2isqrt (example 3.6)

Percentage Accurate: 69.0% → 99.7%

Time: 13.0s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{\left(0.5 + \left(x + \frac{0.0625}{{x}^{2}}\right)\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 5e-7)
     (/
      (/ 1.0 (+ (sqrt x) t_0))
      (- (+ 0.5 (+ x (/ 0.0625 (pow x 2.0)))) (/ 0.125 x)))
     (- (pow x -0.5) (pow (+ 1.0 x) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 5e-7) {
		tmp = (1.0 / (sqrt(x) + t_0)) / ((0.5 + (x + (0.0625 / pow(x, 2.0)))) - (0.125 / x));
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 5d-7) then
        tmp = (1.0d0 / (sqrt(x) + t_0)) / ((0.5d0 + (x + (0.0625d0 / (x ** 2.0d0)))) - (0.125d0 / x))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 5e-7) {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) / ((0.5 + (x + (0.0625 / Math.pow(x, 2.0)))) - (0.125 / x));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / t_0)) <= 5e-7:
		tmp = (1.0 / (math.sqrt(x) + t_0)) / ((0.5 + (x + (0.0625 / math.pow(x, 2.0)))) - (0.125 / x))
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 5e-7)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) / Float64(Float64(0.5 + Float64(x + Float64(0.0625 / (x ^ 2.0)))) - Float64(0.125 / x)));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 5e-7)
		tmp = (1.0 / (sqrt(x) + t_0)) / ((0.5 + (x + (0.0625 / (x ^ 2.0)))) - (0.125 / x));
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 + N[(x + N[(0.0625 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 4.99999999999999977e-7

   1. Initial program 37.1%

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

      1. *-un-lft-identity 37.1%

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

      2. clear-num 37.1%

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

      3. associate-/r/ 37.1%

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

      4. prod-diff 37.1%

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

      5. *-un-lft-identity 37.1%

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

      6. fma-neg 37.1%

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

      7. *-un-lft-identity 37.1%

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

      8. pow1/2 37.1%

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

      9. pow-flip 31.0%

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

      10. metadata-eval 31.0%

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

      11. pow1/2 31.0%

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

      12. pow-flip 37.1%

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

      13. +-commutative 37.1%

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

      14. metadata-eval 37.1%

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

   4. Applied egg-rr 37.1%

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

      1. associate-+l- 37.1%

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

      2. expm1-log1p 37.1%

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

      3. expm1-def 6.0%

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

      4. associate--l- 6.0%

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

      5. fma-udef 6.0%

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

      6. distribute-lft1-in 6.0%

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

      7. metadata-eval 6.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 6.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 6.0%

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

      10. expm1-def 37.1%

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

      11. expm1-log1p 37.1%

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

   6. Simplified 37.1%

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

      1. metadata-eval 37.1%

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

      2. sqrt-pow2 31.2%

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

      3. inv-pow 31.2%

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

      4. metadata-eval 31.2%

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

      5. pow-flip 37.1%

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

      6. metadata-eval 37.1%

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

      7. pow-pow 5.4%

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

      8. pow1/3 7.6%

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

      9. frac-sub 6.3%

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

      10. *-un-lft-identity 6.3%

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

      11. pow1/3 4.1%

          \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

      12. pow-pow 37.2%

          \[\leadsto \frac{\color{blue}{{\left(1 + x\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

      13. metadata-eval 37.2%

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

      14. pow1/2 37.2%

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

      15. +-commutative 37.2%

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

      16. *-rgt-identity 37.2%

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

      17. pow1/3 37.2%

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

      18. pow-pow 37.2%

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

      19. metadata-eval 37.2%

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

      20. pow1/2 37.2%

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

      21. sqrt-unprod 37.2%

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

   8. Applied egg-rr 37.2%

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

      1. flip-- 38.6%

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

      2. div-inv 38.6%

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

      3. add-sqr-sqrt 39.4%

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

      4. +-commutative 39.4%

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

      5. add-sqr-sqrt 40.4%

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

      6. associate--l+ 84.5%

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

   10. Applied egg-rr 84.5%

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

       1. +-inverses 84.5%

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

       2. metadata-eval 84.5%

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

       3. *-lft-identity 84.5%

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

       4. +-commutative 84.5%

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

   12. Simplified 84.5%

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

   13. Taylor expanded in x around inf 99.6%

       \[\leadsto \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{\left(0.5 + \left(x + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right) - 0.125 \cdot \frac{1}{x}}} \]

