2isqrt (example 3.6)

Percentage Accurate: 69.3% → 99.7%

Time: 11.7s

Alternatives: 14

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.3% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0 + \sqrt{x}}}{\sqrt{x \cdot \left(x + 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 0.0)
     (* 0.5 (pow x -1.5))
     (/ (/ 1.0 (+ t_0 (sqrt x))) (sqrt (* x (+ x 1.0)))))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = (1.0 / (t_0 + sqrt(x))) / sqrt((x * (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = (1.0d0 / (t_0 + sqrt(x))) / sqrt((x * (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = (1.0 / (t_0 + Math.sqrt(x))) / Math.sqrt((x * (x + 1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / t_0)) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = (1.0 / (t_0 + math.sqrt(x))) / math.sqrt((x * (x + 1.0)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 0.0)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64(Float64(1.0 / Float64(t_0 + sqrt(x))) / sqrt(Float64(x * Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = (1.0 / (t_0 + sqrt(x))) / sqrt((x * (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 38.6%

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

      1. *-un-lft-identity 38.6%

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

      2. clear-num 38.6%

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

      3. associate-/r/ 38.6%

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

      4. prod-diff 38.6%

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

      5. *-un-lft-identity 38.6%

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

      6. fma-neg 38.6%

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

      7. *-un-lft-identity 38.6%

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

      8. pow1/2 38.6%

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

      9. pow-flip 26.3%

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

      10. metadata-eval 26.3%

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

      11. pow1/2 26.3%

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

      12. pow-flip 38.6%

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

      13. +-commutative 38.6%

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

      14. metadata-eval 38.6%

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

   3. Applied egg-rr 38.6%

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

      1. +-commutative 38.6%

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

      2. sub-neg 38.6%

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

      3. fma-udef 38.6%

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

      4. distribute-lft1-in 38.6%

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

      5. metadata-eval 38.6%

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

      6. mul0-lft 38.6%

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

      7. +-commutative 38.6%

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

      8. associate-+r+ 38.6%

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

      9. sub-neg 38.6%

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

      10. neg-sub0 38.6%

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

      11. +-commutative 38.6%

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

      12. sub-neg 38.6%

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

   5. Simplified 38.6%

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

      1. flip-- 38.6%

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

      2. pow-prod-up 21.6%

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

      3. metadata-eval 21.6%

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

      4. inv-pow 21.6%

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

      5. pow-prod-up 38.6%

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

      6. metadata-eval 38.6%

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

      7. inv-pow 38.6%

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

      8. +-commutative 38.6%

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

      9. +-commutative 38.6%

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

   7. Applied egg-rr 38.6%

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

   8. Taylor expanded in x around inf 66.1%

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

      1. unpow-1 66.1%

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

      2. exp-to-pow 63.9%

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

      3. *-commutative 63.9%

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

      4. log-pow 64.7%

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

      5. associate-*r* 64.7%

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

      6. metadata-eval 64.7%

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

      7. *-commutative 64.7%

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

      8. exp-to-pow 67.0%

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

      9. metadata-eval 67.0%

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

      10. pow-sqr 67.1%

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

      11. rem-sqrt-square 100.0%

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

      12. rem-square-sqrt 99.5%

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

      13. fabs-sqr 99.5%

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

      14. rem-square-sqrt 100.0%

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

   10. Simplified 100.0%

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

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

