   3. Applied egg-rr 40.0%

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

      1. frac-sub 40.9%

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

      2. *-un-lft-identity 40.9%

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

   5. Applied egg-rr 40.9%

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

      1. *-rgt-identity 40.9%

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

      2. associate--l+ 84.3%

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

      3. +-inverses 84.3%

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

      4. metadata-eval 84.3%

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

      5. distribute-lft-in 84.3%

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

      6. *-rgt-identity 84.3%

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

   7. Simplified 84.3%

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

   8. Taylor expanded in x around inf 60.5%

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

      1. unpow-1 60.5%

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

      2. exp-to-pow 58.4%

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

      3. *-commutative 58.4%

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

      4. log-pow 58.9%

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

      5. associate-*r* 58.9%

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

      6. metadata-eval 58.9%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

      7. *-commutative 58.9%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

      8. exp-to-pow 61.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-3}}} \]

      9. metadata-eval 61.1%

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

      10. pow-sqr 61.2%

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

      11. rem-sqrt-square 96.7%

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

      12. rem-square-sqrt 96.3%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

      13. fabs-sqr 96.3%

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

      14. rem-square-sqrt 96.7%

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

   10. Simplified 96.7%

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

4. Final simplification 97.2%

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

Alternative 9: 98.4% 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 99.5%

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

      1. inv-pow 99.5%

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

      2. pow1/2 99.5%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.1%

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

      5. pow-exp 92.1%

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

      6. metadata-eval 92.1%

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

   3. Applied egg-rr 92.1%

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

   4. Taylor expanded in x around 0 99.4%

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

   if 0.680000000000000049 < x

   1. Initial program 39.2%

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

      1. flip-- 39.3%

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

      2. div-inv 39.3%

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

      3. frac-times 25.8%

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

      4. metadata-eval 25.8%

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

      5. add-sqr-sqrt 23.6%

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

      6. frac-times 21.4%

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

      7. metadata-eval 21.4%

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

      8. add-sqr-sqrt 39.5%

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

      9. +-commutative 39.5%

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

      10. pow1/2 39.5%

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

      11. pow-flip 39.5%

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

      12. metadata-eval 39.5%

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

      13. inv-pow 39.5%

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

      14. sqrt-pow2 39.5%

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

      15. +-commutative 39.5%

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

      16. metadata-eval 39.5%

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

   3. Applied egg-rr 39.5%

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

      1. frac-sub 40.4%

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

      2. *-un-lft-identity 40.4%

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

   5. Applied egg-rr 40.4%

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

      1. *-rgt-identity 40.4%

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

      2. associate--l+ 84.2%

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

      3. +-inverses 84.2%

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

      4. metadata-eval 84.2%

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

      5. distribute-lft-in 84.2%

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

      6. *-rgt-identity 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in x around inf 60.9%

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

      1. unpow-1 60.9%

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

      2. exp-to-pow 58.8%

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

      3. *-commutative 58.8%

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

      4. log-pow 59.2%

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

      5. associate-*r* 59.2%

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

      6. metadata-eval 59.2%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

      7. *-commutative 59.2%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

      8. exp-to-pow 61.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-3}}} \]

      9. metadata-eval 61.5%

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

      10. pow-sqr 61.6%

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

      11. rem-sqrt-square 97.5%

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

      12. rem-square-sqrt 97.0%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

      13. fabs-sqr 97.0%

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

      14. rem-square-sqrt 97.5%

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

   10. Simplified 97.5%

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

4. Final simplification 98.6%

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

Alternative 10: 50.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 74.1%

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

   1. inv-pow 74.1%

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

   2. pow1/2 74.1%

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

   3. pow-pow 70.9%

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

   4. add-exp-log 56.4%

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

   5. pow-exp 56.4%

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

   6. metadata-eval 56.4%

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

3. Applied egg-rr 56.4%

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

4. Taylor expanded in x around inf 58.2%

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

   1. pow1/2 58.2%

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

   2. inv-pow 58.2%

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

   3. pow-pow 58.4%

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

   4. metadata-eval 58.4%

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

   5. expm1-log1p-u 54.0%

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

   6. expm1-udef 67.1%

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

6. Applied egg-rr 67.1%

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

   1. expm1-def 54.0%

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

   2. expm1-log1p 58.4%

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

8. Simplified 58.4%

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

9. Final simplification 58.4%

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

Alternative 11: 1.9% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 74.1%

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

2. Taylor expanded in x around 0 58.3%

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

3. Taylor expanded in x around inf 1.8%

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

4. Final simplification 1.8%

   \[\leadsto -1 \]

Developer target: 99.1% 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 2023275 
(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)))))