2isqrt (example 3.6)

Percentage Accurate: 39.2% → 99.9%

Time: 9.4s

Alternatives: 7

Speedup: 1.9×

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: 39.2% 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

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ 1.0 x)))) 0.0)
   (* (pow x -1.5) (- (/ -0.5 x) -0.5))
   (/ (/ 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 = pow(x, -1.5) * ((-0.5 / x) - -0.5);
	} 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((x ^ -1.5) * Float64(Float64(-0.5 / x) - -0.5));
	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[(N[Power[x, -1.5], $MachinePrecision] * N[(N[(-0.5 / x), $MachinePrecision] - -0.5), $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 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.0

   1. Initial program 38.1%

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

   3. Taylor expanded in x around inf 85.0%

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

      1. distribute-lft-out-- 85.0%

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

   5. Simplified 85.0%

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

      1. *-commutative 85.0%

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

      2. unpow2 85.0%

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

      3. times-frac 99.5%

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

      4. inv-pow 99.5%

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

      5. sqrt-pow1 99.5%

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

      6. metadata-eval 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. clear-num 99.5%

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

      2. clear-num 99.5%

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

      3. frac-times 98.3%

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

      4. metadata-eval 98.3%

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

      5. clear-num 98.2%

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

      6. div-sub 98.2%

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

      7. pow1 98.2%

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

      8. pow-div 98.2%

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

      9. metadata-eval 98.2%

         \[\leadsto \frac{1}{\frac{1}{{x}^{\color{blue}{-1.5}} - \frac{\sqrt{x}}{x}} \cdot \frac{x}{-0.5}} \]

      10. pow1/2 98.2%

          \[\leadsto \frac{1}{\frac{1}{{x}^{-1.5} - \frac{\color{blue}{{x}^{0.5}}}{x}} \cdot \frac{x}{-0.5}} \]

      11. pow1 98.2%

          \[\leadsto \frac{1}{\frac{1}{{x}^{-1.5} - \frac{{x}^{0.5}}{\color{blue}{{x}^{1}}}} \cdot \frac{x}{-0.5}} \]

      12. pow-div 98.3%

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

      13. metadata-eval 98.3%

          \[\leadsto \frac{1}{\frac{1}{{x}^{-1.5} - {x}^{\color{blue}{-0.5}}} \cdot \frac{x}{-0.5}} \]

      14. div-inv 98.3%

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

      15. metadata-eval 98.3%

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

   9. Applied egg-rr 98.3%

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

       1. associate-/r* 99.6%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{1}{{x}^{-1.5} - {x}^{-0.5}}}}{x \cdot -2}} \]