   1. Initial program 98.2%

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

      1. frac-sub 98.3%

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

      2. *-un-lft-identity 98.3%

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

      3. *-rgt-identity 98.3%

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

      4. +-commutative 98.3%

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

      5. sqrt-unprod 98.3%

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

      6. +-commutative 98.3%

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

   3. Applied egg-rr 98.3%

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

      1. flip-- 99.1%

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

      2. div-inv 99.1%

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

      3. add-sqr-sqrt 98.7%

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

      4. add-sqr-sqrt 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

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

      5. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 5e-10)
     (/ (/ 1.0 (+ t_0 (sqrt x))) (+ x 0.5))
     (+ (pow x -0.5) (/ -1.0 t_0)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 5e-10) {
		tmp = (1.0 / (t_0 + sqrt(x))) / (x + 0.5);
	} else {
		tmp = pow(x, -0.5) + (-1.0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 5d-10) then
        tmp = (1.0d0 / (t_0 + sqrt(x))) / (x + 0.5d0)
    else
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 5e-10) {
		tmp = (1.0 / (t_0 + Math.sqrt(x))) / (x + 0.5);
	} else {
		tmp = Math.pow(x, -0.5) + (-1.0 / t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / t_0)) <= 5e-10:
		tmp = (1.0 / (t_0 + math.sqrt(x))) / (x + 0.5)
	else:
		tmp = math.pow(x, -0.5) + (-1.0 / t_0)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 5e-10)
		tmp = Float64(Float64(1.0 / Float64(t_0 + sqrt(x))) / Float64(x + 0.5));
	else
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 5e-10)
		tmp = (1.0 / (t_0 + sqrt(x))) / (x + 0.5);
	else
		tmp = (x ^ -0.5) + (-1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 5e-10], N[(N[(1.0 / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.00000000000000031e-10

   1. Initial program 39.0%

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

      1. frac-sub 39.0%

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

      2. *-un-lft-identity 39.0%

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

      3. *-rgt-identity 39.0%

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

      4. +-commutative 39.0%

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

      5. sqrt-unprod 39.0%

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

      6. +-commutative 39.0%

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

   3. Applied egg-rr 39.0%

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

      1. flip-- 39.7%

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

      2. div-inv 39.7%

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

      3. add-sqr-sqrt 39.8%

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

      4. add-sqr-sqrt 40.0%

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

   5. Applied egg-rr 40.0%

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

      1. associate-*r/ 40.0%

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

      2. *-rgt-identity 40.0%

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

      3. associate--l+ 83.9%

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

      4. +-inverses 83.9%

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

      5. metadata-eval 83.9%

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

   7. Simplified 83.9%

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

   8. Taylor expanded in x around inf 99.5%

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

      1. +-commutative 99.5%

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

   10. Simplified 99.5%

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

   if 5.00000000000000031e-10 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.3%

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

      1. expm1-log1p-u 92.4%

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

      2. expm1-udef 92.2%

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

      3. pow1/2 92.2%

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

      4. pow-flip 92.2%

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

      5. metadata-eval 92.2%

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

   3. Applied egg-rr 92.2%

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

      1. expm1-def 92.4%

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

      2. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 5e-23)
     (* 0.5 (pow x -1.5))
     (+ (pow x -0.5) (/ -1.0 t_0)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 5e-23) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = pow(x, -0.5) + (-1.0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 5d-23) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 5e-23) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = Math.pow(x, -0.5) + (-1.0 / t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / t_0)) <= 5e-23:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = math.pow(x, -0.5) + (-1.0 / t_0)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 5e-23)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 5e-23)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = (x ^ -0.5) + (-1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 5e-23], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.0000000000000002e-23

