2isqrt (example 3.6)

Percentage Accurate: 37.8% → 99.9%

Time: 16.4s

Alternatives: 5

Speedup: 2.0×

Specification

\[x > 1 \land x < 10^{+308}\]

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: 37.8% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(x + {x}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 0.0)
   (* 0.5 (pow x -1.5))
   (/ 1.0 (* (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)) (+ x (pow x 2.0))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = 1.0 / ((pow(x, -0.5) + pow((1.0 + x), -0.5)) * (x + pow(x, 2.0)));
	}
	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)))) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = 1.0d0 / (((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0))) * (x + (x ** 2.0d0)))
    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)))) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = 1.0 / ((Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5)) * (x + Math.pow(x, 2.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = 1.0 / ((math.pow(x, -0.5) + math.pow((1.0 + x), -0.5)) * (x + math.pow(x, 2.0)))
	return tmp

Julia

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

MATLAB

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

   1. Initial program 38.1%

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

      1. flip-- 38.1%

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

      2. clear-num 38.1%

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

      3. pow1/2 38.1%

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

      4. pow-flip 38.1%

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

      5. metadata-eval 38.1%

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

      6. inv-pow 38.1%

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

      7. sqrt-pow2 38.1%

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

      8. +-commutative 38.1%

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

      9. metadata-eval 38.1%

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

      10. frac-times 20.3%

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

      11. metadata-eval 20.3%

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

      12. add-sqr-sqrt 16.8%

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

      13. frac-times 20.3%

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

      14. metadata-eval 20.3%

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

      15. add-sqr-sqrt 38.2%

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

   4. Applied egg-rr 38.2%

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

   5. Taylor expanded in x around inf 66.7%

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

      1. expm1-log1p-u 66.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 \cdot \sqrt{{x}^{3}}}\right)\right)} \]

      2. expm1-udef 38.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{2 \cdot \sqrt{{x}^{3}}}\right)} - 1} \]

      3. associate-/r* 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}}\right)} - 1 \]

      4. metadata-eval 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.5}}{\sqrt{{x}^{3}}}\right)} - 1 \]

      5. sqrt-pow1 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{\color{blue}{{x}^{\left(\frac{3}{2}\right)}}}\right)} - 1 \]

      6. metadata-eval 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{{x}^{\color{blue}{1.5}}}\right)} - 1 \]

   7. Applied egg-rr 38.1%

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

      1. expm1-def 97.8%

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

      2. expm1-log1p 97.8%

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

   9. Simplified 97.8%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 97.6%

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

       2. sqrt-unprod 66.9%

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

       3. frac-times 66.7%

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

       4. metadata-eval 66.7%

          \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{{x}^{1.5} \cdot {x}^{1.5}}} \]

       5. pow-prod-up 66.8%

          \[\leadsto \sqrt{\frac{0.25}{\color{blue}{{x}^{\left(1.5 + 1.5\right)}}}} \]

       6. metadata-eval 66.8%

          \[\leadsto \sqrt{\frac{0.25}{{x}^{\color{blue}{3}}}} \]

       7. expm1-log1p-u 66.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{0.25}{{x}^{3}}}\right)\right)} \]

       8. expm1-udef 38.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{0.25}{{x}^{3}}}\right)} - 1} \]

   11. Applied egg-rr 38.1%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

   13. Simplified 100.0%

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

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 60.9%

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

      1. add-cbrt-cube 58.3%

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

      2. pow1/3 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}}} \]

      3. add-sqr-sqrt 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{\left(x + 1\right)} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

      4. pow1 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{{\left(x + 1\right)}^{1}} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

      5. pow1/2 54.4%

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

      6. pow-prod-up 54.4%

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

      7. +-commutative 54.4%

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

      8. metadata-eval 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\left(1 + x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   4. Applied egg-rr 54.4%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   5. Step-by-step derivation

      1. unpow1/3 57.9%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]

   6. Simplified 57.9%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]

   8. Applied egg-rr 99.6%

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

      1. +-inverses 99.6%

         \[\leadsto \frac{1 + \color{blue}{0}}{\left({\left(1 + x\right)}^{-0.5} + {x}^{-0.5}\right) \cdot \left(x + {x}^{2}\right)} \]