   15. Simplified 99.6%

       \[\leadsto \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{\left(0.5 + \left(x + \frac{0.0625}{{x}^{2}}\right)\right) - \frac{0.125}{x}}} \]

   if 4.99999999999999977e-7 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. pow1/2 99.5%

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

      9. pow-flip 99.9%

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

      10. metadata-eval 99.9%

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

      11. pow1/2 99.9%

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

      12. pow-flip 99.9%

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

      13. +-commutative 99.9%

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

      14. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-+l- 99.9%

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

      2. expm1-log1p 99.9%

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

      3. expm1-def 99.8%

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

      4. associate--l- 99.8%

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

      5. fma-udef 99.8%

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

      6. distribute-lft1-in 99.8%

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

      7. metadata-eval 99.8%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 99.8%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 99.8%

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

      10. expm1-def 99.9%

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

      11. expm1-log1p 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\left(0.5 + \left(x + \frac{0.0625}{{x}^{2}}\right)\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{x + \left(0.5 + \frac{-0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 5e-7)
     (/ (/ 1.0 (+ (sqrt x) t_0)) (+ x (+ 0.5 (/ -0.125 x))))
     (- (pow x -0.5) (pow (+ 1.0 x) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 5e-7) {
		tmp = (1.0 / (sqrt(x) + t_0)) / (x + (0.5 + (-0.125 / x)));
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 5d-7) then
        tmp = (1.0d0 / (sqrt(x) + t_0)) / (x + (0.5d0 + ((-0.125d0) / x)))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 5e-7) {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) / (x + (0.5 + (-0.125 / x)));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / t_0)) <= 5e-7:
		tmp = (1.0 / (math.sqrt(x) + t_0)) / (x + (0.5 + (-0.125 / x)))
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 5e-7)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) / Float64(x + Float64(0.5 + Float64(-0.125 / x))));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 5e-7)
		tmp = (1.0 / (sqrt(x) + t_0)) / (x + (0.5 + (-0.125 / x)));
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + N[(0.5 + N[(-0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 4.99999999999999977e-7

   1. Initial program 37.1%

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

      1. *-un-lft-identity 37.1%

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

      2. clear-num 37.1%

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

      3. associate-/r/ 37.1%

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

      4. prod-diff 37.1%

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

      5. *-un-lft-identity 37.1%

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

      6. fma-neg 37.1%

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

      7. *-un-lft-identity 37.1%

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

      8. pow1/2 37.1%

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

      9. pow-flip 31.0%

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

      10. metadata-eval 31.0%

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

      11. pow1/2 31.0%

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

      12. pow-flip 37.1%

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

      13. +-commutative 37.1%

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

      14. metadata-eval 37.1%

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

   4. Applied egg-rr 37.1%

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

      1. associate-+l- 37.1%

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

      2. expm1-log1p 37.1%

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

      3. expm1-def 6.0%

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

      4. associate--l- 6.0%

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

      5. fma-udef 6.0%

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

      6. distribute-lft1-in 6.0%

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

      7. metadata-eval 6.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 6.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 6.0%

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

      10. expm1-def 37.1%

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

      11. expm1-log1p 37.1%

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

   6. Simplified 37.1%

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

      1. metadata-eval 37.1%

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

      2. sqrt-pow2 31.2%

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

      3. inv-pow 31.2%

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

      4. metadata-eval 31.2%

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

      5. pow-flip 37.1%

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

      6. metadata-eval 37.1%

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

      7. pow-pow 5.4%

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

      8. pow1/3 7.6%

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

      9. frac-sub 6.3%

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

      10. *-un-lft-identity 6.3%

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

      11. pow1/3 4.1%

          \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

      12. pow-pow 37.2%

          \[\leadsto \frac{\color{blue}{{\left(1 + x\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