   1. Initial program 38.5%

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

      1. *-un-lft-identity 38.5%

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

      2. clear-num 38.5%

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

      3. associate-/r/ 38.5%

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

      4. prod-diff 38.5%

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

      5. *-un-lft-identity 38.5%

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

      6. fma-neg 38.5%

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

      7. *-un-lft-identity 38.5%

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

      8. pow1/2 38.5%

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

      9. pow-flip 26.3%

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

      10. metadata-eval 26.3%

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

      11. pow1/2 26.3%

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

      12. pow-flip 38.5%

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

      13. +-commutative 38.5%

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

      14. metadata-eval 38.5%

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

   3. Applied egg-rr 38.5%

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

      1. +-commutative 38.5%

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

      2. sub-neg 38.5%

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

      3. fma-udef 38.5%

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

      4. distribute-lft1-in 38.5%

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

      5. metadata-eval 38.5%

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

      6. mul0-lft 38.5%

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

      7. +-commutative 38.5%

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

      8. associate-+r+ 38.5%

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

      9. sub-neg 38.5%

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

      10. neg-sub0 38.5%

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

      11. +-commutative 38.5%

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

      12. sub-neg 38.5%

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

   5. Simplified 38.5%

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

      1. flip-- 38.5%

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

      2. pow-prod-up 21.6%

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

      3. metadata-eval 21.6%

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

      4. inv-pow 21.6%

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

      5. pow-prod-up 38.5%

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

      6. metadata-eval 38.5%

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

      7. inv-pow 38.5%

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

      8. +-commutative 38.5%

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

      9. +-commutative 38.5%

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

   7. Applied egg-rr 38.5%

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

   8. Taylor expanded in x around inf 66.3%

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

      1. unpow-1 66.3%

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

      2. exp-to-pow 64.1%

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

      3. *-commutative 64.1%

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

      4. log-pow 64.9%

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

      5. associate-*r* 64.9%

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

      6. metadata-eval 64.9%

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

      7. *-commutative 64.9%

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

      8. exp-to-pow 67.2%

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

      9. metadata-eval 67.2%

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

      10. pow-sqr 67.3%

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

      11. rem-sqrt-square 99.9%

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

      12. rem-square-sqrt 99.5%

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

      13. fabs-sqr 99.5%

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

      14. rem-square-sqrt 99.9%

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

   10. Simplified 99.9%

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

   if 5.0000000000000002e-23 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 98.8%

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

      1. expm1-log1p-u 92.0%

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

      2. expm1-udef 91.6%

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

      3. pow1/2 91.6%

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

      4. pow-flip 91.6%

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

      5. metadata-eval 91.6%

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

   3. Applied egg-rr 91.6%

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

      1. expm1-def 92.0%

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

      2. expm1-log1p 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{\sqrt{x + 1}}\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \frac{{x}^{-0.5}}{t_0 \cdot \left(t_0 + \sqrt{x}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0)))) (/ (pow x -0.5) (* t_0 (+ t_0 (sqrt x))))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	return pow(x, -0.5) / (t_0 * (t_0 + sqrt(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((x + 1.0d0))
    code = (x ** (-0.5d0)) / (t_0 * (t_0 + sqrt(x)))
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	return Math.pow(x, -0.5) / (t_0 * (t_0 + Math.sqrt(x)));
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	return math.pow(x, -0.5) / (t_0 * (t_0 + math.sqrt(x)))

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	return Float64((x ^ -0.5) / Float64(t_0 * Float64(t_0 + sqrt(x))))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = (x ^ -0.5) / (t_0 * (t_0 + sqrt(x)));
end

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. frac-sub 68.2%

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

   2. *-un-lft-identity 68.2%

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

   3. *-rgt-identity 68.2%

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

   4. +-commutative 68.2%

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

   5. sqrt-unprod 68.2%

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

   6. +-commutative 68.2%

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

3. Applied egg-rr 68.2%

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

   1. flip-- 68.6%

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

   2. div-inv 68.6%

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

   3. add-sqr-sqrt 68.6%

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

   4. add-sqr-sqrt 68.8%

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

5. Applied egg-rr 68.8%

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

   1. associate-*r/ 68.8%

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

   2. *-rgt-identity 68.8%

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

   3. associate--l+ 91.5%

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

   4. +-inverses 91.5%

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

   5. metadata-eval 91.5%

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

7. Simplified 91.5%

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

   1. *-un-lft-identity 91.5%

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

   2. sqrt-prod 99.4%

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

   3. times-frac 99.4%

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

   4. pow1/2 99.4%

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

   5. pow-flip 99.6%

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

   6. metadata-eval 99.6%

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

9. Applied egg-rr 99.6%

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

    1. associate-/l/ 99.6%

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

    2. associate-*r/ 99.6%

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

    3. *-rgt-identity 99.6%

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

    4. +-commutative 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 5: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. frac-sub 68.2%

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

   2. *-un-lft-identity 68.2%

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

   3. *-rgt-identity 68.2%

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

   4. +-commutative 68.2%

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

   5. sqrt-unprod 68.2%

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

   6. +-commutative 68.2%

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

3. Applied egg-rr 68.2%

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

   1. flip-- 68.6%

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

   2. div-inv 68.6%

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

   3. add-sqr-sqrt 68.6%

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

   4. add-sqr-sqrt 68.8%

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

5. Applied egg-rr 68.8%

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

   1. associate-*r/ 68.8%

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

   2. *-rgt-identity 68.8%

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

   3. associate--l+ 91.5%

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

   4. +-inverses 91.5%

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

   5. metadata-eval 91.5%

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

7. Simplified 91.5%

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

   1. expm1-log1p-u 88.2%

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

   2. expm1-udef 64.6%

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

   3. associate-/l/ 64.6%

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

   4. *-commutative 64.6%

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

   5. distribute-rgt-in 64.6%

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

   6. *-un-lft-identity 64.6%

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

   7. add-sqr-sqrt 64.6%

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

   8. hypot-def 64.6%

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

9. Applied egg-rr 64.6%

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

    1. expm1-def 95.7%

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

    2. expm1-log1p 98.9%

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

    3. +-commutative 98.9%

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 108000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 108000000.0)
   (- (pow x -0.5) (pow (+ x 1.0) -0.5))
   (* 0.5 (pow x -1.5))))

