2isqrt (example 3.6)

Percentage Accurate: 69.4% → 99.7%

Time: 11.3s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5500:\\ \;\;\;\;\left(1 - \frac{\sqrt{x}}{\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x + \left(0.5 - \frac{0.125}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5500.0)
   (* (- 1.0 (/ (sqrt x) (hypot 1.0 (sqrt x)))) (pow x -0.5))
   (/ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (+ x (- 0.5 (/ 0.125 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 5500.0) {
		tmp = (1.0 - (sqrt(x) / hypot(1.0, sqrt(x)))) * pow(x, -0.5);
	} else {
		tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) / (x + (0.5 - (0.125 / x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5500.0) {
		tmp = (1.0 - (Math.sqrt(x) / Math.hypot(1.0, Math.sqrt(x)))) * Math.pow(x, -0.5);
	} else {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) / (x + (0.5 - (0.125 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5500.0:
		tmp = (1.0 - (math.sqrt(x) / math.hypot(1.0, math.sqrt(x)))) * math.pow(x, -0.5)
	else:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) / (x + (0.5 - (0.125 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5500.0)
		tmp = Float64(Float64(1.0 - Float64(sqrt(x) / hypot(1.0, sqrt(x)))) * (x ^ -0.5));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) / Float64(x + Float64(0.5 - Float64(0.125 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5500.0)
		tmp = (1.0 - (sqrt(x) / hypot(1.0, sqrt(x)))) * (x ^ -0.5);
	else
		tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) / (x + (0.5 - (0.125 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5500.0], N[(N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + N[(0.5 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5500

   1. Initial program 99.4%

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

      1. frac-sub 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. +-commutative 99.4%

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

      5. *-rgt-identity 99.4%

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

      6. metadata-eval 99.4%

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

      7. frac-times 99.4%

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

      8. un-div-inv 99.4%

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

      9. pow1/2 99.4%

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

      10. pow-flip 99.8%

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

      11. metadata-eval 99.8%

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

      12. +-commutative 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. times-frac 99.8%

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

      4. div-sub 99.8%

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

      5. *-inverses 99.8%

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

      6. unpow1 99.8%

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

      7. sqr-pow 99.8%

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

      8. metadata-eval 99.8%

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

      9. exp-to-pow 99.8%

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

      10. metadata-eval 99.8%

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

      11. exp-to-pow 99.8%

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

      12. hypot-1-def 99.8%

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

      13. exp-to-pow 99.8%

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

      14. unpow1/2 99.8%

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

      15. /-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

   if 5500 < x

   1. Initial program 40.1%

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

      1. frac-sub 40.1%

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

      2. clear-num 40.1%

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

      3. sqrt-unprod 40.1%

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

      4. +-commutative 40.1%

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

      5. *-un-lft-identity 40.1%

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

      6. *-rgt-identity 40.1%

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

      7. +-commutative 40.1%

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

   3. Applied egg-rr 40.1%

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

      1. associate-/r/ 40.1%

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

      2. associate-*l/ 40.1%

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

      3. *-lft-identity 40.1%

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

      4. distribute-rgt-in 40.1%

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

      5. *-lft-identity 40.1%

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

   5. Simplified 40.1%

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

      1. pow1/2 40.1%

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

      2. metadata-eval 40.1%

         \[\leadsto \frac{{\left(1 + x\right)}^{\color{blue}{\left(0.25 \cdot 2\right)}} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]

      3. pow-pow 40.1%

         \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]

      4. flip-- 40.0%

         \[\leadsto \frac{\color{blue}{\frac{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}}{\sqrt{x + x \cdot x}} \]

      5. pow-pow 40.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      6. metadata-eval 40.7%

         \[\leadsto \frac{\frac{{\left(1 + x\right)}^{\color{blue}{0.5}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      7. pow1/2 40.7%

         \[\leadsto \frac{\frac{\color{blue}{\sqrt{1 + x}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      8. pow-pow 40.8%

         \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot \color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      9. metadata-eval 40.8%

         \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot {\left(1 + x\right)}^{\color{blue}{0.5}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      10. pow1/2 40.8%

          \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot \color{blue}{\sqrt{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      11. add-sqr-sqrt 41.8%

          \[\leadsto \frac{\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      12. add-sqr-sqrt 43.2%

          \[\leadsto \frac{\frac{\left(1 + x\right) - \color{blue}{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      13. pow-pow 43.2%

          \[\leadsto \frac{\frac{\left(1 + x\right) - x}{\color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      14. metadata-eval 43.2%

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

      15. pow1/2 43.2%

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

   7. Applied egg-rr 43.2%

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

      1. associate--l+ 77.2%

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

      2. +-inverses 77.2%

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

      3. metadata-eval 77.2%

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

      4. +-commutative 77.2%

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

   9. Simplified 77.2%

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

   10. Taylor expanded in x around inf 99.7%

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

       1. +-commutative 99.7%

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

       2. associate--l+ 99.7%

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

       3. associate-*r/ 99.7%

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

       4. metadata-eval 99.7%

          \[\leadsto \frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x + \left(0.5 - \frac{\color{blue}{0.125}}{x}\right)} \]

