2isqrt (example 3.6)

Percentage Accurate: 69.7% → 99.3%

Time: 14.2s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.7% 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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ 1.0 x)))) 5e-17)
   (* (- (/ 0.5 x) (/ 0.375 (* x x))) (pow x -0.5))
   (- (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)))) <= 5e-17) {
		tmp = ((0.5 / x) - (0.375 / (x * x))) * pow(x, -0.5);
	} 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)))) <= 5d-17) then
        tmp = ((0.5d0 / x) - (0.375d0 / (x * x))) * (x ** (-0.5d0))
    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)))) <= 5e-17) {
		tmp = ((0.5 / x) - (0.375 / (x * x))) * Math.pow(x, -0.5);
	} 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)))) <= 5e-17:
		tmp = ((0.5 / x) - (0.375 / (x * x))) * math.pow(x, -0.5)
	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)))) <= 5e-17)
		tmp = Float64(Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * x))) * (x ^ -0.5));
	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)))) <= 5e-17)
		tmp = ((0.5 / x) - (0.375 / (x * x))) * (x ^ -0.5);
	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], 5e-17], N[(N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.5], $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)))) < 4.9999999999999999e-17

   1. Initial program 27.5%

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

      1. frac-sub 27.5%

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

      2. div-inv 27.5%

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

      3. *-un-lft-identity 27.5%

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

      4. +-commutative 27.5%

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

      5. *-rgt-identity 27.5%

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

      6. metadata-eval 27.5%

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

      7. frac-times 27.5%

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

      8. un-div-inv 27.5%

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

      9. pow1/2 27.5%

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

      10. pow-flip 27.5%

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

      11. metadata-eval 27.5%

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

      12. +-commutative 27.5%

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

   3. Applied egg-rr 27.5%

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

      1. associate-*r/ 27.5%

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

      2. *-rgt-identity 27.5%

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

      3. times-frac 27.5%

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

      4. div-sub 27.5%

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

      5. *-inverses 27.5%

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

      6. unpow1 27.5%

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

      7. sqr-pow 27.5%

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

      8. metadata-eval 27.5%

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

      9. exp-to-pow 5.0%

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

      10. metadata-eval 5.0%

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

      11. exp-to-pow 4.9%

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

      12. hypot-1-def 4.9%

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

      13. exp-to-pow 27.5%

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

      14. unpow1/2 27.5%

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

      15. /-rgt-identity 27.5%

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

   5. Simplified 27.5%

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

      1. *-un-lft-identity 27.5%

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

      2. hypot-1-def 27.5%

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

      3. add-sqr-sqrt 27.5%

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

      4. sqrt-undiv 27.5%

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

   7. Applied egg-rr 27.5%

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

      1. *-lft-identity 27.5%

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

   9. Simplified 27.5%

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

   10. Taylor expanded in x around inf 99.7%

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

       1. associate-*r/ 99.7%

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

       2. metadata-eval 99.7%

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

       3. associate-*r/ 99.7%

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

       4. metadata-eval 99.7%

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

       5. unpow2 99.7%

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

   12. Simplified 99.7%

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

   if 4.9999999999999999e-17 < (-.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 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow1/2 99.8%

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

      12. pow-flip 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. neg-mul-1 99.8%

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

      3. rem-log-exp 99.7%

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

      4. log-rec 99.7%

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

      5. +-commutative 99.7%

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

      6. log-rec 99.7%

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

      7. rem-log-exp 99.8%

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

      8. sub-neg 99.8%

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

      9. +-inverses 99.8%

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

      10. +-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right) \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.45)
   (+ (/ 1.0 (sqrt x)) (/ -1.0 (+ 1.0 (* x 0.5))))
   (* (- (/ 0.5 x) (/ 0.375 (* x x))) (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = (1.0 / sqrt(x)) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = ((0.5 / x) - (0.375 / (x * x))) * pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.45d0) then
        tmp = (1.0d0 / sqrt(x)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = ((0.5d0 / x) - (0.375d0 / (x * x))) * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = (1.0 / Math.sqrt(x)) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = ((0.5 / x) - (0.375 / (x * x))) * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.45:
		tmp = (1.0 / math.sqrt(x)) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = ((0.5 / x) - (0.375 / (x * x))) * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45)
		tmp = Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * x))) * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.45)
		tmp = (1.0 / sqrt(x)) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = ((0.5 / x) - (0.375 / (x * x))) * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.45], N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