C

double code(double x) {
	double tmp;
	if (x <= 108000000.0) {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	} else {
		tmp = 0.5 * pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 108000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    else
        tmp = 0.5d0 * (x ** (-1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 108000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	} else {
		tmp = 0.5 * Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 108000000.0:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	else:
		tmp = 0.5 * math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 108000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	else
		tmp = Float64(0.5 * (x ^ -1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 108000000.0)
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	else
		tmp = 0.5 * (x ^ -1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 108000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.08e8

   1. Initial program 98.8%

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

      1. *-un-lft-identity 98.8%

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

      2. clear-num 98.8%

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

      3. associate-/r/ 98.8%

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

      4. prod-diff 98.8%

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

      5. *-un-lft-identity 98.8%

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

      6. fma-neg 98.8%

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

      7. *-un-lft-identity 98.8%

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

      8. pow1/2 98.8%

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

      9. pow-flip 99.1%

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

      10. metadata-eval 99.1%

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

      11. pow1/2 99.1%

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

      12. pow-flip 99.1%

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

      13. +-commutative 99.1%

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

      14. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

      2. sub-neg 99.1%

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

      3. fma-udef 99.1%

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

      4. distribute-lft1-in 99.1%

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

      5. metadata-eval 99.1%

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

      6. mul0-lft 99.1%

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

      7. +-commutative 99.1%

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

      8. associate-+r+ 99.1%

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

      9. sub-neg 99.1%

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

      10. neg-sub0 99.1%

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

      11. +-commutative 99.1%

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

      12. sub-neg 99.1%

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

   5. Simplified 99.1%

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

   if 1.08e8 < x

   1. Initial program 38.5%

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

      1. *-un-lft-identity 38.5%

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

      2. clear-num 38.5%

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

      3. associate-/r/ 38.5%

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

      4. prod-diff 38.5%

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

      5. *-un-lft-identity 38.5%

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

      6. fma-neg 38.5%

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

      7. *-un-lft-identity 38.5%

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

      8. pow1/2 38.5%

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

      9. pow-flip 26.3%

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

      10. metadata-eval 26.3%

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

      11. pow1/2 26.3%

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

      12. pow-flip 38.5%

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

      13. +-commutative 38.5%

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

      14. metadata-eval 38.5%

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

   3. Applied egg-rr 38.5%

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

      1. +-commutative 38.5%

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

      2. sub-neg 38.5%

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

      3. fma-udef 38.5%

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

      4. distribute-lft1-in 38.5%

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

      5. metadata-eval 38.5%

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

      6. mul0-lft 38.5%

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

      7. +-commutative 38.5%

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

      8. associate-+r+ 38.5%

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

      9. sub-neg 38.5%

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

      10. neg-sub0 38.5%

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

      11. +-commutative 38.5%

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

      12. sub-neg 38.5%

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

   5. Simplified 38.5%

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

      1. flip-- 38.5%

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

      2. pow-prod-up 21.6%

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

      3. metadata-eval 21.6%

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

      4. inv-pow 21.6%

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

      5. pow-prod-up 38.5%

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

      6. metadata-eval 38.5%

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

      7. inv-pow 38.5%

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

      8. +-commutative 38.5%

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

      9. +-commutative 38.5%

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

   7. Applied egg-rr 38.5%

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

   8. Taylor expanded in x around inf 66.3%

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

      1. unpow-1 66.3%

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

      2. exp-to-pow 64.1%

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

      3. *-commutative 64.1%

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

      4. log-pow 64.9%

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

      5. associate-*r* 64.9%

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

      6. metadata-eval 64.9%

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

      7. *-commutative 64.9%

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

      8. exp-to-pow 67.2%

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

      9. metadata-eval 67.2%

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

      10. pow-sqr 67.3%

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

      11. rem-sqrt-square 99.9%

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

      12. rem-square-sqrt 99.5%

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

      13. fabs-sqr 99.5%

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

      14. rem-square-sqrt 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 108000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 7: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      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. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul0-lft 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+r+ 100.0%