   12. Simplified 99.7%

       \[\leadsto \frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\color{blue}{x + \left(0.5 - \frac{0.125}{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5500:\\ \;\;\;\;\left(1 - \frac{\sqrt{x}}{\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x + \left(0.5 - \frac{0.125}{x}\right)}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{x + \left(0.5 - \frac{0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 4e-11)
     (/ (/ 1.0 (+ (sqrt x) t_0)) (+ x (- 0.5 (/ 0.125 x))))
     (- (pow x -0.5) (pow (+ 1.0 x) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 4e-11) {
		tmp = (1.0 / (sqrt(x) + t_0)) / (x + (0.5 - (0.125 / x)));
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 4d-11) then
        tmp = (1.0d0 / (sqrt(x) + t_0)) / (x + (0.5d0 - (0.125d0 / x)))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 4e-11) {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) / (x + (0.5 - (0.125 / x)));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 4e-11:
		tmp = (1.0 / (math.sqrt(x) + t_0)) / (x + (0.5 - (0.125 / x)))
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 4e-11)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) / Float64(x + Float64(0.5 - Float64(0.125 / x))));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 4e-11)
		tmp = (1.0 / (sqrt(x) + t_0)) / (x + (0.5 - (0.125 / x)));
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 4e-11], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + N[(0.5 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 3.99999999999999976e-11

   1. Initial program 40.1%

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

      1. frac-sub 40.1%

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

      2. clear-num 40.1%

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

      3. sqrt-unprod 40.1%

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

      4. +-commutative 40.1%

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

      5. *-un-lft-identity 40.1%

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

      6. *-rgt-identity 40.1%

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

      7. +-commutative 40.1%

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

   3. Applied egg-rr 40.1%

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

      1. associate-/r/ 40.1%

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

      2. associate-*l/ 40.1%

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

      3. *-lft-identity 40.1%

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

      4. distribute-rgt-in 40.1%

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

      5. *-lft-identity 40.1%

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

   5. Simplified 40.1%

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

      1. pow1/2 40.1%

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

      2. metadata-eval 40.1%

         \[\leadsto \frac{{\left(1 + x\right)}^{\color{blue}{\left(0.25 \cdot 2\right)}} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]

      3. pow-pow 40.1%

         \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]

      4. flip-- 40.0%

         \[\leadsto \frac{\color{blue}{\frac{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}}{\sqrt{x + x \cdot x}} \]

      5. pow-pow 40.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      6. metadata-eval 40.7%

         \[\leadsto \frac{\frac{{\left(1 + x\right)}^{\color{blue}{0.5}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      7. pow1/2 40.7%

         \[\leadsto \frac{\frac{\color{blue}{\sqrt{1 + x}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      8. pow-pow 40.8%

         \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot \color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      9. metadata-eval 40.8%

         \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot {\left(1 + x\right)}^{\color{blue}{0.5}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      10. pow1/2 40.8%

          \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot \color{blue}{\sqrt{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      11. add-sqr-sqrt 41.8%

          \[\leadsto \frac{\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      12. add-sqr-sqrt 43.2%

          \[\leadsto \frac{\frac{\left(1 + x\right) - \color{blue}{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      13. pow-pow 43.2%

          \[\leadsto \frac{\frac{\left(1 + x\right) - x}{\color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      14. metadata-eval 43.2%

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

      15. pow1/2 43.2%

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

   7. Applied egg-rr 43.2%

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

      1. associate--l+ 77.2%

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

      2. +-inverses 77.2%

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

      3. metadata-eval 77.2%

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

      4. +-commutative 77.2%

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

   9. Simplified 77.2%

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

   10. Taylor expanded in x around inf 99.7%

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

       1. +-commutative 99.7%

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

       2. associate--l+ 99.7%

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

       3. associate-*r/ 99.7%

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

       4. metadata-eval 99.7%

          \[\leadsto \frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x + \left(0.5 - \frac{\color{blue}{0.125}}{x}\right)} \]

   12. Simplified 99.7%

       \[\leadsto \frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\color{blue}{x + \left(0.5 - \frac{0.125}{x}\right)}} \]

   if 3.99999999999999976e-11 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. clear-num 99.4%

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

      3. associate-/r/ 99.4%

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

      4. prod-diff 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. fma-neg 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. inv-pow 99.4%

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

      9. sqrt-pow2 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow1/2 99.8%

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

      12. pow-flip 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. neg-mul-1 99.8%

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

      3. rem-log-exp 99.7%

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

      4. log-rec 99.7%

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

      5. +-commutative 99.7%

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

      6. log-rec 99.7%

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

      7. rem-log-exp 99.8%

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

      8. sub-neg 99.8%

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

      9. +-inverses 99.8%

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

      10. +-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x + \left(0.5 - \frac{0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 4e-11)
     (/ (/ 1.0 (+ (sqrt x) t_0)) (+ x 0.5))
     (- (pow x -0.5) (pow (+ 1.0 x) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 4e-11) {
		tmp = (1.0 / (sqrt(x) + t_0)) / (x + 0.5);
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 4d-11) then
        tmp = (1.0d0 / (sqrt(x) + t_0)) / (x + 0.5d0)
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 4e-11) {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) / (x + 0.5);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 4e-11:
		tmp = (1.0 / (math.sqrt(x) + t_0)) / (x + 0.5)
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 4e-11)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) / Float64(x + 0.5));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 4e-11)
		tmp = (1.0 / (sqrt(x) + t_0)) / (x + 0.5);
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 4e-11], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 3.99999999999999976e-11