   4. Simplified 98.3%

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

   if 1.44999999999999996 < x

   1. Initial program 29.6%

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

      1. frac-sub 29.6%

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

      2. div-inv 29.6%

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

      3. *-un-lft-identity 29.6%

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

      4. +-commutative 29.6%

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

      5. *-rgt-identity 29.6%

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

      6. metadata-eval 29.6%

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

      7. frac-times 29.7%

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

      8. un-div-inv 29.7%

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

      9. pow1/2 29.7%

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

      10. pow-flip 29.7%

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

      11. metadata-eval 29.7%

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

      12. +-commutative 29.7%

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

   3. Applied egg-rr 29.7%

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

      1. associate-*r/ 29.7%

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

      2. *-rgt-identity 29.7%

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

      3. times-frac 29.7%

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

      4. div-sub 29.6%

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

      5. *-inverses 29.6%

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

      6. unpow1 29.6%

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

      7. sqr-pow 29.6%

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

      8. metadata-eval 29.6%

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

      9. exp-to-pow 7.9%

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

      10. metadata-eval 7.9%

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

      11. exp-to-pow 7.7%

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

      12. hypot-1-def 7.7%

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

      13. exp-to-pow 29.6%

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

      14. unpow1/2 29.6%

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

      15. /-rgt-identity 29.6%

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

   5. Simplified 29.6%

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

      1. *-un-lft-identity 29.6%

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

      2. hypot-1-def 29.6%

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

      3. add-sqr-sqrt 29.6%

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

      4. sqrt-undiv 29.6%

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

   7. Applied egg-rr 29.6%

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

      1. *-lft-identity 29.6%

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

   9. Simplified 29.6%

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

   10. Taylor expanded in x around inf 97.5%

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

       1. associate-*r/ 97.5%

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

       2. metadata-eval 97.5%

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

       3. associate-*r/ 97.5%

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

       4. metadata-eval 97.5%

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

       5. unpow2 97.5%

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

   12. Simplified 97.5%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right) \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