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

      9. sub-neg 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 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 < x

   1. Initial program 40.0%

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

      1. *-un-lft-identity 40.0%

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

      2. clear-num 40.0%

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

      3. associate-/r/ 40.0%

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

      4. prod-diff 40.0%

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

      5. *-un-lft-identity 40.0%

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

      6. fma-neg 40.0%

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

      7. *-un-lft-identity 40.0%

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

      8. pow1/2 40.0%

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

      9. pow-flip 28.2%

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

      10. metadata-eval 28.2%

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

      11. pow1/2 28.2%

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

      12. pow-flip 39.9%

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

      13. +-commutative 39.9%

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

      14. metadata-eval 39.9%

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

   3. Applied egg-rr 39.9%

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

      1. +-commutative 39.9%

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

      2. sub-neg 39.9%

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

      3. fma-udef 39.9%

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

      4. distribute-lft1-in 39.9%

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

      5. metadata-eval 39.9%

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

      6. mul0-lft 39.9%

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

      7. +-commutative 39.9%

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

      8. associate-+r+ 39.9%

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

      9. sub-neg 39.9%

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

      10. neg-sub0 39.9%

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

      11. +-commutative 39.9%

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

      12. sub-neg 39.9%

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

   5. Simplified 39.9%

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

      1. flip-- 40.0%

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

      2. pow-prod-up 23.8%

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

      3. metadata-eval 23.8%

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

      4. inv-pow 23.8%

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

      5. pow-prod-up 40.1%

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

      6. metadata-eval 40.1%

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

      7. inv-pow 40.1%

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

      8. +-commutative 40.1%

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

      9. +-commutative 40.1%

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

   7. Applied egg-rr 40.1%

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

   8. Taylor expanded in x around inf 65.3%

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

      1. unpow-1 65.3%

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

      2. exp-to-pow 63.3%

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

      3. *-commutative 63.3%

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

      4. log-pow 64.0%

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

      5. associate-*r* 64.0%

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

      6. metadata-eval 64.0%

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

      7. *-commutative 64.0%

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

      8. exp-to-pow 66.3%

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

      9. metadata-eval 66.3%

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

      10. pow-sqr 66.3%

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

      11. rem-sqrt-square 97.7%

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

      12. rem-square-sqrt 97.3%

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

      13. fabs-sqr 97.3%

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

      14. rem-square-sqrt 97.7%

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

   10. Simplified 97.7%

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

4. Final simplification 98.0%

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

Alternative 8: 96.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.6%

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

      1. inv-pow 99.6%

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

      2. add-sqr-sqrt 99.1%

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

      3. unpow-prod-down 99.0%

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

      4. pow1/2 99.0%

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

      5. sqrt-pow1 99.2%

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

      6. metadata-eval 99.2%

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

      7. pow1/2 99.2%

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

      8. sqrt-pow1 99.1%

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

      9. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{{\left({x}^{0.25}\right)}^{-1} \cdot {\left({x}^{0.25}\right)}^{-1}} - \frac{1}{\sqrt{x + 1}} \]
   4. Step-by-step derivation