   1. Initial program 40.1%

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

      1. frac-sub 40.1%

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

      2. clear-num 40.1%

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

      3. sqrt-unprod 40.1%

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

      4. +-commutative 40.1%

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

      5. *-un-lft-identity 40.1%

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

      6. *-rgt-identity 40.1%

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

      7. +-commutative 40.1%

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

   3. Applied egg-rr 40.1%

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

      1. associate-/r/ 40.1%

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

      2. associate-*l/ 40.1%

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

      3. *-lft-identity 40.1%

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

      4. distribute-rgt-in 40.1%

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

      5. *-lft-identity 40.1%

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

   5. Simplified 40.1%

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

      1. pow1/2 40.1%

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

      2. metadata-eval 40.1%

         \[\leadsto \frac{{\left(1 + x\right)}^{\color{blue}{\left(0.25 \cdot 2\right)}} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]

      3. pow-pow 40.1%

         \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]

      4. flip-- 40.0%

         \[\leadsto \frac{\color{blue}{\frac{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}}{\sqrt{x + x \cdot x}} \]

      5. pow-pow 40.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      6. metadata-eval 40.7%

         \[\leadsto \frac{\frac{{\left(1 + x\right)}^{\color{blue}{0.5}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      7. pow1/2 40.7%

         \[\leadsto \frac{\frac{\color{blue}{\sqrt{1 + x}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      8. pow-pow 40.8%

         \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot \color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      9. metadata-eval 40.8%

         \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot {\left(1 + x\right)}^{\color{blue}{0.5}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      10. pow1/2 40.8%

          \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot \color{blue}{\sqrt{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      11. add-sqr-sqrt 41.8%

          \[\leadsto \frac{\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      12. add-sqr-sqrt 43.2%

          \[\leadsto \frac{\frac{\left(1 + x\right) - \color{blue}{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      13. pow-pow 43.2%

          \[\leadsto \frac{\frac{\left(1 + x\right) - x}{\color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      14. metadata-eval 43.2%

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

      15. pow1/2 43.2%

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

   7. Applied egg-rr 43.2%

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

      1. associate--l+ 77.2%

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

      2. +-inverses 77.2%

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

      3. metadata-eval 77.2%

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

      4. +-commutative 77.2%

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

   9. Simplified 77.2%

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

   10. Taylor expanded in x around inf 99.6%

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

       1. +-commutative 99.6%

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

   12. Simplified 99.6%

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

   if 3.99999999999999976e-11 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. clear-num 99.4%

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

      3. associate-/r/ 99.4%

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

      4. prod-diff 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. fma-neg 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. inv-pow 99.4%

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

      9. sqrt-pow2 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow1/2 99.8%

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

      12. pow-flip 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. neg-mul-1 99.8%

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

      3. rem-log-exp 99.7%

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

      4. log-rec 99.7%

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

      5. +-commutative 99.7%

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

      6. log-rec 99.7%

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

      7. rem-log-exp 99.8%

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

      8. sub-neg 99.8%

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

      9. +-inverses 99.8%

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

      10. +-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 5e-13)
     (/ (/ 1.0 (+ (sqrt x) t_0)) x)
     (- (pow x -0.5) (pow (+ 1.0 x) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-13) {
		tmp = (1.0 / (sqrt(x) + t_0)) / x;
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 5d-13) then
        tmp = (1.0d0 / (sqrt(x) + t_0)) / x
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 5e-13) {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) / x;
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 5e-13:
		tmp = (1.0 / (math.sqrt(x) + t_0)) / x
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 5e-13)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) / x);
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-13)
		tmp = (1.0 / (sqrt(x) + t_0)) / x;
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 5e-13], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 4.9999999999999999e-13

   1. Initial program 39.6%

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

      1. frac-sub 39.6%

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

      2. clear-num 39.6%

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

      3. sqrt-unprod 39.6%

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

      4. +-commutative 39.6%

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

      5. *-un-lft-identity 39.6%

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

      6. *-rgt-identity 39.6%

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

      7. +-commutative 39.6%

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

   3. Applied egg-rr 39.6%

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

      1. associate-/r/ 39.6%

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

      2. associate-*l/ 39.6%

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

      3. *-lft-identity 39.6%

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

      4. distribute-rgt-in 39.6%

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

      5. *-lft-identity 39.6%

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

   5. Simplified 39.6%

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

      1. pow1/2 39.6%

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

      2. metadata-eval 39.6%

         \[\leadsto \frac{{\left(1 + x\right)}^{\color{blue}{\left(0.25 \cdot 2\right)}} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]

      3. pow-pow 39.6%

         \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]

      4. flip-- 39.5%

         \[\leadsto \frac{\color{blue}{\frac{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}}{\sqrt{x + x \cdot x}} \]

      5. pow-pow 39.9%

         \[\leadsto \frac{\frac{\color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      6. metadata-eval 39.9%

         \[\leadsto \frac{\frac{{\left(1 + x\right)}^{\color{blue}{0.5}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      7. pow1/2 39.9%

         \[\leadsto \frac{\frac{\color{blue}{\sqrt{1 + x}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      8. pow-pow 40.0%

         \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot \color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      9. metadata-eval 40.0%

         \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot {\left(1 + x\right)}^{\color{blue}{0.5}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      10. pow1/2 40.0%