      13. metadata-eval 37.2%

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

      14. pow1/2 37.2%

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

      15. +-commutative 37.2%

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

      16. *-rgt-identity 37.2%

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

      17. pow1/3 37.2%

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

      18. pow-pow 37.2%

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

      19. metadata-eval 37.2%

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

      20. pow1/2 37.2%

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

      21. sqrt-unprod 37.2%

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

   8. Applied egg-rr 37.2%

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

      1. flip-- 38.6%

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

      2. div-inv 38.6%

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

      3. add-sqr-sqrt 39.4%

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

      4. +-commutative 39.4%

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

      5. add-sqr-sqrt 40.4%

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

      6. associate--l+ 84.5%

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

   10. Applied egg-rr 84.5%

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

       1. +-inverses 84.5%

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

       2. metadata-eval 84.5%

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

       3. *-lft-identity 84.5%

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

       4. +-commutative 84.5%

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

   12. Simplified 84.5%

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

   13. Taylor expanded in x around inf 99.5%

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

       1. sub-neg 99.5%

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

       2. +-commutative 99.5%

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

       3. associate-+l+ 99.5%

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

       4. associate-*r/ 99.5%

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

       5. metadata-eval 99.5%

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

       6. distribute-neg-frac 99.5%

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

       7. metadata-eval 99.5%

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

   15. Simplified 99.5%

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

   if 4.99999999999999977e-7 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. pow1/2 99.5%

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

      9. pow-flip 99.9%

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

      10. metadata-eval 99.9%

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

      11. pow1/2 99.9%

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

      12. pow-flip 99.9%

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

      13. +-commutative 99.9%

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

      14. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-+l- 99.9%

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

      2. expm1-log1p 99.9%

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

      3. expm1-def 99.8%

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

      4. associate--l- 99.8%

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

      5. fma-udef 99.8%

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

      6. distribute-lft1-in 99.8%

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

      7. metadata-eval 99.8%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 99.8%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 99.8%

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

      10. expm1-def 99.9%

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

      11. expm1-log1p 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x + \left(0.5 + \frac{-0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 36.7%

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

      1. *-un-lft-identity 36.7%

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

      2. clear-num 36.7%

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

      3. associate-/r/ 36.7%

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

      4. prod-diff 36.7%

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

      5. *-un-lft-identity 36.7%

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

      6. fma-neg 36.7%

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

      7. *-un-lft-identity 36.7%

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

      8. pow1/2 36.7%

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

      9. pow-flip 30.5%

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

      10. metadata-eval 30.5%

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

      11. pow1/2 30.5%

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

      12. pow-flip 36.7%

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

      13. +-commutative 36.7%

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

      14. metadata-eval 36.7%

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

   4. Applied egg-rr 36.7%

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

      1. associate-+l- 36.7%

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

      2. expm1-log1p 36.7%

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

      3. expm1-def 5.5%

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

      4. associate--l- 5.5%

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

      5. fma-udef 5.5%

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

      6. distribute-lft1-in 5.5%

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

      7. metadata-eval 5.5%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 5.5%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 5.5%

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

      10. expm1-def 36.7%

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

      11. expm1-log1p 36.7%

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

   6. Simplified 36.7%

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

      1. metadata-eval 36.7%

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

      2. sqrt-pow2 30.7%

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

      3. inv-pow 30.7%

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

      4. metadata-eval 30.7%

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

      5. pow-flip 36.7%

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

      6. metadata-eval 36.7%

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

      7. pow-pow 4.8%

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

      8. pow1/3 6.9%

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

      9. frac-sub 5.6%

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

      10. *-un-lft-identity 5.6%

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

      11. pow1/3 3.4%

          \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

      12. pow-pow 36.7%

          \[\leadsto \frac{\color{blue}{{\left(1 + x\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

      13. metadata-eval 36.7%

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

      14. pow1/2 36.7%

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

      15. +-commutative 36.7%

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

      16. *-rgt-identity 36.7%

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

      17. pow1/3 36.7%

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

      18. pow-pow 36.7%

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

      19. metadata-eval 36.7%

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

      20. pow1/2 36.7%

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

      21. sqrt-unprod 36.7%

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

   8. Applied egg-rr 36.7%

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

      1. flip-- 38.0%

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

      2. div-inv 38.0%

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

      3. add-sqr-sqrt 38.9%

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

      4. +-commutative 38.9%

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

      5. add-sqr-sqrt 39.9%

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

      6. associate--l+ 84.4%

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

   10. Applied egg-rr 84.4%

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

       1. +-inverses 84.4%

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

       2. metadata-eval 84.4%

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

       3. *-lft-identity 84.4%

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

       4. +-commutative 84.4%

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

   12. Simplified 84.4%

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

   13. Taylor expanded in x around inf 99.6%

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

       1. +-commutative 7.8%

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

   15. Simplified 99.6%

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

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

   1. Initial program 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. clear-num 99.4%

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

      3. associate-/r/ 99.4%

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

      4. prod-diff 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. fma-neg 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. pow1/2 99.4%