      1. pow-sqr 99.2%

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

      2. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around inf 93.4%

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

      1. sqrt-div 94.1%

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

      2. metadata-eval 94.1%

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

      3. frac-2neg 94.1%

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

      4. metadata-eval 94.1%

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

      5. div-inv 94.1%

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

   8. Applied egg-rr 94.1%

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

      1. associate-*r/ 94.1%

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

      2. metadata-eval 94.1%

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

      3. neg-mul-1 94.1%

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

      4. associate-/r* 94.1%

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

      5. metadata-eval 94.1%

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

   10. Simplified 94.1%

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

   if 0.5 < x

   1. Initial program 40.9%

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

      1. *-un-lft-identity 40.9%

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

      2. clear-num 40.9%

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

      3. associate-/r/ 40.9%

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

      4. prod-diff 40.9%

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

      5. *-un-lft-identity 40.9%

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

      6. fma-neg 40.9%

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

      7. *-un-lft-identity 40.9%

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

      8. pow1/2 40.9%

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

      9. pow-flip 29.2%

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

      10. metadata-eval 29.2%

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

      11. pow1/2 29.2%

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

      12. pow-flip 40.8%

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

      13. +-commutative 40.8%

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

      14. metadata-eval 40.8%

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

   3. Applied egg-rr 40.8%

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

      1. +-commutative 40.8%

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

      2. sub-neg 40.8%

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

      3. fma-udef 40.8%

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

      4. distribute-lft1-in 40.8%

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

      5. metadata-eval 40.8%

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

      6. mul0-lft 40.8%

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

      7. +-commutative 40.8%

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

      8. associate-+r+ 40.8%

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

      9. sub-neg 40.8%

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

      10. neg-sub0 40.8%

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

      11. +-commutative 40.8%

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

      12. sub-neg 40.8%

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

   5. Simplified 40.8%

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

      1. flip-- 40.9%

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

      2. pow-prod-up 24.9%

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

      3. metadata-eval 24.9%

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

      4. inv-pow 24.9%

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

      5. pow-prod-up 40.9%

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

      6. metadata-eval 40.9%

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

      7. inv-pow 40.9%

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

      8. +-commutative 40.9%

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

      9. +-commutative 40.9%

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

   7. Applied egg-rr 40.9%

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

   8. Taylor expanded in x around inf 64.7%

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

      1. unpow-1 64.7%

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

      2. exp-to-pow 62.6%

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

      3. *-commutative 62.6%

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

      4. log-pow 63.4%

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

      5. associate-*r* 63.4%

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

      6. metadata-eval 63.4%

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

      7. *-commutative 63.4%

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

      8. exp-to-pow 65.6%

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

      9. metadata-eval 65.6%

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

      10. pow-sqr 65.6%

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

      11. rem-sqrt-square 96.6%

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

      12. rem-square-sqrt 96.1%

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

      13. fabs-sqr 96.1%

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

      14. rem-square-sqrt 96.6%

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

   10. Simplified 96.6%

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

4. Final simplification 95.4%

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

Alternative 9: 98.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.68) (+ (pow x -0.5) -1.0) (* 0.5 (pow x -1.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 0.5 * pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = 0.5d0 * (x ** (-1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = 0.5 * Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.68:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = 0.5 * math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(0.5 * (x ^ -1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 0.5 * (x ^ -1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.68], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      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. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul0-lft 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+r+ 100.0%

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

      9. sub-neg 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 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 98.4%

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

   if 0.680000000000000049 < x

   1. Initial program 40.9%

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

      1. *-un-lft-identity 40.9%

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

      2. clear-num 40.9%

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

      3. associate-/r/ 40.9%

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

      4. prod-diff 40.9%

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

      5. *-un-lft-identity 40.9%

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

      6. fma-neg 40.9%

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

      7. *-un-lft-identity 40.9%

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

      8. pow1/2 40.9%

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

      9. pow-flip 29.2%

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

      10. metadata-eval 29.2%

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

      11. pow1/2 29.2%

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

      12. pow-flip 40.8%

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

      13. +-commutative 40.8%

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

      14. metadata-eval 40.8%

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

   3. Applied egg-rr 40.8%

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

      1. +-commutative 40.8%

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

      2. sub-neg 40.8%

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

      3. fma-udef 40.8%

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

      4. distribute-lft1-in 40.8%

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

      5. metadata-eval 40.8%

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

      6. mul0-lft 40.8%

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

      7. +-commutative 40.8%

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

      8. associate-+r+ 40.8%

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

      9. sub-neg 40.8%

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

      10. neg-sub0 40.8%

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

      11. +-commutative 40.8%

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

      12. sub-neg 40.8%

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

   5. Simplified 40.8%

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

      1. flip-- 40.9%

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

      2. pow-prod-up 24.9%

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

      3. metadata-eval 24.9%

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

      4. inv-pow 24.9%

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

      5. pow-prod-up 40.9%

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

      6. metadata-eval 40.9%

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

      7. inv-pow 40.9%

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

      8. +-commutative 40.9%

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

      9. +-commutative 40.9%

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

   7. Applied egg-rr 40.9%

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

   8. Taylor expanded in x around inf 64.7%

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

      1. unpow-1 64.7%

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

      2. exp-to-pow 62.6%

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

      3. *-commutative 62.6%

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

      4. log-pow 63.4%

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

      5. associate-*r* 63.4%

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

      6. metadata-eval 63.4%

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

      7. *-commutative 63.4%

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

      8. exp-to-pow 65.6%

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

      9. metadata-eval 65.6%

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

      10. pow-sqr 65.6%

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

      11. rem-sqrt-square 96.6%

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

      12. rem-square-sqrt 96.1%

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

      13. fabs-sqr 96.1%

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

      14. rem-square-sqrt 96.6%

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

   10. Simplified 96.6%

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

4. Final simplification 97.4%

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

Alternative 10: 2.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. add-exp-log 51.1%