          \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot \color{blue}{\sqrt{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      11. add-sqr-sqrt 41.0%

          \[\leadsto \frac{\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      12. add-sqr-sqrt 41.8%

          \[\leadsto \frac{\frac{\left(1 + x\right) - \color{blue}{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      13. pow-pow 41.8%

          \[\leadsto \frac{\frac{\left(1 + x\right) - x}{\color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

      14. metadata-eval 41.8%

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

      15. pow1/2 41.8%

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

   7. Applied egg-rr 41.8%

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

      1. associate--l+ 76.6%

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

      2. +-inverses 76.6%

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

      3. metadata-eval 76.6%

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

      4. +-commutative 76.6%

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

   9. Simplified 76.6%

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

   10. Taylor expanded in x around inf 98.8%

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

   if 4.9999999999999999e-13 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 98.6%

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

      1. *-un-lft-identity 98.6%

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

      2. clear-num 98.6%

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

      3. associate-/r/ 98.6%

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

      4. prod-diff 98.6%

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

      5. *-un-lft-identity 98.6%

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

      6. fma-neg 98.6%

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

      7. *-un-lft-identity 98.6%

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

      8. inv-pow 98.6%

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

      9. sqrt-pow2 99.0%

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

      10. metadata-eval 99.0%

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

      11. pow1/2 99.0%

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

      12. pow-flip 99.0%

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

      13. +-commutative 99.0%

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

      14. metadata-eval 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. fma-udef 99.0%

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

      2. neg-mul-1 99.0%

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

      3. rem-log-exp 98.5%

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

      4. log-rec 98.5%

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

      5. +-commutative 98.5%

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

      6. log-rec 98.5%

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

      7. rem-log-exp 99.0%

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

      8. sub-neg 99.0%

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

      9. +-inverses 99.0%

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

      10. +-rgt-identity 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 5: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{x}^{-2}}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 5e-13)
   (* 0.5 (sqrt (/ (pow x -2.0) x)))
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 5e-13) {
		tmp = 0.5 * sqrt((pow(x, -2.0) / x));
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 5d-13) then
        tmp = 0.5d0 * sqrt(((x ** (-2.0d0)) / x))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 5e-13) {
		tmp = 0.5 * Math.sqrt((Math.pow(x, -2.0) / x));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 5e-13:
		tmp = 0.5 * math.sqrt((math.pow(x, -2.0) / x))
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 5e-13)
		tmp = Float64(0.5 * sqrt(Float64((x ^ -2.0) / x)));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 5e-13)
		tmp = 0.5 * sqrt(((x ^ -2.0) / x));
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-13], N[(0.5 * N[Sqrt[N[(N[Power[x, -2.0], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 4.9999999999999999e-13

   1. Initial program 39.6%

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

      1. flip-- 39.5%

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

      2. div-inv 39.5%

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

      3. frac-times 24.3%

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

      4. metadata-eval 24.3%

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

      5. add-sqr-sqrt 21.5%

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

      6. frac-times 29.0%

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

      7. metadata-eval 29.0%

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

      8. add-sqr-sqrt 39.7%

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

      9. +-commutative 39.7%

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

      10. pow1/2 39.7%

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

      11. pow-flip 39.7%

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

      12. metadata-eval 39.7%

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

      13. inv-pow 39.7%

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

      14. sqrt-pow2 39.7%

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

      15. +-commutative 39.7%

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

      16. metadata-eval 39.7%

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

   3. Applied egg-rr 39.7%

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

   4. Taylor expanded in x around inf 67.3%

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

      1. unpow3 67.3%

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

      2. unpow2 67.3%

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

      3. associate-/r* 67.4%

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

      4. unpow2 67.4%

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

      5. associate-/r* 67.4%

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

      6. unpow-1 67.4%

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

      7. *-lft-identity 67.4%

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

      8. associate-*l/ 67.4%

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

      9. unpow-1 67.4%

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

      10. pow-sqr 67.5%

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

      11. metadata-eval 67.5%

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

   6. Simplified 67.5%

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

   if 4.9999999999999999e-13 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 98.6%

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

      1. *-un-lft-identity 98.6%

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

      2. clear-num 98.6%

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

      3. associate-/r/ 98.6%

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

      4. prod-diff 98.6%

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

      5. *-un-lft-identity 98.6%

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

      6. fma-neg 98.6%

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

      7. *-un-lft-identity 98.6%

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

      8. inv-pow 98.6%

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

      9. sqrt-pow2 99.0%

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

      10. metadata-eval 99.0%

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

      11. pow1/2 99.0%

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

      12. pow-flip 99.0%

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

      13. +-commutative 99.0%

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

      14. metadata-eval 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. fma-udef 99.0%

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

      2. neg-mul-1 99.0%

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

      3. rem-log-exp 98.5%

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

      4. log-rec 98.5%

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

      5. +-commutative 98.5%

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

      6. log-rec 98.5%

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

      7. rem-log-exp 99.0%

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

      8. sub-neg 99.0%

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

      9. +-inverses 99.0%

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

      10. +-rgt-identity 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{x}^{-2}}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 6: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[x ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.2%

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

   1. frac-sub 70.3%

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

   2. clear-num 70.3%

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

   3. sqrt-unprod 70.2%

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

   4. +-commutative 70.2%

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

   5. *-un-lft-identity 70.2%

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

   6. *-rgt-identity 70.2%

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

   7. +-commutative 70.2%

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

3. Applied egg-rr 70.2%

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

   1. associate-/r/ 70.3%

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

   2. associate-*l/ 70.3%

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

   3. *-lft-identity 70.3%

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

   4. distribute-rgt-in 70.2%

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

   5. *-lft-identity 70.2%

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

5. Simplified 70.2%

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

   1. pow1/2 70.2%

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

   2. metadata-eval 70.2%

      \[\leadsto \frac{{\left(1 + x\right)}^{\color{blue}{\left(0.25 \cdot 2\right)}} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]