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

      9. pow-flip 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow1/2 99.8%

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

      12. pow-flip 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-+l- 99.8%

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

      2. expm1-log1p 99.8%

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

      3. expm1-def 99.6%

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

      4. associate--l- 99.6%

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

      5. fma-udef 99.6%

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

      6. distribute-lft1-in 99.6%

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

      7. metadata-eval 99.6%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 99.6%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 99.6%

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

      10. expm1-def 99.8%

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

      11. expm1-log1p 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ 1.0 x)))) 5e-13)
   (* 0.5 (pow x -1.5))
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 5e-13) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) - (1.0d0 / sqrt((1.0d0 + x)))) <= 5d-13) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((1.0 + x)))) <= 5e-13) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / math.sqrt((1.0 + x)))) <= 5e-13:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(1.0 + x)))) <= 5e-13)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 5e-13)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-13], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 36.7%

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

      1. *-un-lft-identity 36.7%

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

      2. clear-num 36.7%

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

      3. associate-/r/ 36.7%

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

      4. prod-diff 36.7%

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

      5. *-un-lft-identity 36.7%

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

      6. fma-neg 36.7%

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

      7. *-un-lft-identity 36.7%

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

      8. pow1/2 36.7%

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

      9. pow-flip 30.5%

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

      10. metadata-eval 30.5%

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

      11. pow1/2 30.5%

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

      12. pow-flip 36.7%

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

      13. +-commutative 36.7%

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

      14. metadata-eval 36.7%

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

   4. Applied egg-rr 36.7%

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

      1. associate-+l- 36.7%

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

      2. expm1-log1p 36.7%

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

      3. expm1-def 5.5%

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

      4. associate--l- 5.5%

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

      5. fma-udef 5.5%

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

      6. distribute-lft1-in 5.5%

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

      7. metadata-eval 5.5%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 5.5%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 5.5%

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

      10. expm1-def 36.7%

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

      11. expm1-log1p 36.7%

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

   6. Simplified 36.7%

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

   7. Taylor expanded in x around inf 69.3%

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

      1. expm1-log1p-u 69.3%

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

      2. expm1-udef 35.3%

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

      3. pow-flip 35.3%

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

      4. sqrt-pow1 35.3%

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

      5. metadata-eval 35.3%

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

      6. metadata-eval 35.3%

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

   9. Applied egg-rr 35.3%

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

       1. expm1-def 98.9%

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

       2. expm1-log1p 98.9%

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

   11. Simplified 98.9%

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

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

   1. Initial program 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. clear-num 99.4%

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

      3. associate-/r/ 99.4%

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

      4. prod-diff 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. fma-neg 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. pow1/2 99.4%

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

      9. pow-flip 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow1/2 99.8%

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

      12. pow-flip 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-+l- 99.8%

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

      2. expm1-log1p 99.8%

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

      3. expm1-def 99.6%

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

      4. associate--l- 99.6%

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

      5. fma-udef 99.6%

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

      6. distribute-lft1-in 99.6%

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

      7. metadata-eval 99.6%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 99.6%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 99.6%