      2. metadata-eval 99.6%

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

      3. +-commutative 99.6%

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

   10. Simplified 99.6%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(x + {x}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}{\mathsf{fma}\left(x, x, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 0.0)
   (* 0.5 (pow x -1.5))
   (/ (/ 1.0 (+ (pow x -0.5) (pow (+ 1.0 x) -0.5))) (fma x x x))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = (1.0 / (pow(x, -0.5) + pow((1.0 + x), -0.5))) / fma(x, x, 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)))) <= 0.0)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64(Float64(1.0 / Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5))) / fma(x, x, x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 0.0

   1. Initial program 38.1%

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

      1. flip-- 38.1%

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

      2. clear-num 38.1%

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

      3. pow1/2 38.1%

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

      4. pow-flip 38.1%

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

      5. metadata-eval 38.1%

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

      6. inv-pow 38.1%

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

      7. sqrt-pow2 38.1%

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

      8. +-commutative 38.1%

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

      9. metadata-eval 38.1%

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

      10. frac-times 20.3%

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

      11. metadata-eval 20.3%

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

      12. add-sqr-sqrt 16.8%

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

      13. frac-times 20.3%

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

      14. metadata-eval 20.3%

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

      15. add-sqr-sqrt 38.2%

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

   4. Applied egg-rr 38.2%

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

   5. Taylor expanded in x around inf 66.7%

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

      1. expm1-log1p-u 66.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 \cdot \sqrt{{x}^{3}}}\right)\right)} \]

      2. expm1-udef 38.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{2 \cdot \sqrt{{x}^{3}}}\right)} - 1} \]

      3. associate-/r* 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}}\right)} - 1 \]

      4. metadata-eval 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.5}}{\sqrt{{x}^{3}}}\right)} - 1 \]

      5. sqrt-pow1 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{\color{blue}{{x}^{\left(\frac{3}{2}\right)}}}\right)} - 1 \]

      6. metadata-eval 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{{x}^{\color{blue}{1.5}}}\right)} - 1 \]

   7. Applied egg-rr 38.1%

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

      1. expm1-def 97.8%

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

      2. expm1-log1p 97.8%

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

   9. Simplified 97.8%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 97.6%

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

       2. sqrt-unprod 66.9%

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

       3. frac-times 66.7%

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

       4. metadata-eval 66.7%

          \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{{x}^{1.5} \cdot {x}^{1.5}}} \]

       5. pow-prod-up 66.8%

          \[\leadsto \sqrt{\frac{0.25}{\color{blue}{{x}^{\left(1.5 + 1.5\right)}}}} \]

       6. metadata-eval 66.8%

          \[\leadsto \sqrt{\frac{0.25}{{x}^{\color{blue}{3}}}} \]

       7. expm1-log1p-u 66.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{0.25}{{x}^{3}}}\right)\right)} \]

       8. expm1-udef 38.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{0.25}{{x}^{3}}}\right)} - 1} \]

   11. Applied egg-rr 38.1%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

   13. Simplified 100.0%

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

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 60.9%

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

      1. add-cbrt-cube 58.3%

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

      2. pow1/3 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}}} \]

      3. add-sqr-sqrt 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{\left(x + 1\right)} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

      4. pow1 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left(\color{blue}{{\left(x + 1\right)}^{1}} \cdot \sqrt{x + 1}\right)}^{0.3333333333333333}} \]

      5. pow1/2 54.4%

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

      6. pow-prod-up 54.4%

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

      7. +-commutative 54.4%

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

      8. metadata-eval 54.4%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{{\left({\left(1 + x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   4. Applied egg-rr 54.4%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   5. Step-by-step derivation

      1. unpow1/3 57.9%

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]

   6. Simplified 57.9%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{1.5}}}} \]

   8. Applied egg-rr 99.6%

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

      1. associate-/r* 99.3%

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

      2. +-inverses 99.3%

         \[\leadsto \frac{\frac{1 + \color{blue}{0}}{{\left(1 + x\right)}^{-0.5} + {x}^{-0.5}}}{x + {x}^{2}} \]

      3. metadata-eval 99.3%

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

      4. /-rgt-identity 99.3%

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

      5. metadata-eval 99.3%

         \[\leadsto \frac{\frac{1}{{\left(1 + x\right)}^{-0.5} + {x}^{-0.5}}}{\frac{x + {x}^{2}}{\color{blue}{1 + 0}}} \]