       2. remove-double-div 99.7%

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

   11. Simplified 99.7%

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

       1. div-sub 99.7%

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

       2. sub-neg 99.7%

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

       3. div-inv 99.7%

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

       4. *-commutative 99.7%

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

       5. associate-/r* 99.7%

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

       6. metadata-eval 99.7%

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

       7. *-un-lft-identity 99.7%

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

       8. *-commutative 99.7%

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

       9. times-frac 99.7%

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

       10. metadata-eval 99.7%

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

       11. pow1 99.7%

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

       12. pow-div 100.0%

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

       13. metadata-eval 100.0%

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

   13. Applied egg-rr 100.0%

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

       1. sub-neg 100.0%

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

       2. *-commutative 100.0%

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

       3. distribute-lft-out-- 100.0%

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

   15. Simplified 100.0%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 57.0%

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

      1. sub-neg 57.0%

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

      2. inv-pow 57.0%

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

      3. sqrt-pow2 56.9%

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

      4. metadata-eval 56.9%

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

      5. distribute-neg-frac 56.9%

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

      6. metadata-eval 56.9%

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

      7. +-commutative 56.9%

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

   4. Applied egg-rr 56.9%

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

      1. *-rgt-identity 56.9%

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

      2. cancel-sign-sub 56.9%

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

      3. distribute-lft-neg-in 56.9%

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

      4. *-rgt-identity 56.9%

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

      5. distribute-neg-frac 56.9%

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

      6. metadata-eval 56.9%

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

      7. unpow1/2 56.9%

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

      8. exp-to-pow 52.8%

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

      9. log1p-undefine 52.8%

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

      10. *-commutative 52.8%

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

      11. exp-neg 53.5%

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

      12. *-commutative 53.5%

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

      13. distribute-rgt-neg-in 53.5%

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

      14. log1p-undefine 53.5%

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

      15. metadata-eval 53.5%

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

      16. exp-to-pow 57.0%

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

   6. Simplified 57.0%

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

      1. flip-- 56.9%

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

      2. pow-prod-up 57.4%

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

      3. metadata-eval 57.4%

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

      4. inv-pow 57.4%

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

      5. pow-prod-up 58.7%

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

      6. metadata-eval 58.7%

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

      7. inv-pow 58.7%

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

      8. un-div-inv 58.7%

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

      9. *-commutative 58.7%

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

      10. frac-sub 99.0%

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

      11. frac-times 99.3%

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

      12. *-un-lft-identity 99.3%

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

      13. *-un-lft-identity 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. *-rgt-identity 99.3%

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

      2. associate--l+ 99.3%

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

      3. distribute-lft-in 99.2%

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

      4. *-rgt-identity 99.2%

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

      5. unpow2 99.2%

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

   10. Simplified 99.2%

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

       1. *-un-lft-identity 99.2%

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

       2. associate-/r* 99.1%

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

       3. +-inverses 99.1%

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

       4. metadata-eval 99.1%

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

       5. +-commutative 99.1%

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

       6. unpow2 99.1%

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

       7. fma-define 99.4%

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

   12. Applied egg-rr 99.4%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ 1.0 x)))) 0.0)
   (* (pow x -1.5) (- (/ -0.5 x) -0.5))
   (/ -1.0 (* (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)) (* 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 = pow(x, -1.5) * ((-0.5 / x) - -0.5);
	} else {
		tmp = -1.0 / ((pow(x, -0.5) + pow((1.0 + x), -0.5)) * (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 = (x ** (-1.5d0)) * (((-0.5d0) / x) - (-0.5d0))
    else
        tmp = (-1.0d0) / (((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0))) * (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 = Math.pow(x, -1.5) * ((-0.5 / x) - -0.5);
	} else {
		tmp = -1.0 / ((Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5)) * (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 = math.pow(x, -1.5) * ((-0.5 / x) - -0.5)
	else:
		tmp = -1.0 / ((math.pow(x, -0.5) + math.pow((1.0 + x), -0.5)) * (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((x ^ -1.5) * Float64(Float64(-0.5 / x) - -0.5));
	else
		tmp = Float64(-1.0 / Float64(Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5)) * 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 = (x ^ -1.5) * ((-0.5 / x) - -0.5);
	else
		tmp = -1.0 / (((x ^ -0.5) + ((1.0 + x) ^ -0.5)) * (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[(N[Power[x, -1.5], $MachinePrecision] * N[(N[(-0.5 / x), $MachinePrecision] - -0.5), $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[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.0

   1. Initial program 38.1%

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

   3. Taylor expanded in x around inf 85.0%

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

      1. distribute-lft-out-- 85.0%

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

   5. Simplified 85.0%

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

      1. *-commutative 85.0%

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

      2. unpow2 85.0%

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

      3. times-frac 99.5%

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

      4. inv-pow 99.5%

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

      5. sqrt-pow1 99.5%

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

      6. metadata-eval 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. clear-num 99.5%