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

      10. expm1-def 99.8%

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

      11. expm1-log1p 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

   1. *-un-lft-identity 71.7%

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

   2. clear-num 71.7%

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

   3. associate-/r/ 71.7%

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

   4. prod-diff 71.7%

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

   5. *-un-lft-identity 71.7%

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

   6. fma-neg 71.7%

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

   7. *-un-lft-identity 71.7%

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

   8. pow1/2 71.7%

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

   9. pow-flip 69.2%

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

   10. metadata-eval 69.2%

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

   11. pow1/2 69.2%

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

   12. pow-flip 71.9%

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

   13. +-commutative 71.9%

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

   14. metadata-eval 71.9%

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

4. Applied egg-rr 71.9%

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

   1. associate-+l- 71.9%

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

   2. expm1-log1p 71.9%

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

   3. expm1-def 58.1%

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

   4. associate--l- 58.1%

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

   5. fma-udef 58.1%

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

   6. distribute-lft1-in 58.1%

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

   7. metadata-eval 58.1%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

   8. mul0-lft 58.1%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

   9. metadata-eval 58.1%

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

   10. expm1-def 71.9%

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

   11. expm1-log1p 71.9%

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

6. Simplified 71.9%

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

   1. metadata-eval 71.9%

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

   2. sqrt-pow2 69.1%

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

   3. inv-pow 69.1%

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

   4. metadata-eval 69.1%

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

   5. pow-flip 71.7%

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

   6. metadata-eval 71.7%

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

   7. pow-pow 57.6%

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

   8. pow1/3 58.6%

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

   9. frac-sub 58.0%

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

   10. *-un-lft-identity 58.0%

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

   11. pow1/3 56.9%

       \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

   12. pow-pow 71.7%

       \[\leadsto \frac{\color{blue}{{\left(1 + x\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

   13. metadata-eval 71.7%

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

   14. pow1/2 71.7%

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

   15. +-commutative 71.7%

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

   16. *-rgt-identity 71.7%

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

   17. pow1/3 71.7%

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

   18. pow-pow 71.7%

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

   19. metadata-eval 71.7%

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

   20. pow1/2 71.7%

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

   21. sqrt-unprod 71.7%

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

8. Applied egg-rr 71.7%

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

   1. flip-- 72.4%

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

   2. div-inv 72.4%

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

   3. add-sqr-sqrt 72.7%

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

   4. +-commutative 72.7%

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

   5. add-sqr-sqrt 73.2%

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

   6. associate--l+ 92.8%

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

10. Applied egg-rr 92.8%

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

    1. +-inverses 92.8%

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

    2. metadata-eval 92.8%

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

    3. *-lft-identity 92.8%

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

    4. +-commutative 92.8%

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

12. Simplified 92.8%

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

    1. expm1-log1p-u 89.1%

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

    2. expm1-udef 67.2%

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

    3. associate-/l/ 67.2%

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

    4. associate-/r* 67.2%

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

    5. distribute-rgt-in 67.2%

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

    6. *-un-lft-identity 67.2%

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

    7. add-sqr-sqrt 67.2%

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

    8. hypot-def 67.2%

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

14. Applied egg-rr 67.2%

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

    1. expm1-def 95.8%

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

    2. expm1-log1p 99.5%

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

    3. associate-/r* 99.1%

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

    4. +-commutative 99.1%

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

16. Simplified 99.1%

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

17. Final simplification 99.1%

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

Alternative 6: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64((x ^ -0.5) - Float64(1.0 + Float64(x * -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 <= 1.0)
		tmp = (x ^ -0.5) - (1.0 + (x * -0.5));
	else
		tmp = 0.5 * (x ^ -1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $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. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 98.9%

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

      1. *-commutative 98.9%

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

   9. Simplified 98.9%

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

   if 1 < x

   1. Initial program 37.5%

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

      1. *-un-lft-identity 37.5%

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

      2. clear-num 37.5%

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

      3. associate-/r/ 37.5%

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

      4. prod-diff 37.5%

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

      5. *-un-lft-identity 37.5%

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

      6. fma-neg 37.5%

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

      7. *-un-lft-identity 37.5%

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

      8. pow1/2 37.5%

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

      9. pow-flip 31.5%

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

      10. metadata-eval 31.5%

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

      11. pow1/2 31.5%

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

      12. pow-flip 37.6%

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

      13. +-commutative 37.6%

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

      14. metadata-eval 37.6%

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

   4. Applied egg-rr 37.6%

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

      1. associate-+l- 37.6%

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

      2. expm1-log1p 37.6%

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

      3. expm1-def 6.7%

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

      4. associate--l- 6.7%

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

      5. fma-udef 6.7%

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

      6. distribute-lft1-in 6.7%

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

      7. metadata-eval 6.7%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 6.7%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 6.7%