      6. +-inverses 99.3%

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

      7. associate-/l* 99.3%

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

      8. +-inverses 99.3%

         \[\leadsto \frac{\frac{1}{{\left(1 + x\right)}^{-0.5} + {x}^{-0.5}} \cdot \left(1 + \color{blue}{0}\right)}{x + {x}^{2}} \]

      9. metadata-eval 99.3%

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

      10. *-rgt-identity 99.3%

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

      11. +-commutative 99.3%

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

      12. +-commutative 99.3%

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

      13. unpow2 99.3%

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

      14. fma-def 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}{\mathsf{fma}\left(x, x, x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{\frac{-1}{x}}{-1 - x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 0.0)
   (* 0.5 (pow x -1.5))
   (/
    1.0
    (/ (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)) (/ (/ -1.0 x) (- -1.0 x))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = 1.0 / ((pow(x, -0.5) + pow((1.0 + x), -0.5)) / ((-1.0 / 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)))) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = 1.0d0 / (((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0))) / (((-1.0d0) / 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)))) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = 1.0 / ((Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5)) / ((-1.0 / 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)))) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = 1.0 / ((math.pow(x, -0.5) + math.pow((1.0 + x), -0.5)) / ((-1.0 / 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)))) <= 0.0)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64(1.0 / Float64(Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5)) / Float64(Float64(-1.0 / 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)))) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = 1.0 / (((x ^ -0.5) + ((1.0 + x) ^ -0.5)) / ((-1.0 / 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], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 / x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 0.0

   1. Initial program 38.1%

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

      1. flip-- 38.1%

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

      2. clear-num 38.1%

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

      3. pow1/2 38.1%

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

      4. pow-flip 38.1%

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

      5. metadata-eval 38.1%

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

      6. inv-pow 38.1%

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

      7. sqrt-pow2 38.1%

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

      8. +-commutative 38.1%

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

      9. metadata-eval 38.1%

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

      10. frac-times 20.3%

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

      11. metadata-eval 20.3%

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

      12. add-sqr-sqrt 16.8%

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

      13. frac-times 20.3%

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

      14. metadata-eval 20.3%

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

      15. add-sqr-sqrt 38.2%

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

   4. Applied egg-rr 38.2%

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

   5. Taylor expanded in x around inf 66.7%

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

      1. expm1-log1p-u 66.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 \cdot \sqrt{{x}^{3}}}\right)\right)} \]

      2. expm1-udef 38.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{2 \cdot \sqrt{{x}^{3}}}\right)} - 1} \]

      3. associate-/r* 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}}\right)} - 1 \]

      4. metadata-eval 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.5}}{\sqrt{{x}^{3}}}\right)} - 1 \]

      5. sqrt-pow1 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{\color{blue}{{x}^{\left(\frac{3}{2}\right)}}}\right)} - 1 \]

      6. metadata-eval 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{{x}^{\color{blue}{1.5}}}\right)} - 1 \]

   7. Applied egg-rr 38.1%

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

      1. expm1-def 97.8%

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

      2. expm1-log1p 97.8%

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

   9. Simplified 97.8%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 97.6%

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

       2. sqrt-unprod 66.9%

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

       3. frac-times 66.7%

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

       4. metadata-eval 66.7%

          \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{{x}^{1.5} \cdot {x}^{1.5}}} \]

       5. pow-prod-up 66.8%

          \[\leadsto \sqrt{\frac{0.25}{\color{blue}{{x}^{\left(1.5 + 1.5\right)}}}} \]

       6. metadata-eval 66.8%

          \[\leadsto \sqrt{\frac{0.25}{{x}^{\color{blue}{3}}}} \]

       7. expm1-log1p-u 66.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{0.25}{{x}^{3}}}\right)\right)} \]

       8. expm1-udef 38.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{0.25}{{x}^{3}}}\right)} - 1} \]

   11. Applied egg-rr 38.1%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

   13. Simplified 100.0%

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

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 60.9%

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

      1. flip-- 59.6%

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

      2. clear-num 59.6%

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

      3. pow1/2 59.6%

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

      4. pow-flip 59.6%

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

      5. metadata-eval 59.6%

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

      6. inv-pow 59.6%

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

      7. sqrt-pow2 59.6%

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

      8. +-commutative 59.6%

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

      9. metadata-eval 59.6%

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

      10. frac-times 59.3%

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

      11. metadata-eval 59.3%

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

      12. add-sqr-sqrt 58.9%

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

      13. frac-times 60.2%

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

      14. metadata-eval 60.2%

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

      15. add-sqr-sqrt 60.9%

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

   4. Applied egg-rr 60.9%

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

      1. frac-2neg 60.9%

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

      2. metadata-eval 60.9%

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

      3. frac-sub 99.4%

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

      4. *-un-lft-identity 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. associate-/r* 99.3%