   3. pow-pow 70.2%

      \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]

   4. flip-- 70.2%

      \[\leadsto \frac{\color{blue}{\frac{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}}{\sqrt{x + x \cdot x}} \]

   5. pow-pow 70.5%

      \[\leadsto \frac{\frac{\color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   6. metadata-eval 70.5%

      \[\leadsto \frac{\frac{{\left(1 + x\right)}^{\color{blue}{0.5}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   7. pow1/2 70.5%

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{1 + x}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   8. pow-pow 70.6%

      \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot \color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   9. metadata-eval 70.6%

      \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot {\left(1 + x\right)}^{\color{blue}{0.5}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   10. pow1/2 70.6%

       \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot \color{blue}{\sqrt{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   11. add-sqr-sqrt 71.1%

       \[\leadsto \frac{\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   12. add-sqr-sqrt 71.8%

       \[\leadsto \frac{\frac{\left(1 + x\right) - \color{blue}{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   13. pow-pow 71.8%

       \[\leadsto \frac{\frac{\left(1 + x\right) - x}{\color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   14. metadata-eval 71.8%

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

   15. pow1/2 71.8%

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

7. Applied egg-rr 71.8%

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

   1. associate--l+ 88.5%

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

   2. +-inverses 88.5%

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

   3. metadata-eval 88.5%

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

   4. +-commutative 88.5%

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

9. Simplified 88.5%

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

    1. expm1-log1p-u 85.1%

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

    2. expm1-udef 66.0%

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

    3. associate-/l/ 66.0%

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

    4. +-commutative 66.0%

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

    5. add-sqr-sqrt 66.0%

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

    6. hypot-def 66.0%

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

11. Applied egg-rr 66.0%

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

    1. expm1-def 95.1%

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

    2. expm1-log1p 98.6%

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

    3. associate-/r* 99.6%

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

13. Simplified 99.6%

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

14. Final simplification 99.6%

    \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}}{\sqrt{x} + \sqrt{1 + x}} \]

Alternative 7: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = 1.0 / (hypot(x, sqrt(x)) * (sqrt(x) + sqrt((1.0 + x))));
end

Wolfram

code[x_] := N[(1.0 / N[(N[Sqrt[x ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.2%

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

   1. frac-sub 70.3%

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

   2. clear-num 70.3%

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

   3. sqrt-unprod 70.2%

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

   4. +-commutative 70.2%

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

   5. *-un-lft-identity 70.2%

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

   6. *-rgt-identity 70.2%

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

   7. +-commutative 70.2%

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

3. Applied egg-rr 70.2%

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

   1. associate-/r/ 70.3%

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

   2. associate-*l/ 70.3%

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

   3. *-lft-identity 70.3%

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

   4. distribute-rgt-in 70.2%

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

   5. *-lft-identity 70.2%

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

5. Simplified 70.2%

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

   1. pow1/2 70.2%

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

   2. metadata-eval 70.2%

      \[\leadsto \frac{{\left(1 + x\right)}^{\color{blue}{\left(0.25 \cdot 2\right)}} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]

   3. pow-pow 70.2%

      \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]

   4. flip-- 70.2%

      \[\leadsto \frac{\color{blue}{\frac{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}}{\sqrt{x + x \cdot x}} \]

   5. pow-pow 70.5%

      \[\leadsto \frac{\frac{\color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   6. metadata-eval 70.5%

      \[\leadsto \frac{\frac{{\left(1 + x\right)}^{\color{blue}{0.5}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   7. pow1/2 70.5%

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{1 + x}} \cdot {\left({\left(1 + x\right)}^{0.25}\right)}^{2} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   8. pow-pow 70.6%

      \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot \color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   9. metadata-eval 70.6%

      \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot {\left(1 + x\right)}^{\color{blue}{0.5}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   10. pow1/2 70.6%

       \[\leadsto \frac{\frac{\sqrt{1 + x} \cdot \color{blue}{\sqrt{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   11. add-sqr-sqrt 71.1%

       \[\leadsto \frac{\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   12. add-sqr-sqrt 71.8%

       \[\leadsto \frac{\frac{\left(1 + x\right) - \color{blue}{x}}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   13. pow-pow 71.8%

       \[\leadsto \frac{\frac{\left(1 + x\right) - x}{\color{blue}{{\left(1 + x\right)}^{\left(0.25 \cdot 2\right)}} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   14. metadata-eval 71.8%

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

   15. pow1/2 71.8%

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

7. Applied egg-rr 71.8%

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

   1. associate--l+ 88.5%

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

   2. +-inverses 88.5%

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

   3. metadata-eval 88.5%

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

   4. +-commutative 88.5%

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

9. Simplified 88.5%

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

    1. expm1-log1p-u 85.1%

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

    2. expm1-udef 66.0%

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

    3. associate-/l/ 66.0%

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

    4. +-commutative 66.0%

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

    5. add-sqr-sqrt 66.0%

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

    6. hypot-def 66.0%

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

11. Applied egg-rr 66.0%

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

    1. expm1-def 95.1%

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

    2. expm1-log1p 98.6%

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

13. Simplified 98.6%

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

14. Final simplification 98.6%

    \[\leadsto \frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]