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

      10. expm1-def 37.6%

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

      11. expm1-log1p 37.6%

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

   6. Simplified 37.6%

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

   7. Taylor expanded in x around inf 68.7%

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

      1. expm1-log1p-u 68.7%

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

      2. expm1-udef 35.3%

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

      3. pow-flip 35.3%

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

      4. sqrt-pow1 35.3%

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

      5. metadata-eval 35.3%

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

      6. metadata-eval 35.3%

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

   9. Applied egg-rr 35.3%

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

       1. expm1-def 97.8%

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

       2. expm1-log1p 97.8%

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

   11. Simplified 97.8%

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

4. Final simplification 98.4%

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

Alternative 7: 96.7% accurate, 1.9× 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. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

      1. metadata-eval 100.0%

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

      2. sqrt-pow2 99.6%

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

      3. inv-pow 99.6%

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

      4. metadata-eval 99.6%

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

      5. pow-flip 99.6%

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

      6. metadata-eval 99.6%

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

      7. pow-pow 99.6%

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

      8. pow1/3 99.6%

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

      9. frac-sub 99.5%

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

      10. *-un-lft-identity 99.5%

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

      11. pow1/3 99.4%

          \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

      12. pow-pow 99.5%

          \[\leadsto \frac{\color{blue}{{\left(1 + x\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

      13. metadata-eval 99.5%

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

      14. pow1/2 99.5%

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

      15. +-commutative 99.5%

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

      16. *-rgt-identity 99.5%

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

      17. pow1/3 99.5%

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

      18. pow-pow 99.5%

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

      19. metadata-eval 99.5%

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

      20. pow1/2 99.5%

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

      21. sqrt-unprod 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in x around 0 93.7%

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

   10. Taylor expanded in x around 0 93.7%

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

   if 0.5 < x

   1. Initial program 37.5%

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

      1. *-un-lft-identity 37.5%

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

      2. clear-num 37.5%

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

      3. associate-/r/ 37.5%

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

      4. prod-diff 37.5%

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

      5. *-un-lft-identity 37.5%

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

      6. fma-neg 37.5%

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

      7. *-un-lft-identity 37.5%

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

      8. pow1/2 37.5%

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

      9. pow-flip 31.5%

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

      10. metadata-eval 31.5%

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

      11. pow1/2 31.5%

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

      12. pow-flip 37.6%

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

      13. +-commutative 37.6%

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

      14. metadata-eval 37.6%

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

   4. Applied egg-rr 37.6%

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

      1. associate-+l- 37.6%

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

      2. expm1-log1p 37.6%

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

      3. expm1-def 6.7%

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

      4. associate--l- 6.7%

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

      5. fma-udef 6.7%

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

      6. distribute-lft1-in 6.7%

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

      7. metadata-eval 6.7%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 6.7%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 6.7%

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

      10. expm1-def 37.6%

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

      11. expm1-log1p 37.6%

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

   6. Simplified 37.6%

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

   7. Taylor expanded in x around inf 68.7%

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

      1. expm1-log1p-u 68.7%

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

      2. expm1-udef 35.3%

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

      3. pow-flip 35.3%

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

      4. sqrt-pow1 35.3%

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

      5. metadata-eval 35.3%

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

      6. metadata-eval 35.3%

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

   9. Applied egg-rr 35.3%

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

       1. expm1-def 97.8%

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

       2. expm1-log1p 97.8%

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

   11. Simplified 97.8%

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

4. Final simplification 95.6%

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

Alternative 8: 98.4% accurate, 1.9× 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. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 97.9%

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

   if 0.680000000000000049 < x

   1. Initial program 37.5%

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

      1. *-un-lft-identity 37.5%

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

      2. clear-num 37.5%

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

      3. associate-/r/ 37.5%

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

      4. prod-diff 37.5%

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

      5. *-un-lft-identity 37.5%

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

      6. fma-neg 37.5%

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

      7. *-un-lft-identity 37.5%

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

      8. pow1/2 37.5%

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

      9. pow-flip 31.5%

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

      10. metadata-eval 31.5%

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

      11. pow1/2 31.5%

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

      12. pow-flip 37.6%

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

      13. +-commutative 37.6%

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

      14. metadata-eval 37.6%

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

   4. Applied egg-rr 37.6%

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

      1. associate-+l- 37.6%

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

      2. expm1-log1p 37.6%

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

      3. expm1-def 6.7%

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

      4. associate--l- 6.7%

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

      5. fma-udef 6.7%

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

      6. distribute-lft1-in 6.7%

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

      7. metadata-eval 6.7%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 6.7%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 6.7%