   4. Simplified 98.3%

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

   if 1.69999999999999996 < x

   1. Initial program 29.6%

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

      1. frac-sub 29.6%

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

      2. div-inv 29.6%

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

      3. *-un-lft-identity 29.6%

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

      4. +-commutative 29.6%

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

      5. *-rgt-identity 29.6%

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

      6. metadata-eval 29.6%

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

      7. frac-times 29.7%

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

      8. un-div-inv 29.7%

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

      9. pow1/2 29.7%

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

      10. pow-flip 29.7%

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

      11. metadata-eval 29.7%

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

      12. +-commutative 29.7%

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

   3. Applied egg-rr 29.7%

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

      1. associate-*r/ 29.7%

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

      2. *-rgt-identity 29.7%

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

      3. times-frac 29.7%

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

      4. div-sub 29.6%

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

      5. *-inverses 29.6%

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

      6. unpow1 29.6%

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

      7. sqr-pow 29.6%

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

      8. metadata-eval 29.6%

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

      9. exp-to-pow 7.9%

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

      10. metadata-eval 7.9%

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

      11. exp-to-pow 7.7%

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

      12. hypot-1-def 7.7%

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

      13. exp-to-pow 29.6%

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

      14. unpow1/2 29.6%

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

      15. /-rgt-identity 29.6%

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

   5. Simplified 29.6%

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

      1. *-un-lft-identity 29.6%

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

      2. hypot-1-def 29.6%

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

      3. add-sqr-sqrt 29.6%

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

      4. sqrt-undiv 29.6%

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

   7. Applied egg-rr 29.6%

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

      1. *-lft-identity 29.6%

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

   9. Simplified 29.6%

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

   10. Taylor expanded in x around inf 97.1%

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

       1. associate-*l/ 97.1%

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

   12. Applied egg-rr 97.1%

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

4. Final simplification 97.7%

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

Alternative 4: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 98.2%

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

   if 1 < x

   1. Initial program 29.6%

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

      1. frac-sub 29.6%

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

      2. div-inv 29.6%

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

      3. *-un-lft-identity 29.6%

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

      4. +-commutative 29.6%

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

      5. *-rgt-identity 29.6%

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

      6. metadata-eval 29.6%

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

      7. frac-times 29.7%

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

      8. un-div-inv 29.7%

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

      9. pow1/2 29.7%

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

      10. pow-flip 29.7%

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

      11. metadata-eval 29.7%

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

      12. +-commutative 29.7%

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

   3. Applied egg-rr 29.7%

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

      1. associate-*r/ 29.7%

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

      2. *-rgt-identity 29.7%

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

      3. times-frac 29.7%

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

      4. div-sub 29.6%

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

      5. *-inverses 29.6%

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

      6. unpow1 29.6%

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

      7. sqr-pow 29.6%

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

      8. metadata-eval 29.6%

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

      9. exp-to-pow 7.9%

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

      10. metadata-eval 7.9%

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

      11. exp-to-pow 7.7%

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

      12. hypot-1-def 7.7%

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

      13. exp-to-pow 29.6%

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

      14. unpow1/2 29.6%

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

      15. /-rgt-identity 29.6%

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

   5. Simplified 29.6%

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

      1. *-un-lft-identity 29.6%

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

      2. hypot-1-def 29.6%

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

      3. add-sqr-sqrt 29.6%

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

      4. sqrt-undiv 29.6%

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

   7. Applied egg-rr 29.6%

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

      1. *-lft-identity 29.6%

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

   9. Simplified 29.6%

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

   10. Taylor expanded in x around inf 97.1%

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

       1. associate-*l/ 97.1%

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

   12. Applied egg-rr 97.1%

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

4. Final simplification 97.7%

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

Alternative 5: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) * pow(x, -0.5);
	}
	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) * (x ** (-0.5d0))
    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 / x) * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(Float64(0.5 / x) * (x ^ -0.5));
	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) * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 97.5%

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

      1. add-log-exp 5.4%

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

      2. *-un-lft-identity 5.4%

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

      3. log-prod 5.4%

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

      4. metadata-eval 5.4%

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

      5. add-log-exp 97.5%

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

      6. pow1/2 97.5%

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

      7. pow-flip 97.9%

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

      8. metadata-eval 97.9%

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

   4. Applied egg-rr 97.9%

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

      1. +-lft-identity 97.9%

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

   6. Simplified 97.9%

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

   if 0.680000000000000049 < x

   1. Initial program 29.6%

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

      1. frac-sub 29.6%

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

      2. div-inv 29.6%

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

      3. *-un-lft-identity 29.6%

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

      4. +-commutative 29.6%

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

      5. *-rgt-identity 29.6%

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

      6. metadata-eval 29.6%

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

      7. frac-times 29.7%

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

      8. un-div-inv 29.7%

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

      9. pow1/2 29.7%

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

      10. pow-flip 29.7%

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

      11. metadata-eval 29.7%

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

      12. +-commutative 29.7%

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

   3. Applied egg-rr 29.7%

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

      1. associate-*r/ 29.7%

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

      2. *-rgt-identity 29.7%

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

      3. times-frac 29.7%

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

      4. div-sub 29.6%

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

      5. *-inverses 29.6%

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

      6. unpow1 29.6%

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

      7. sqr-pow 29.6%

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

      8. metadata-eval 29.6%

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

      9. exp-to-pow 7.9%

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

      10. metadata-eval 7.9%

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

      11. exp-to-pow 7.7%

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

      12. hypot-1-def 7.7%

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

      13. exp-to-pow 29.6%

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

      14. unpow1/2 29.6%

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

      15. /-rgt-identity 29.6%

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

   5. Simplified 29.6%

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

      1. *-un-lft-identity 29.6%

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

      2. hypot-1-def 29.6%

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

      3. add-sqr-sqrt 29.6%

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

      4. sqrt-undiv 29.6%

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

   7. Applied egg-rr 29.6%

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

      1. *-lft-identity 29.6%

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

   9. Simplified 29.6%

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

   10. Taylor expanded in x around inf 97.1%

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

4. Final simplification 97.5%

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

Alternative 6: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 * pow(x, -0.5)) / x;
	}
	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 ** (-0.5d0))) / x
    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, -0.5)) / x;
	}
	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, -0.5)) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(Float64(0.5 * (x ^ -0.5)) / x);
	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 ^ -0.5)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 97.5%