Alternative 8: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.7d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.7], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.6%

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

      1. add-log-exp 7.5%

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

      2. *-un-lft-identity 7.5%

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

      3. log-prod 7.5%

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

      4. metadata-eval 7.5%

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

      5. add-log-exp 99.6%

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

      6. pow1/2 99.6%

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

      7. pow-flip 100.0%

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

      8. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

   8. Simplified 98.3%

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

   if 1.69999999999999996 < x

   1. Initial program 40.5%

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

      1. *-un-lft-identity 40.5%

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

      2. clear-num 40.5%

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

      3. associate-/r/ 40.5%

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

      4. prod-diff 40.5%

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

      5. *-un-lft-identity 40.5%

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

      6. fma-neg 40.5%

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

      7. *-un-lft-identity 40.5%

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

      8. inv-pow 40.5%

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

      9. sqrt-pow2 32.8%

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

      10. metadata-eval 32.8%

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

      11. pow1/2 32.8%

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

      12. pow-flip 40.6%

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

      13. +-commutative 40.6%

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

      14. metadata-eval 40.6%

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

   3. Applied egg-rr 40.6%

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

      1. fma-udef 40.6%

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

      2. neg-mul-1 40.6%

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

      3. rem-log-exp 7.1%

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

      4. log-rec 7.1%

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

      5. +-commutative 7.1%

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

      6. log-rec 7.1%

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

      7. rem-log-exp 40.6%

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

      8. sub-neg 40.6%

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

      9. +-inverses 40.6%

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

      10. +-rgt-identity 40.6%

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

   5. Simplified 40.6%

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

   6. Taylor expanded in x around inf 66.8%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 9: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{x}^{-2}}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (* 0.5 (sqrt (/ (pow x -2.0) x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 0.5 * sqrt((pow(x, -2.0) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.7d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = 0.5d0 * sqrt(((x ** (-2.0d0)) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 0.5 * Math.sqrt((Math.pow(x, -2.0) / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = 0.5 * math.sqrt((math.pow(x, -2.0) / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(0.5 * sqrt(Float64((x ^ -2.0) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = 0.5 * sqrt(((x ^ -2.0) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.7], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(N[Power[x, -2.0], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.6%

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

      1. add-log-exp 7.5%

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

      2. *-un-lft-identity 7.5%

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

      3. log-prod 7.5%

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

      4. metadata-eval 7.5%

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

      5. add-log-exp 99.6%

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

      6. pow1/2 99.6%

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

      7. pow-flip 100.0%

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

      8. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

   8. Simplified 98.3%

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

   if 1.69999999999999996 < x

   1. Initial program 40.5%

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

      1. flip-- 40.5%

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

      2. div-inv 40.5%

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

      3. frac-times 25.7%

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

      4. metadata-eval 25.7%

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

      5. add-sqr-sqrt 23.0%

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

      6. frac-times 30.3%

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

      7. metadata-eval 30.3%

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

      8. add-sqr-sqrt 40.6%

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

      9. +-commutative 40.6%

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

      10. pow1/2 40.6%

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

      11. pow-flip 40.6%

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

      12. metadata-eval 40.6%

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

      13. inv-pow 40.6%

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

      14. sqrt-pow2 40.6%

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

      15. +-commutative 40.6%

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

      16. metadata-eval 40.6%

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

   3. Applied egg-rr 40.6%

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

   4. Taylor expanded in x around inf 66.8%

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

      1. unpow3 66.8%

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

      2. unpow2 66.8%

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

      3. associate-/r* 67.0%

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

      4. unpow2 67.0%

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

      5. associate-/r* 67.0%

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

      6. unpow-1 67.0%

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

      7. *-lft-identity 67.0%

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

      8. associate-*l/ 66.9%

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

      9. unpow-1 66.9%

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

      10. pow-sqr 67.0%

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

      11. metadata-eval 67.0%

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

   6. Simplified 67.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{x}^{-2}}{x}}\\ \end{array} \]

Alternative 10: 68.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{-1}{1 + x}\right) \cdot \left(\sqrt{x} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.2)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (* (+ (/ 1.0 x) (/ -1.0 (+ 1.0 x))) (* (sqrt x) 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = ((1.0 / x) + (-1.0 / (1.0 + x))) * (sqrt(x) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.2d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = ((1.0d0 / x) + ((-1.0d0) / (1.0d0 + x))) * (sqrt(x) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = ((1.0 / x) + (-1.0 / (1.0 + x))) * (Math.sqrt(x) * 0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.2:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = ((1.0 / x) + (-1.0 / (1.0 + x))) * (math.sqrt(x) * 0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.2)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(-1.0 / Float64(1.0 + x))) * Float64(sqrt(x) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.2)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = ((1.0 / x) + (-1.0 / (1.0 + x))) * (sqrt(x) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.2], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 99.6%