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

   2. log-rec 51.1%

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

   3. pow1/2 51.1%

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

   4. log-pow 51.1%

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

   5. +-commutative 51.1%

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

   6. log1p-udef 51.1%

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

3. Applied egg-rr 51.1%

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

   1. distribute-lft-neg-in 51.1%

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

   2. metadata-eval 51.1%

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

5. Simplified 51.1%

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

6. Taylor expanded in x around inf 2.2%

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

   1. unpow1/2 2.2%

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

   2. sqr-pow 2.2%

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

   3. sqr-pow 2.2%

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

   4. rem-exp-log 2.2%

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

   5. exp-neg 2.2%

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

   6. exp-prod 2.2%

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

   7. distribute-lft-neg-out 2.2%

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

   8. distribute-rgt-neg-in 2.2%

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

   9. metadata-eval 2.2%

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

   10. exp-to-pow 2.2%

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

   11. neg-mul-1 2.2%

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

8. Simplified 2.2%

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

9. Final simplification 2.2%

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

Alternative 11: 11.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.1111111111111111}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (/ 0.1111111111111111 x)))

C

double code(double x) {
	return sqrt((0.1111111111111111 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((0.1111111111111111d0 / x))
end function

Java

public static double code(double x) {
	return Math.sqrt((0.1111111111111111 / x));
}

Python

def code(x):
	return math.sqrt((0.1111111111111111 / x))

Julia

function code(x)
	return sqrt(Float64(0.1111111111111111 / x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((0.1111111111111111 / x));
end

Wolfram

code[x_] := N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 68.2%

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

   1. sub-neg 68.2%

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

   2. flip3-+ 55.4%

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

   3. inv-pow 55.4%

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

   4. pow-pow 55.2%

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

   5. metadata-eval 55.2%

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

   6. distribute-neg-frac 55.2%

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

   7. metadata-eval 55.2%

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

   8. +-commutative 55.2%

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

   9. frac-times 55.3%

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

   10. metadata-eval 55.3%

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

   11. add-sqr-sqrt 55.1%

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

3. Applied egg-rr 55.1%

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

   1. cube-div 53.8%

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

   2. metadata-eval 53.8%

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

   3. cancel-sign-sub-inv 53.8%

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

   4. associate-*r/ 53.8%

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

   5. associate-*l/ 53.8%

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

   6. metadata-eval 53.8%

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

   7. associate-/l/ 53.8%

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

   8. rem-square-sqrt 53.8%

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

   9. associate-*r/ 53.8%

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

5. Simplified 53.8%

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

   1. expm1-log1p-u 51.2%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt{x}\right)}^{-3}\right)\right)} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]

   2. expm1-udef 51.0%

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

   3. sqrt-pow2 51.0%

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

   4. metadata-eval 51.0%

      \[\leadsto \frac{\left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-1.5}}\right)} - 1\right) + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]

7. Applied egg-rr 51.0%

   \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-1.5}\right)} - 1\right)} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]
8. Step-by-step derivation

   1. expm1-def 51.3%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-1.5}\right)\right)} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]

   2. expm1-log1p 54.1%

      \[\leadsto \frac{\color{blue}{{x}^{-1.5}} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]

9. Simplified 54.1%

   \[\leadsto \frac{\color{blue}{{x}^{-1.5}} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]

10. Taylor expanded in x around inf 11.4%

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
11. Step-by-step derivation

    1. *-commutative 11.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.3333333333333333} \]

12. Simplified 11.4%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.3333333333333333} \]
13. Step-by-step derivation

    1. add-sqr-sqrt 11.4%

       \[\leadsto \color{blue}{\sqrt{\sqrt{\frac{1}{x}} \cdot 0.3333333333333333} \cdot \sqrt{\sqrt{\frac{1}{x}} \cdot 0.3333333333333333}} \]