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

      2. sub-neg 99.3%

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

      3. *-commutative 99.3%

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

      4. neg-mul-1 99.3%

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

      5. distribute-neg-out 99.3%

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

      6. remove-double-neg 99.3%

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

      7. mul-1-neg 99.3%

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

      8. unsub-neg 99.3%

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

      9. mul-1-neg 99.3%

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

      10. remove-double-neg 99.3%

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

      11. associate-+r- 99.3%

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

      12. +-inverses 99.3%

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

      13. metadata-eval 99.3%

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

      14. metadata-eval 99.3%

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

      15. distribute-neg-in 99.3%

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

      16. metadata-eval 99.3%

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

      17. unsub-neg 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{\frac{-1}{x}}{-1 - x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{-1}{x \cdot \left(-1 - x\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 0.0)
   (* 0.5 (pow x -1.5))
   (/
    1.0
    (/ (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)) (/ -1.0 (* x (- -1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = 1.0 / ((pow(x, -0.5) + pow((1.0 + x), -0.5)) / (-1.0 / (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)))) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = 1.0d0 / (((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0))) / ((-1.0d0) / (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)))) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = 1.0 / ((Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5)) / (-1.0 / (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)))) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = 1.0 / ((math.pow(x, -0.5) + math.pow((1.0 + x), -0.5)) / (-1.0 / (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)))) <= 0.0)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64(1.0 / Float64(Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5)) / Float64(-1.0 / 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)))) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = 1.0 / (((x ^ -0.5) + ((1.0 + x) ^ -0.5)) / (-1.0 / (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], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[(-1.0 / 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)))) < 0.0

   1. Initial program 38.1%

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

      1. flip-- 38.1%

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

      2. clear-num 38.1%

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

      3. pow1/2 38.1%

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

      4. pow-flip 38.1%

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

      5. metadata-eval 38.1%

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

      6. inv-pow 38.1%

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

      7. sqrt-pow2 38.1%

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

      8. +-commutative 38.1%

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

      9. metadata-eval 38.1%

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

      10. frac-times 20.3%

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

      11. metadata-eval 20.3%

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

      12. add-sqr-sqrt 16.8%

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

      13. frac-times 20.3%

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

      14. metadata-eval 20.3%

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

      15. add-sqr-sqrt 38.2%

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

   4. Applied egg-rr 38.2%

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

   5. Taylor expanded in x around inf 66.7%

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

      1. expm1-log1p-u 66.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 \cdot \sqrt{{x}^{3}}}\right)\right)} \]

      2. expm1-udef 38.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{2 \cdot \sqrt{{x}^{3}}}\right)} - 1} \]

      3. associate-/r* 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}}\right)} - 1 \]

      4. metadata-eval 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.5}}{\sqrt{{x}^{3}}}\right)} - 1 \]

      5. sqrt-pow1 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{\color{blue}{{x}^{\left(\frac{3}{2}\right)}}}\right)} - 1 \]

      6. metadata-eval 38.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{{x}^{\color{blue}{1.5}}}\right)} - 1 \]

   7. Applied egg-rr 38.1%

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

      1. expm1-def 97.8%

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

      2. expm1-log1p 97.8%

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

   9. Simplified 97.8%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 97.6%

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

       2. sqrt-unprod 66.9%

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

       3. frac-times 66.7%

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

       4. metadata-eval 66.7%

          \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{{x}^{1.5} \cdot {x}^{1.5}}} \]

       5. pow-prod-up 66.8%

          \[\leadsto \sqrt{\frac{0.25}{\color{blue}{{x}^{\left(1.5 + 1.5\right)}}}} \]

       6. metadata-eval 66.8%

          \[\leadsto \sqrt{\frac{0.25}{{x}^{\color{blue}{3}}}} \]

       7. expm1-log1p-u 66.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{0.25}{{x}^{3}}}\right)\right)} \]

       8. expm1-udef 38.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{0.25}{{x}^{3}}}\right)} - 1} \]