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

      2. clear-num 99.5%

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

      3. frac-times 98.3%

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

      4. metadata-eval 98.3%

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

      5. clear-num 98.2%

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

      6. div-sub 98.2%

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

      7. pow1 98.2%

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

      8. pow-div 98.2%

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

      9. metadata-eval 98.2%

         \[\leadsto \frac{1}{\frac{1}{{x}^{\color{blue}{-1.5}} - \frac{\sqrt{x}}{x}} \cdot \frac{x}{-0.5}} \]

      10. pow1/2 98.2%

          \[\leadsto \frac{1}{\frac{1}{{x}^{-1.5} - \frac{\color{blue}{{x}^{0.5}}}{x}} \cdot \frac{x}{-0.5}} \]

      11. pow1 98.2%

          \[\leadsto \frac{1}{\frac{1}{{x}^{-1.5} - \frac{{x}^{0.5}}{\color{blue}{{x}^{1}}}} \cdot \frac{x}{-0.5}} \]

      12. pow-div 98.3%

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

      13. metadata-eval 98.3%

          \[\leadsto \frac{1}{\frac{1}{{x}^{-1.5} - {x}^{\color{blue}{-0.5}}} \cdot \frac{x}{-0.5}} \]

      14. div-inv 98.3%

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

      15. metadata-eval 98.3%

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

   9. Applied egg-rr 98.3%

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

       1. associate-/r* 99.6%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{1}{{x}^{-1.5} - {x}^{-0.5}}}}{x \cdot -2}} \]

       2. remove-double-div 99.7%

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

   11. Simplified 99.7%

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

       1. div-sub 99.7%

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

       2. sub-neg 99.7%

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

       3. div-inv 99.7%

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

       4. *-commutative 99.7%

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

       5. associate-/r* 99.7%

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

       6. metadata-eval 99.7%

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

       7. *-un-lft-identity 99.7%

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

       8. *-commutative 99.7%

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

       9. times-frac 99.7%

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

       10. metadata-eval 99.7%

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

       11. pow1 99.7%

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

       12. pow-div 100.0%

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

       13. metadata-eval 100.0%

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

   13. Applied egg-rr 100.0%

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

       1. sub-neg 100.0%

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

       2. *-commutative 100.0%

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

       3. distribute-lft-out-- 100.0%

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

   15. Simplified 100.0%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 57.0%

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

      1. sub-neg 57.0%

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

      2. inv-pow 57.0%

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

      3. sqrt-pow2 56.9%

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

      4. metadata-eval 56.9%

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

      5. distribute-neg-frac 56.9%

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

      6. metadata-eval 56.9%

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

      7. +-commutative 56.9%

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

   4. Applied egg-rr 56.9%

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

      1. *-rgt-identity 56.9%

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

      2. cancel-sign-sub 56.9%

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

      3. distribute-lft-neg-in 56.9%

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

      4. *-rgt-identity 56.9%

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

      5. distribute-neg-frac 56.9%

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

      6. metadata-eval 56.9%

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

      7. unpow1/2 56.9%

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

      8. exp-to-pow 52.8%

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

      9. log1p-undefine 52.8%

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

      10. *-commutative 52.8%

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

      11. exp-neg 53.5%

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

      12. *-commutative 53.5%

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

      13. distribute-rgt-neg-in 53.5%

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

      14. log1p-undefine 53.5%

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

      15. metadata-eval 53.5%

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

      16. exp-to-pow 57.0%

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

   6. Simplified 57.0%

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

      1. flip-- 56.9%

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

      2. pow-prod-up 57.4%

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

      3. metadata-eval 57.4%

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

      4. inv-pow 57.4%

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

      5. pow-prod-up 58.7%

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

      6. metadata-eval 58.7%

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

      7. inv-pow 58.7%

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

      8. un-div-inv 58.7%

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

      9. *-commutative 58.7%

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

      10. frac-sub 99.0%

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

      11. frac-times 99.3%

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

      12. *-un-lft-identity 99.3%

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

      13. *-un-lft-identity 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. associate-/r* 99.4%