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

      10. expm1-def 37.6%

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

      11. expm1-log1p 37.6%

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

   6. Simplified 37.6%

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

   7. Taylor expanded in x around inf 68.7%

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

      1. expm1-log1p-u 68.7%

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

      2. expm1-udef 35.3%

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

      3. pow-flip 35.3%

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

      4. sqrt-pow1 35.3%

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

      5. metadata-eval 35.3%

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

      6. metadata-eval 35.3%

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

   9. Applied egg-rr 35.3%

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

       1. expm1-def 97.8%

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

       2. expm1-log1p 97.8%

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

   11. Simplified 97.8%

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

4. Final simplification 97.9%

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

Alternative 9: 51.1% 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 71.7%

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

   1. *-un-lft-identity 71.7%

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

   2. clear-num 71.7%

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

   3. associate-/r/ 71.7%

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

   4. prod-diff 71.7%

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

   5. *-un-lft-identity 71.7%

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

   6. fma-neg 71.7%

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

   7. *-un-lft-identity 71.7%

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

   8. pow1/2 71.7%

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

   9. pow-flip 69.2%

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

   10. metadata-eval 69.2%

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

   11. pow1/2 69.2%

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

   12. pow-flip 71.9%

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

   13. +-commutative 71.9%

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

   14. metadata-eval 71.9%

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

4. Applied egg-rr 71.9%

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

   1. associate-+l- 71.9%

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

   2. expm1-log1p 71.9%

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

   3. expm1-def 58.1%

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

   4. associate--l- 58.1%

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

   5. fma-udef 58.1%

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

   6. distribute-lft1-in 58.1%

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

   7. metadata-eval 58.1%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

   8. mul0-lft 58.1%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

   9. metadata-eval 58.1%

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

   10. expm1-def 71.9%

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

   11. expm1-log1p 71.9%

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

6. Simplified 71.9%

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

   1. metadata-eval 71.9%

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

   2. sqrt-pow2 69.1%

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

   3. inv-pow 69.1%

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

   4. metadata-eval 69.1%

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

   5. pow-flip 71.7%

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

   6. metadata-eval 71.7%

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

   7. pow-pow 57.6%

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

   8. pow1/3 58.6%

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

   9. frac-sub 58.0%

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

   10. *-un-lft-identity 58.0%

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

   11. pow1/3 56.9%

       \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

   12. pow-pow 71.7%

       \[\leadsto \frac{\color{blue}{{\left(1 + x\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

   13. metadata-eval 71.7%

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

   14. pow1/2 71.7%

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

   15. +-commutative 71.7%

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

   16. *-rgt-identity 71.7%

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

   17. pow1/3 71.7%

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

   18. pow-pow 71.7%

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

   19. metadata-eval 71.7%

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

   20. pow1/2 71.7%

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

   21. sqrt-unprod 71.7%

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

8. Applied egg-rr 71.7%

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

9. Taylor expanded in x around 0 68.0%

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

10. Taylor expanded in x around 0 54.2%

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

11. Final simplification 54.2%

    \[\leadsto \frac{1}{\sqrt{x}} \]
12. Add Preprocessing

Alternative 10: 7.4% accurate, 41.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ x 0.5)))

C

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

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (x + 0.5);
}

Python

def code(x):
	return 1.0 / (x + 0.5)

Julia

function code(x)
	return Float64(1.0 / Float64(x + 0.5))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