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

      1. add-log-exp 5.4%

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

      2. *-un-lft-identity 5.4%

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

      3. log-prod 5.4%

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

      4. metadata-eval 5.4%

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

      5. add-log-exp 97.5%

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

      6. pow1/2 97.5%

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

      7. pow-flip 97.9%

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

      8. metadata-eval 97.9%

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

   4. Applied egg-rr 97.9%

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

      1. +-lft-identity 97.9%

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

   6. Simplified 97.9%

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

   if 0.680000000000000049 < x

   1. Initial program 29.6%

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

      1. frac-sub 29.6%

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

      2. div-inv 29.6%

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

      3. *-un-lft-identity 29.6%

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

      4. +-commutative 29.6%

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

      5. *-rgt-identity 29.6%

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

      6. metadata-eval 29.6%

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

      7. frac-times 29.7%

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

      8. un-div-inv 29.7%

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

      9. pow1/2 29.7%

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

      10. pow-flip 29.7%

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

      11. metadata-eval 29.7%

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

      12. +-commutative 29.7%

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

   3. Applied egg-rr 29.7%

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

      1. associate-*r/ 29.7%

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

      2. *-rgt-identity 29.7%

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

      3. times-frac 29.7%

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

      4. div-sub 29.6%

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

      5. *-inverses 29.6%

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

      6. unpow1 29.6%

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

      7. sqr-pow 29.6%

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

      8. metadata-eval 29.6%

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

      9. exp-to-pow 7.9%

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

      10. metadata-eval 7.9%

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

      11. exp-to-pow 7.7%

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

      12. hypot-1-def 7.7%

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

      13. exp-to-pow 29.6%

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

      14. unpow1/2 29.6%

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

      15. /-rgt-identity 29.6%

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

   5. Simplified 29.6%

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

      1. *-un-lft-identity 29.6%

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

      2. hypot-1-def 29.6%

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

      3. add-sqr-sqrt 29.6%

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

      4. sqrt-undiv 29.6%

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

   7. Applied egg-rr 29.6%

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

      1. *-lft-identity 29.6%

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

   9. Simplified 29.6%

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

   10. Taylor expanded in x around inf 97.1%

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

       1. associate-*l/ 97.1%

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

   12. Applied egg-rr 97.1%

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

4. Final simplification 97.5%

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

Alternative 7: 67.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;-{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.0) (+ (pow x -0.5) -1.0) (- (pow (* x x) -0.25))))

C

double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = -pow((x * x), -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.0d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = -((x * x) ** (-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = -Math.pow((x * x), -0.25);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.0:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = -math.pow((x * x), -0.25)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(-(Float64(x * x) ^ -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.0)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = -((x * x) ^ -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.0], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], (-N[Power[N[(x * x), $MachinePrecision], -0.25], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 96.7%

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

      1. add-log-exp 5.4%

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

      2. *-un-lft-identity 5.4%

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

      3. log-prod 5.4%

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

      4. metadata-eval 5.4%

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

      5. add-log-exp 96.7%

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

      6. pow1/2 96.7%

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

      7. pow-flip 97.1%

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

      8. metadata-eval 97.1%

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

   4. Applied egg-rr 97.1%

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

      1. +-lft-identity 97.1%

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

   6. Simplified 97.1%

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

   if 4 < x

   1. Initial program 29.0%

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

      1. pow1/2 29.0%

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

      2. pow-to-exp 6.9%

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

      3. +-commutative 6.9%

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

      4. log1p-udef 6.9%

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

   3. Applied egg-rr 6.9%

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

   4. Taylor expanded in x around inf 3.1%

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

      1. mul-1-neg 3.1%

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

   6. Simplified 3.1%

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

      1. inv-pow 3.1%

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

      2. sqrt-pow1 3.1%

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

      3. metadata-eval 3.1%

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

      4. sqr-pow 3.1%

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

      5. pow-prod-down 25.2%

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

      6. metadata-eval 25.2%

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

   8. Applied egg-rr 25.2%

      \[\leadsto -\color{blue}{{\left(x \cdot x\right)}^{-0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;-{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \]