    2. sqrt-unprod 11.4%

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

    3. *-commutative 11.4%

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

    4. inv-pow 11.4%

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

    5. sqrt-pow1 11.4%

       \[\leadsto \sqrt{\left(\sqrt{\frac{1}{x}} \cdot 0.3333333333333333\right) \cdot \left(0.3333333333333333 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} \]

    6. metadata-eval 11.4%

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

    7. *-commutative 11.4%

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

    8. inv-pow 11.4%

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

    9. sqrt-pow1 11.4%

       \[\leadsto \sqrt{\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot 0.3333333333333333\right) \cdot \left({x}^{-0.5} \cdot 0.3333333333333333\right)} \]

    10. metadata-eval 11.4%

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

    11. swap-sqr 11.4%

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

    12. pow-prod-up 11.4%

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

    13. metadata-eval 11.4%

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

    14. inv-pow 11.4%

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

    15. metadata-eval 11.4%

        \[\leadsto \sqrt{\frac{1}{x} \cdot \color{blue}{0.1111111111111111}} \]

14. Applied egg-rr 11.4%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{x} \cdot 0.1111111111111111}} \]
15. Step-by-step derivation

    1. associate-*l/ 11.4%

       \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 0.1111111111111111}{x}}} \]

    2. metadata-eval 11.4%

       \[\leadsto \sqrt{\frac{\color{blue}{0.1111111111111111}}{x}} \]

16. Simplified 11.4%

    \[\leadsto \color{blue}{\sqrt{\frac{0.1111111111111111}{x}}} \]

17. Final simplification 11.4%

    \[\leadsto \sqrt{\frac{0.1111111111111111}{x}} \]

Alternative 12: 51.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. inv-pow 68.2%

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

   2. add-sqr-sqrt 59.8%

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

   3. unpow-prod-down 58.2%

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

   4. pow1/2 58.2%

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

   5. sqrt-pow1 59.0%

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

   6. metadata-eval 59.0%

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

   7. pow1/2 59.0%

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

   8. sqrt-pow1 57.8%

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

   9. metadata-eval 57.8%

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

3. Applied egg-rr 57.8%

   \[\leadsto \color{blue}{{\left({x}^{0.25}\right)}^{-1} \cdot {\left({x}^{0.25}\right)}^{-1}} - \frac{1}{\sqrt{x + 1}} \]
4. Step-by-step derivation

   1. pow-sqr 60.7%

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

   2. metadata-eval 60.7%

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

5. Simplified 60.7%

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

6. Taylor expanded in x around inf 46.5%

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

7. Final simplification 46.5%

   \[\leadsto \sqrt{\frac{1}{x}} \]

Alternative 13: 51.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. inv-pow 68.2%

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

   2. add-sqr-sqrt 59.8%

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

   3. unpow-prod-down 58.2%

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

   4. pow1/2 58.2%

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

   5. sqrt-pow1 59.0%

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

   6. metadata-eval 59.0%

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

   7. pow1/2 59.0%

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

   8. sqrt-pow1 57.8%

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

   9. metadata-eval 57.8%

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

3. Applied egg-rr 57.8%

   \[\leadsto \color{blue}{{\left({x}^{0.25}\right)}^{-1} \cdot {\left({x}^{0.25}\right)}^{-1}} - \frac{1}{\sqrt{x + 1}} \]
4. Step-by-step derivation

   1. pow-sqr 60.7%

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

   2. metadata-eval 60.7%

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

5. Simplified 60.7%

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

6. Taylor expanded in x around inf 46.5%

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

   1. sqrt-div 46.9%

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

   2. metadata-eval 46.9%

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

   3. frac-2neg 46.9%

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

   4. metadata-eval 46.9%

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

   5. div-inv 46.9%

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

8. Applied egg-rr 46.9%

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

   1. associate-*r/ 46.9%

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

   2. metadata-eval 46.9%

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

   3. neg-mul-1 46.9%

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

   4. associate-/r* 46.9%

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

   5. metadata-eval 46.9%

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

10. Simplified 46.9%

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

11. Final simplification 46.9%

    \[\leadsto \frac{1}{\sqrt{x}} \]

Alternative 14: 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 68.2%

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

2. Taylor expanded in x around 0 47.0%

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

3. Taylor expanded in x around inf 2.0%

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

4. Final simplification 2.0%

   \[\leadsto -1 \]

Developer target: 99.0% 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 2023334 
(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)))))