   1. *-un-lft-identity 71.7%

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

   2. clear-num 71.7%

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

   3. associate-/r/ 71.7%

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

   4. prod-diff 71.7%

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

   5. *-un-lft-identity 71.7%

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

   6. fma-neg 71.7%

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

   7. *-un-lft-identity 71.7%

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

   8. pow1/2 71.7%

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

   9. pow-flip 69.2%

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

   10. metadata-eval 69.2%

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

   11. pow1/2 69.2%

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

   12. pow-flip 71.9%

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

   13. +-commutative 71.9%

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

   14. metadata-eval 71.9%

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

4. Applied egg-rr 71.9%

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

   1. associate-+l- 71.9%

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

   2. expm1-log1p 71.9%

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

   3. expm1-def 58.1%

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

   4. associate--l- 58.1%

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

   5. fma-udef 58.1%

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

   6. distribute-lft1-in 58.1%

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

   7. metadata-eval 58.1%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

   8. mul0-lft 58.1%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

   9. metadata-eval 58.1%

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

   10. expm1-def 71.9%

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

   11. expm1-log1p 71.9%

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

6. Simplified 71.9%

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

   1. metadata-eval 71.9%

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

   2. sqrt-pow2 69.1%

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

   3. inv-pow 69.1%

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

   4. metadata-eval 69.1%

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

   5. pow-flip 71.7%

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

   6. metadata-eval 71.7%

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

   7. pow-pow 57.6%

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

   8. pow1/3 58.6%

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

   9. frac-sub 58.0%

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

   10. *-un-lft-identity 58.0%

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

   11. pow1/3 56.9%

       \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

   12. pow-pow 71.7%

       \[\leadsto \frac{\color{blue}{{\left(1 + x\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

   13. metadata-eval 71.7%

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

   14. pow1/2 71.7%

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

   15. +-commutative 71.7%

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

   16. *-rgt-identity 71.7%

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

   17. pow1/3 71.7%

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

   18. pow-pow 71.7%

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

   19. metadata-eval 71.7%

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

   20. pow1/2 71.7%

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

   21. sqrt-unprod 71.7%

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

8. Applied egg-rr 71.7%

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

9. Taylor expanded in x around 0 68.0%

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

10. Taylor expanded in x around inf 7.5%

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

    1. +-commutative 7.5%

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

12. Simplified 7.5%

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

13. Final simplification 7.5%

    \[\leadsto \frac{1}{x + 0.5} \]
14. Add Preprocessing

Alternative 11: 7.4% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

   1. *-un-lft-identity 71.7%

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

   2. clear-num 71.7%

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

   3. associate-/r/ 71.7%

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

   4. prod-diff 71.7%

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

   5. *-un-lft-identity 71.7%

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

   6. fma-neg 71.7%

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

   7. *-un-lft-identity 71.7%

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

   8. pow1/2 71.7%

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

   9. pow-flip 69.2%

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

   10. metadata-eval 69.2%

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

   11. pow1/2 69.2%

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

   12. pow-flip 71.9%

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

   13. +-commutative 71.9%

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

   14. metadata-eval 71.9%

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

4. Applied egg-rr 71.9%

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

   1. associate-+l- 71.9%

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

   2. expm1-log1p 71.9%

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

   3. expm1-def 58.1%

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

   4. associate--l- 58.1%

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

   5. fma-udef 58.1%

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

   6. distribute-lft1-in 58.1%

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

   7. metadata-eval 58.1%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

   8. mul0-lft 58.1%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

   9. metadata-eval 58.1%

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

   10. expm1-def 71.9%

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

   11. expm1-log1p 71.9%

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

6. Simplified 71.9%

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

   1. metadata-eval 71.9%

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

   2. sqrt-pow2 69.1%

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

   3. inv-pow 69.1%

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

   4. metadata-eval 69.1%

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

   5. pow-flip 71.7%

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

   6. metadata-eval 71.7%

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

   7. pow-pow 57.6%

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

   8. pow1/3 58.6%

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

   9. frac-sub 58.0%

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

   10. *-un-lft-identity 58.0%

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

   11. pow1/3 56.9%

       \[\leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

   12. pow-pow 71.7%

       \[\leadsto \frac{\color{blue}{{\left(1 + x\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt[3]{{\left(1 + x\right)}^{1.5}}} \]

   13. metadata-eval 71.7%

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

   14. pow1/2 71.7%

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

   15. +-commutative 71.7%

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

   16. *-rgt-identity 71.7%

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

   17. pow1/3 71.7%

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

   18. pow-pow 71.7%

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

   19. metadata-eval 71.7%

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

   20. pow1/2 71.7%

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

   21. sqrt-unprod 71.7%

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

8. Applied egg-rr 71.7%

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

9. Taylor expanded in x around 0 68.0%

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

10. Taylor expanded in x around inf 7.4%

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

11. Final simplification 7.4%

    \[\leadsto \frac{1}{x} \]
12. Add Preprocessing

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 2024023 
(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)))))