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

      2. *-lft-identity 99.4%

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

      3. *-lft-identity 99.4%

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

      4. times-frac 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

      7. times-frac 99.4%

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

      8. neg-mul-1 99.4%

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

      9. neg-mul-1 99.4%

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

      10. associate-/r* 99.3%

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

   10. Simplified 99.3%

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

4. Final simplification 99.9%

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

Alternative 3: 37.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \mathbf{if}\;x \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x))))
   (if (<= x 6.5e+153) (* t_0 0.5) (* (/ -0.5 x) t_0))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / x));
	double tmp;
	if (x <= 6.5e+153) {
		tmp = t_0 * 0.5;
	} else {
		tmp = (-0.5 / x) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / x))
    if (x <= 6.5d+153) then
        tmp = t_0 * 0.5d0
    else
        tmp = ((-0.5d0) / x) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / x));
	double tmp;
	if (x <= 6.5e+153) {
		tmp = t_0 * 0.5;
	} else {
		tmp = (-0.5 / x) * t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / x))
	tmp = 0
	if x <= 6.5e+153:
		tmp = t_0 * 0.5
	else:
		tmp = (-0.5 / x) * t_0
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / x))
	tmp = 0.0
	if (x <= 6.5e+153)
		tmp = Float64(t_0 * 0.5);
	else
		tmp = Float64(Float64(-0.5 / x) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / x));
	tmp = 0.0;
	if (x <= 6.5e+153)
		tmp = t_0 * 0.5;
	else
		tmp = (-0.5 / x) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 6.5e+153], N[(t$95$0 * 0.5), $MachinePrecision], N[(N[(-0.5 / x), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.49999999999999972e153

   1. Initial program 9.6%

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

      1. flip-- 9.6%

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

      2. div-inv 9.6%

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

      3. frac-times 9.8%

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

      4. metadata-eval 9.8%

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

      5. add-sqr-sqrt 9.7%

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

      6. frac-times 9.7%

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

      7. metadata-eval 9.7%

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

      8. add-sqr-sqrt 9.7%

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

      9. +-commutative 9.7%

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

      10. inv-pow 9.7%

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

      11. sqrt-pow2 9.7%

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

      12. metadata-eval 9.7%

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

      13. pow1/2 9.7%

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

      14. pow-flip 9.7%

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

      15. +-commutative 9.7%

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

      16. metadata-eval 9.7%

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

   4. Applied egg-rr 9.7%

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

   5. Taylor expanded in x around inf 8.3%

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

      1. *-commutative 8.3%

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

   7. Simplified 8.3%

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

   8. Taylor expanded in x around 0 7.0%

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

   if 6.49999999999999972e153 < x

   1. Initial program 70.9%

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

   3. Taylor expanded in x around inf 70.9%

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

      1. distribute-lft-out-- 70.9%

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

   5. Simplified 70.9%

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

      1. *-commutative 70.9%

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

      2. unpow2 70.9%

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

      3. times-frac 99.8%

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

      4. inv-pow 99.8%

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

      5. sqrt-pow1 99.8%

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

      6. metadata-eval 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. /-rgt-identity 99.8%

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

      2. clear-num 99.7%

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

      3. inv-pow 99.7%

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

      4. sqrt-pow2 99.7%

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

      5. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in x around -inf 70.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{-0.5}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x} \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {x}^{-1.5} \cdot \left(\frac{-0.5}{x} - -0.5\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in x around inf 84.0%

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

   1. distribute-lft-out-- 84.0%

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

5. Simplified 84.0%

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

   1. *-commutative 84.0%

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

   2. unpow2 84.0%

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

   3. times-frac 97.7%

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

   4. inv-pow 97.7%

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

   5. sqrt-pow1 97.7%

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

   6. metadata-eval 97.7%

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

7. Applied egg-rr 97.7%

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

   1. clear-num 97.7%

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

   2. clear-num 97.7%

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

   3. frac-times 96.6%

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

   4. metadata-eval 96.6%

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

   5. clear-num 96.5%

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

   6. div-sub 96.5%

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

   7. pow1 96.5%

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

   8. pow-div 96.5%

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

   9. metadata-eval 96.5%

      \[\leadsto \frac{1}{\frac{1}{{x}^{\color{blue}{-1.5}} - \frac{\sqrt{x}}{x}} \cdot \frac{x}{-0.5}} \]