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

      1. add-log-exp 7.5%

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

      2. *-un-lft-identity 7.5%

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

      3. log-prod 7.5%

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

      4. metadata-eval 7.5%

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

      5. add-log-exp 99.6%

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

      6. pow1/2 99.6%

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

      7. pow-flip 100.0%

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

      8. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

   8. Simplified 98.3%

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

   if 1.19999999999999996 < x

   1. Initial program 40.5%

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

      1. flip-- 40.5%

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

      2. div-inv 40.5%

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

      3. frac-times 25.7%

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

      4. metadata-eval 25.7%

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

      5. add-sqr-sqrt 23.0%

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

      6. frac-times 30.3%

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

      7. metadata-eval 30.3%

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

      8. add-sqr-sqrt 40.6%

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

      9. +-commutative 40.6%

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

      10. pow1/2 40.6%

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

      11. pow-flip 40.6%

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

      12. metadata-eval 40.6%

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

      13. inv-pow 40.6%

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

      14. sqrt-pow2 40.6%

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

      15. +-commutative 40.6%

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

      16. metadata-eval 40.6%

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

   3. Applied egg-rr 40.6%

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

   4. Taylor expanded in x around inf 40.1%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{-1}{1 + x}\right) \cdot \left(\sqrt{x} \cdot 0.5\right)\\ \end{array} \]

Alternative 11: 68.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.0)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (/ 1.0 (sqrt (+ x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 6.0) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 1.0 / sqrt((x + (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.0d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = 1.0d0 / sqrt((x + (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.0) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 1.0 / Math.sqrt((x + (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.0:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = 1.0 / math.sqrt((x + (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.0)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(1.0 / sqrt(Float64(x + Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.0)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = 1.0 / sqrt((x + (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.0], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6

   1. Initial program 99.6%

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

      1. add-log-exp 7.5%

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

      2. *-un-lft-identity 7.5%

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

      3. log-prod 7.5%

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

      4. metadata-eval 7.5%

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

      5. add-log-exp 99.6%

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

      6. pow1/2 99.6%

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

      7. pow-flip 100.0%

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

      8. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

   8. Simplified 98.3%

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

   if 6 < x

   1. Initial program 40.5%

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

      1. frac-sub 40.5%

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

      2. clear-num 40.5%

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

      3. sqrt-unprod 40.5%

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

      4. +-commutative 40.5%

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

      5. *-un-lft-identity 40.5%

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

      6. *-rgt-identity 40.5%

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

      7. +-commutative 40.5%

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

   3. Applied egg-rr 40.5%

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

      1. associate-/r/ 40.5%

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

      2. associate-*l/ 40.5%

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

      3. *-lft-identity 40.5%

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

      4. distribute-rgt-in 40.5%

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

      5. *-lft-identity 40.5%

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

   5. Simplified 40.5%

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

   6. Taylor expanded in x around 0 38.8%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x}}\\ \end{array} \]

Alternative 12: 68.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ 1.0 (sqrt (+ x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = 1.0 / sqrt((x + (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = 1.0d0 / sqrt((x + (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = 1.0 / Math.sqrt((x + (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = 1.0 / math.sqrt((x + (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(1.0 / sqrt(Float64(x + Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = 1.0 / sqrt((x + (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 / N[Sqrt[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. neg-mul-1 100.0%

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

      3. rem-log-exp 100.0%

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

      4. log-rec 100.0%

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

      5. +-commutative 100.0%

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

      6. log-rec 100.0%

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

      7. rem-log-exp 100.0%

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

      8. sub-neg 100.0%

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

      9. +-inverses 100.0%

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

      10. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.2%

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

   if 1.3999999999999999 < x

   1. Initial program 40.5%

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

      1. frac-sub 40.5%

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

      2. clear-num 40.5%

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

      3. sqrt-unprod 40.5%

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

      4. +-commutative 40.5%

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

      5. *-un-lft-identity 40.5%

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

      6. *-rgt-identity 40.5%

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

      7. +-commutative 40.5%

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

   3. Applied egg-rr 40.5%

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

      1. associate-/r/ 40.5%

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

      2. associate-*l/ 40.5%

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

      3. *-lft-identity 40.5%

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

      4. distribute-rgt-in 40.5%

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

      5. *-lft-identity 40.5%

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

   5. Simplified 40.5%

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

   6. Taylor expanded in x around 0 38.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x}}\\ \end{array} \]

Alternative 13: 68.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.7) (+ (pow x -0.5) -1.0) (/ 1.0 (sqrt (+ x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.7) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / sqrt((x + (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = 1.0d0 / sqrt((x + (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.7) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / Math.sqrt((x + (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.7:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = 1.0 / math.sqrt((x + (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(1.0 / sqrt(Float64(x + Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 1.0 / sqrt((x + (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.7], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 / N[Sqrt[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. neg-mul-1 100.0%

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

      3. rem-log-exp 100.0%

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

      4. log-rec 100.0%

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

      5. +-commutative 100.0%

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

      6. log-rec 100.0%

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

      7. rem-log-exp 100.0%

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

      8. sub-neg 100.0%

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

      9. +-inverses 100.0%

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

      10. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 97.4%

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

   if 0.69999999999999996 < x

   1. Initial program 40.5%

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

      1. frac-sub 40.5%

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

      2. clear-num 40.5%

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

      3. sqrt-unprod 40.5%

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

      4. +-commutative 40.5%

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

      5. *-un-lft-identity 40.5%

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

      6. *-rgt-identity 40.5%

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

      7. +-commutative 40.5%

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

   3. Applied egg-rr 40.5%

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

      1. associate-/r/ 40.5%

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

      2. associate-*l/ 40.5%

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

      3. *-lft-identity 40.5%

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

      4. distribute-rgt-in 40.5%

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

      5. *-lft-identity 40.5%

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

   5. Simplified 40.5%

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

   6. Taylor expanded in x around 0 38.8%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x}}\\ \end{array} \]

Alternative 14: 68.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.8) (+ (pow x -0.5) -1.0) (pow (* x x) -0.25)))