   11. Applied egg-rr 38.1%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

   13. Simplified 100.0%

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

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 60.9%

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

      1. flip-- 59.6%

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

      2. clear-num 59.6%

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

      3. pow1/2 59.6%

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

      4. pow-flip 59.6%

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

      5. metadata-eval 59.6%

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

      6. inv-pow 59.6%

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

      7. sqrt-pow2 59.6%

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

      8. +-commutative 59.6%

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

      9. metadata-eval 59.6%

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

      10. frac-times 59.3%

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

      11. metadata-eval 59.3%

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

      12. add-sqr-sqrt 58.9%

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

      13. frac-times 60.2%

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

      14. metadata-eval 60.2%

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

      15. add-sqr-sqrt 60.9%

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

   4. Applied egg-rr 60.9%

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

      1. frac-2neg 60.9%

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

      2. metadata-eval 60.9%

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

      3. frac-sub 99.4%

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

      4. *-un-lft-identity 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. neg-mul-1 99.4%

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

      3. *-rgt-identity 99.4%

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

      4. distribute-rgt-neg-out 99.4%

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

      5. distribute-lft-neg-out 99.4%

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

      6. distribute-neg-in 99.4%

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

      7. metadata-eval 99.4%

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

      8. *-rgt-identity 99.4%

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

      9. *-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in x around 0 99.4%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{-1}{x \cdot \left(-1 - x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-1.5} \end{array} \]

FPCore

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

C

double code(double x) {
	return 0.5 * pow(x, -1.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x ** (-1.5d0))
end function

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, -1.5);
}

Python

def code(x):
	return 0.5 * math.pow(x, -1.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -1.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ -1.5);
end

Wolfram

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

Derivation

1. Initial program 39.2%

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

   1. flip-- 39.1%

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

   2. clear-num 39.1%

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

   3. pow1/2 39.1%

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

   4. pow-flip 39.1%

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

   5. metadata-eval 39.1%

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

   6. inv-pow 39.1%

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

   7. sqrt-pow2 39.1%

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

   8. +-commutative 39.1%

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

   9. metadata-eval 39.1%

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

   10. frac-times 22.1%

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

   11. metadata-eval 22.1%

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

   12. add-sqr-sqrt 18.7%

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

   13. frac-times 22.1%

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

   14. metadata-eval 22.1%

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

   15. add-sqr-sqrt 39.3%

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

4. Applied egg-rr 39.3%

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

5. Taylor expanded in x around inf 66.4%

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

   1. expm1-log1p-u 66.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 \cdot \sqrt{{x}^{3}}}\right)\right)} \]

   2. expm1-udef 37.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{2 \cdot \sqrt{{x}^{3}}}\right)} - 1} \]

   3. associate-/r* 37.8%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}}\right)} - 1 \]

   4. metadata-eval 37.8%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.5}}{\sqrt{{x}^{3}}}\right)} - 1 \]

   5. sqrt-pow1 37.8%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{\color{blue}{{x}^{\left(\frac{3}{2}\right)}}}\right)} - 1 \]

   6. metadata-eval 37.8%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{{x}^{\color{blue}{1.5}}}\right)} - 1 \]

7. Applied egg-rr 37.8%

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

   1. expm1-def 96.0%

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

   2. expm1-log1p 96.0%

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

9. Simplified 96.0%

   \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 95.8%

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

    2. sqrt-unprod 66.6%

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

    3. frac-times 66.4%

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

    4. metadata-eval 66.4%

       \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{{x}^{1.5} \cdot {x}^{1.5}}} \]

    5. pow-prod-up 66.5%

       \[\leadsto \sqrt{\frac{0.25}{\color{blue}{{x}^{\left(1.5 + 1.5\right)}}}} \]

    6. metadata-eval 66.5%

       \[\leadsto \sqrt{\frac{0.25}{{x}^{\color{blue}{3}}}} \]

    7. expm1-log1p-u 66.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{0.25}{{x}^{3}}}\right)\right)} \]

    8. expm1-udef 37.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{0.25}{{x}^{3}}}\right)} - 1} \]

11. Applied egg-rr 37.8%

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

    1. expm1-def 98.1%

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

    2. expm1-log1p 98.1%

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

13. Simplified 98.1%

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

14. Final simplification 98.1%

    \[\leadsto 0.5 \cdot {x}^{-1.5} \]
15. Add Preprocessing

Developer target: 98.4% 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 2024040 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))