   10. pow1/2 96.5%

       \[\leadsto \frac{1}{\frac{1}{{x}^{-1.5} - \frac{\color{blue}{{x}^{0.5}}}{x}} \cdot \frac{x}{-0.5}} \]

   11. pow1 96.5%

       \[\leadsto \frac{1}{\frac{1}{{x}^{-1.5} - \frac{{x}^{0.5}}{\color{blue}{{x}^{1}}}} \cdot \frac{x}{-0.5}} \]

   12. pow-div 96.6%

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

   13. metadata-eval 96.6%

       \[\leadsto \frac{1}{\frac{1}{{x}^{-1.5} - {x}^{\color{blue}{-0.5}}} \cdot \frac{x}{-0.5}} \]

   14. div-inv 96.6%

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

   15. metadata-eval 96.6%

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

9. Applied egg-rr 96.6%

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

    1. associate-/r* 97.8%

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{1}{{x}^{-1.5} - {x}^{-0.5}}}}{x \cdot -2}} \]

    2. remove-double-div 98.0%

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

11. Simplified 98.0%

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

    1. div-sub 98.0%

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

    2. sub-neg 98.0%

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

    3. div-inv 98.0%

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

    4. *-commutative 98.0%

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

    5. associate-/r* 98.0%

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

    6. metadata-eval 98.0%

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

    7. *-un-lft-identity 98.0%

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

    8. *-commutative 98.0%

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

    9. times-frac 97.9%

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

    10. metadata-eval 97.9%

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

    11. pow1 97.9%

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

    12. pow-div 98.2%

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

    13. metadata-eval 98.2%

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

13. Applied egg-rr 98.2%

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

    1. sub-neg 98.2%

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

    2. *-commutative 98.2%

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

    3. distribute-lft-out-- 98.2%

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

15. Simplified 98.2%

    \[\leadsto \color{blue}{{x}^{-1.5} \cdot \left(\frac{-0.5}{x} - -0.5\right)} \]
16. Add Preprocessing

Alternative 5: 97.5% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (/ -0.5 x) (- (sqrt (/ 1.0 x)))))

C

double code(double x) {
	return (-0.5 / x) * -sqrt((1.0 / x));
}

Fortran

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

Java

public static double code(double x) {
	return (-0.5 / x) * -Math.sqrt((1.0 / x));
}

Python

def code(x):
	return (-0.5 / x) * -math.sqrt((1.0 / x))

Julia

function code(x)
	return Float64(Float64(-0.5 / x) * Float64(-sqrt(Float64(1.0 / x))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in x around inf 84.0%