C

double code(double x) {
	double tmp;
	if (x <= 0.8) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = pow((x * x), -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.8d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (x * x) ** (-0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.8) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = Math.pow((x * x), -0.25);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.8:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = math.pow((x * x), -0.25)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.8)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(x * x) ^ -0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.8)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (x * x) ^ -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.8], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[Power[N[(x * x), $MachinePrecision], -0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. neg-mul-1 100.0%

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

      3. rem-log-exp 100.0%

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

      4. log-rec 100.0%

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

      5. +-commutative 100.0%

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

      6. log-rec 100.0%

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

      7. rem-log-exp 100.0%

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

      8. sub-neg 100.0%

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

      9. +-inverses 100.0%

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

      10. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 97.4%

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

   if 0.80000000000000004 < x

   1. Initial program 40.5%

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

      1. add-log-exp 5.6%

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

      2. *-un-lft-identity 5.6%

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

      3. log-prod 5.6%

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

      4. metadata-eval 5.6%

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

      5. add-log-exp 40.5%

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

      6. pow1/2 40.5%

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

      7. pow-flip 32.8%

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

      8. metadata-eval 32.8%

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

   3. Applied egg-rr 32.8%

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

      1. +-lft-identity 32.8%

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

   5. Simplified 32.8%

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

   6. Taylor expanded in x around inf 5.6%

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

      1. inv-pow 5.6%

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

      2. sqrt-pow1 5.6%

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

      3. metadata-eval 5.6%

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

      4. sqr-pow 5.6%

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

      5. pow-prod-down 38.1%

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

      6. metadata-eval 38.1%

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

   8. Applied egg-rr 38.1%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \]

Alternative 15: 51.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. *-un-lft-identity 70.2%

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

   2. clear-num 70.2%

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

   3. associate-/r/ 70.2%

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

   4. prod-diff 70.2%

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

   5. *-un-lft-identity 70.2%

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

   6. fma-neg 70.2%

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

   7. *-un-lft-identity 70.2%

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

   8. inv-pow 70.2%

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

   9. sqrt-pow2 66.6%

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

   10. metadata-eval 66.6%

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

   11. pow1/2 66.6%

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

   12. pow-flip 70.5%

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

   13. +-commutative 70.5%

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

   14. metadata-eval 70.5%

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

3. Applied egg-rr 70.5%

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

   1. fma-udef 70.5%

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

   2. neg-mul-1 70.5%

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

   3. rem-log-exp 53.9%

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

   4. log-rec 53.9%

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

   5. +-commutative 53.9%

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

   6. log-rec 53.9%

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

   7. rem-log-exp 70.5%

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

   8. sub-neg 70.5%

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

   9. +-inverses 70.5%

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

   10. +-rgt-identity 70.5%

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

5. Simplified 70.5%

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

6. Taylor expanded in x around 0 50.3%

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

7. Final simplification 50.3%

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

Alternative 16: 51.6% 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 70.2%

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

   1. add-log-exp 6.6%

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

   2. *-un-lft-identity 6.6%

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

   3. log-prod 6.6%

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

   4. metadata-eval 6.6%

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

   5. add-log-exp 70.2%

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

   6. pow1/2 70.2%

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

   7. pow-flip 66.6%

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

   8. metadata-eval 66.6%

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

3. Applied egg-rr 66.6%

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

   1. +-lft-identity 66.6%

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

5. Simplified 66.6%

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

6. Taylor expanded in x around inf 49.4%

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

   1. inv-pow 49.4%

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

   2. sqrt-pow1 49.9%

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

   3. metadata-eval 49.9%

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

   4. expm1-log1p-u 46.4%

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

   5. expm1-udef 62.2%

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

8. Applied egg-rr 62.2%

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

   1. expm1-def 46.4%

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

   2. expm1-log1p 49.9%

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

10. Simplified 49.9%

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

11. Final simplification 49.9%

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

Alternative 17: 3.9% accurate, 69.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x 0.5))

C

double code(double x) {
	return 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 x * 0.5;
}

Python

def code(x):
	return x * 0.5

Julia

function code(x)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.5;
end

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. add-cbrt-cube 54.5%

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

   2. pow1/3 53.5%

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

   3. add-sqr-sqrt 53.5%

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

   4. pow1 53.5%

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

   5. pow1/2 53.5%

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

   6. pow-prod-up 53.5%

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

   7. +-commutative 53.5%

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

   8. metadata-eval 53.5%

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

3. Applied egg-rr 53.5%

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

   1. unpow1/3 54.4%

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

5. Simplified 54.4%

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

6. Taylor expanded in x around 0 51.0%

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

   1. *-commutative 51.0%

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

8. Simplified 51.0%

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

9. Taylor expanded in x around inf 4.1%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

10. Final simplification 4.1%

    \[\leadsto x \cdot 0.5 \]

Alternative 18: 1.9% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 70.2%

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

2. Taylor expanded in x around 0 50.1%

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

3. Taylor expanded in x around inf 1.9%

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

4. Final simplification 1.9%

   \[\leadsto -1 \]

Developer target: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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