Alternative 8: 51.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

2. Taylor expanded in x around 0 50.4%

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

   1. add-log-exp 4.0%

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

   2. *-un-lft-identity 4.0%

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

   3. log-prod 4.0%

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

   4. metadata-eval 4.0%

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

   5. add-log-exp 50.4%

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

   6. pow1/2 50.4%

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

   7. pow-flip 50.6%

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

   8. metadata-eval 50.6%

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

4. Applied egg-rr 50.6%

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

   1. +-lft-identity 50.6%

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

6. Simplified 50.6%

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

7. Final simplification 50.6%

   \[\leadsto {x}^{-0.5} + -1 \]

Alternative 9: 2.1% 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 Float64(-(x ^ -0.5))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

   1. inv-pow 64.9%

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

   2. pow1/2 64.9%

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

   3. pow-pow 62.1%

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

   4. add-exp-log 53.9%

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

   5. +-commutative 53.9%

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

   6. log1p-udef 53.9%

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

   7. pow-exp 53.9%

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

   8. metadata-eval 53.9%

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

3. Applied egg-rr 53.9%

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

4. Taylor expanded in x around inf 2.1%

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

   1. mul-1-neg 2.1%

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

   2. unpow1/2 2.1%

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

   3. exp-to-pow 2.1%

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

   4. log-rec 2.1%

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

   5. distribute-lft-neg-out 2.1%

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

   6. distribute-rgt-neg-in 2.1%

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

   7. metadata-eval 2.1%

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

   8. exp-to-pow 2.1%

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

6. Simplified 2.1%

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

7. Final simplification 2.1%

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

Alternative 10: 51.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

   1. frac-sub 64.9%

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

   2. div-inv 64.9%

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

   3. *-un-lft-identity 64.9%

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

   4. +-commutative 64.9%

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

   5. *-rgt-identity 64.9%

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

   6. metadata-eval 64.9%

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

   7. frac-times 64.9%

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

   8. un-div-inv 64.9%

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

   9. pow1/2 64.9%

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

   10. pow-flip 65.1%

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

   11. metadata-eval 65.1%

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

   12. +-commutative 65.1%

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

3. Applied egg-rr 65.1%

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

   1. associate-*r/ 65.1%

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

   2. *-rgt-identity 65.1%

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

   3. times-frac 65.1%

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

   4. div-sub 65.1%

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

   5. *-inverses 65.1%

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

   6. unpow1 65.1%

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

   7. sqr-pow 65.1%

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

   8. metadata-eval 65.1%

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

   9. exp-to-pow 54.3%

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

   10. metadata-eval 54.3%

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

   11. exp-to-pow 54.2%

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

   12. hypot-1-def 54.2%

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

   13. exp-to-pow 65.1%

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

   14. unpow1/2 65.1%

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

   15. /-rgt-identity 65.1%

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

5. Simplified 65.1%

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

6. Taylor expanded in x around 0 50.2%

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

7. Final simplification 50.2%

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

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 64.9%

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

   1. add-cube-cbrt 55.2%

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

   2. associate-*l* 55.2%

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

   3. fma-neg 53.1%

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

   4. pow1/2 53.1%

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

   5. pow-flip 53.1%

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

   6. metadata-eval 53.1%

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

   7. cbrt-unprod 53.5%

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

   8. frac-times 53.5%

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

   9. metadata-eval 53.5%

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

   10. add-sqr-sqrt 53.5%

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

   11. distribute-neg-frac 53.5%

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

   12. metadata-eval 53.5%

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

   13. +-commutative 53.5%

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

3. Applied egg-rr 53.5%

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

4. Taylor expanded in x around 0 1.9%

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

5. Final simplification 1.9%

   \[\leadsto -1 \]

Developer target: 98.8% 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 2023285 
(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)))))