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

   1. distribute-lft-out-- 84.0%

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

5. Simplified 84.0%

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

   1. *-commutative 84.0%

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

   2. unpow2 84.0%

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

   3. times-frac 97.7%

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

   4. inv-pow 97.7%

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

   5. sqrt-pow1 97.7%

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

   6. metadata-eval 97.7%

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

7. Applied egg-rr 97.7%

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

   1. div-sub 97.7%

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

   2. pow1 97.7%

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

   3. pow-div 97.7%

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

   4. metadata-eval 97.7%

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

   5. metadata-eval 97.7%

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

   6. metadata-eval 97.7%

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

   7. sqrt-pow1 97.7%

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

   8. pow-flip 97.7%

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

   9. pow1/2 97.7%

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

   10. pow1 97.7%

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

   11. pow-div 97.8%

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

   12. metadata-eval 97.8%

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

   13. sub-neg 97.8%

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

   14. pow-flip 97.8%

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

   15. sqrt-pow1 97.8%

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

   16. metadata-eval 97.8%

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

   17. metadata-eval 97.8%

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

9. Applied egg-rr 97.8%

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

    1. sub-neg 97.8%

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

11. Simplified 97.8%

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

12. Taylor expanded in x around inf 97.7%

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

    1. mul-1-neg 97.7%

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

14. Simplified 97.7%

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

15. Final simplification 97.7%

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

Alternative 6: 5.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{x}} \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt (/ 1.0 x)) 0.5))

C

double code(double x) {
	return sqrt((1.0 / x)) * 0.5;
}

Fortran

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

Java

public static double code(double x) {
	return Math.sqrt((1.0 / x)) * 0.5;
}

Python

def code(x):
	return math.sqrt((1.0 / x)) * 0.5

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / x)) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / x)) * 0.5;
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 39.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. div-inv 39.1%

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

   3. frac-times 23.4%

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

   4. metadata-eval 23.4%

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

   5. add-sqr-sqrt 19.0%

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

   6. frac-times 24.6%

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

   7. metadata-eval 24.6%

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

   8. add-sqr-sqrt 39.1%

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

   9. +-commutative 39.1%

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

   10. inv-pow 39.1%

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

   11. sqrt-pow2 39.1%

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

   12. metadata-eval 39.1%

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

   13. pow1/2 39.1%

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

   14. pow-flip 39.1%

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

   15. +-commutative 39.1%

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

   16. metadata-eval 39.1%

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

4. Applied egg-rr 39.1%

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

5. Taylor expanded in x around inf 38.4%

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

   1. *-commutative 38.4%

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

7. Simplified 38.4%

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

8. Taylor expanded in x around 0 5.7%

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

9. Final simplification 5.7%

   \[\leadsto \sqrt{\frac{1}{x}} \cdot 0.5 \]
10. Add Preprocessing

Alternative 7: 5.6% 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 39.0%

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

   1. sub-neg 39.0%

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

   2. inv-pow 39.0%

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

   3. sqrt-pow2 28.4%

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

   4. metadata-eval 28.4%

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

   5. distribute-neg-frac 28.4%

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

   6. metadata-eval 28.4%

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

   7. +-commutative 28.4%

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

4. Applied egg-rr 28.4%

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

   1. *-rgt-identity 28.4%

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

   2. cancel-sign-sub 28.4%

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

   3. distribute-lft-neg-in 28.4%

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

   4. *-rgt-identity 28.4%

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

   5. distribute-neg-frac 28.4%

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

   6. metadata-eval 28.4%

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

   7. unpow1/2 28.4%

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

   8. exp-to-pow 7.0%

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

   9. log1p-undefine 7.0%

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

   10. *-commutative 7.0%

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

   11. exp-neg 7.1%

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

   12. *-commutative 7.1%

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

   13. distribute-rgt-neg-in 7.1%

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

   14. log1p-undefine 7.1%

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

   15. metadata-eval 7.1%

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

   16. exp-to-pow 39.0%

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

6. Simplified 39.0%

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

7. Taylor expanded in x around 0 5.7%

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

   1. unpow-1 5.7%

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

   2. metadata-eval 5.7%

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

   3. pow-sqr 5.7%

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

   4. rem-sqrt-square 5.7%

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

   5. metadata-eval 5.7%

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

   6. pow-sqr 5.7%

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

   7. fabs-sqr 5.7%

      \[\leadsto \color{blue}{{x}^{-0.25} \cdot {x}^{-0.25}} \]

   8. pow-sqr 5.7%

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

   9. metadata-eval 5.7%

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

9. Simplified 5.7%

   \[\leadsto \color{blue}{{x}^{-0.5}} \]
10. Add Preprocessing

Developer Target 1: 98.2% 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 2024139 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

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