2isqrt (example 3.6)

Percentage Accurate: 69.2% → 99.6%

Time: 9.9s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{t_0 + \sqrt{x}}}{x + \left(0.5 + \frac{-0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 5e-11)
     (/ (/ 1.0 (+ t_0 (sqrt x))) (+ x (+ 0.5 (/ -0.125 x))))
     (- (pow x -0.5) (sqrt (/ 1.0 (+ 1.0 x)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-11) {
		tmp = (1.0 / (t_0 + sqrt(x))) / (x + (0.5 + (-0.125 / x)));
	} else {
		tmp = pow(x, -0.5) - sqrt((1.0 / (1.0 + x)));
	}
	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-11) then
        tmp = (1.0d0 / (t_0 + sqrt(x))) / (x + (0.5d0 + ((-0.125d0) / x)))
    else
        tmp = (x ** (-0.5d0)) - sqrt((1.0d0 / (1.0d0 + x)))
    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-11) {
		tmp = (1.0 / (t_0 + Math.sqrt(x))) / (x + (0.5 + (-0.125 / x)));
	} else {
		tmp = Math.pow(x, -0.5) - Math.sqrt((1.0 / (1.0 + x)));
	}
	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-11:
		tmp = (1.0 / (t_0 + math.sqrt(x))) / (x + (0.5 + (-0.125 / x)))
	else:
		tmp = math.pow(x, -0.5) - math.sqrt((1.0 / (1.0 + x)))
	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-11)
		tmp = Float64(Float64(1.0 / Float64(t_0 + sqrt(x))) / Float64(x + Float64(0.5 + Float64(-0.125 / x))));
	else
		tmp = Float64((x ^ -0.5) - sqrt(Float64(1.0 / Float64(1.0 + x))));
	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-11)
		tmp = (1.0 / (t_0 + sqrt(x))) / (x + (0.5 + (-0.125 / x)));
	else
		tmp = (x ^ -0.5) - sqrt((1.0 / (1.0 + x)));
	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-11], N[(N[(1.0 / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + N[(0.5 + N[(-0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Sqrt[N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.00000000000000018e-11

   1. Initial program 38.5%

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

      1. frac-sub 38.6%

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

      2. clear-num 38.6%

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

      3. sqrt-unprod 38.6%

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

      4. +-commutative 38.6%

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

      5. *-un-lft-identity 38.6%

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

      6. *-rgt-identity 38.6%

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

      7. +-commutative 38.6%

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

   3. Applied egg-rr 38.6%

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

      1. associate-/r/ 38.6%

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

      2. associate-*l/ 38.6%

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

      3. *-lft-identity 38.6%

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

      4. distribute-rgt-in 38.6%

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

      5. *-lft-identity 38.6%

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

   5. Simplified 38.6%

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

      1. flip-- 38.8%

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

      2. add-sqr-sqrt 39.5%

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

      3. add-sqr-sqrt 40.2%

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

   7. Applied egg-rr 40.2%

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

      1. associate--l+ 79.6%

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

      2. +-inverses 79.6%

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

      3. metadata-eval 79.6%

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

   9. Simplified 79.6%

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

   10. Taylor expanded in x around inf 99.6%

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

       1. sub-neg 99.6%

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

       2. +-commutative 99.6%

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

       3. associate-+l+ 99.6%

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

       4. associate-*r/ 99.6%

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

       5. metadata-eval 99.6%

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

       6. distribute-neg-frac 99.6%

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

       7. metadata-eval 99.6%

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

   12. Simplified 99.6%

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

   if 5.00000000000000018e-11 < (-.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. 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. inv-pow 99.5%

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

      9. sqrt-pow2 99.9%

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

      10. metadata-eval 99.9%

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

      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%

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

   3. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. distribute-lft1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. mul0-lft 99.9%

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

      5. +-rgt-identity 99.9%

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

   5. Simplified 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. sqrt-unprod 99.9%

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

      3. metadata-eval 99.9%

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

      4. sqrt-pow2 99.9%

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

      5. inv-pow 99.9%

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

      6. metadata-eval 99.9%

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

      7. sqrt-pow2 99.9%

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

      8. inv-pow 99.9%

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

      9. frac-times 99.9%

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

      10. metadata-eval 99.9%

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

      11. add-sqr-sqrt 99.9%

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

      12. +-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{x + \left(0.5 + \frac{-0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{1 + x}}\\ \end{array} \]

Alternative 2: 98.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

   1. frac-sub 72.6%

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

   2. clear-num 72.6%

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

   3. sqrt-unprod 72.6%

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

   4. +-commutative 72.6%

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

   5. *-un-lft-identity 72.6%

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

   6. *-rgt-identity 72.6%

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

   7. +-commutative 72.6%

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

3. Applied egg-rr 72.6%

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

   1. associate-/r/ 72.6%

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

   2. associate-*l/ 72.6%

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

   3. *-lft-identity 72.6%

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

   4. distribute-rgt-in 72.6%

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

   5. *-lft-identity 72.6%

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

5. Simplified 72.6%

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

   1. flip-- 72.7%

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

   2. add-sqr-sqrt 73.0%

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

   3. add-sqr-sqrt 73.3%

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

7. Applied egg-rr 73.3%

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

   1. associate--l+ 90.7%

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

   2. +-inverses 90.7%

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

   3. metadata-eval 90.7%

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

9. Simplified 90.7%

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

    1. associate-/l/ 90.7%

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

    2. inv-pow 90.7%

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

    3. *-commutative 90.7%

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

    4. +-commutative 90.7%

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

    5. add-sqr-sqrt 90.7%

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

    6. hypot-def 99.6%

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{t_0 + \sqrt{x}}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 5e-11)
     (/ (/ 1.0 (+ t_0 (sqrt x))) (+ x 0.5))
     (- (pow x -0.5) (sqrt (/ 1.0 (+ 1.0 x)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-11) {
		tmp = (1.0 / (t_0 + sqrt(x))) / (x + 0.5);
	} else {
		tmp = pow(x, -0.5) - sqrt((1.0 / (1.0 + x)));
	}
	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-11) then
        tmp = (1.0d0 / (t_0 + sqrt(x))) / (x + 0.5d0)
    else
        tmp = (x ** (-0.5d0)) - sqrt((1.0d0 / (1.0d0 + x)))
    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-11) {
		tmp = (1.0 / (t_0 + Math.sqrt(x))) / (x + 0.5);
	} else {
		tmp = Math.pow(x, -0.5) - Math.sqrt((1.0 / (1.0 + x)));
	}
	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-11:
		tmp = (1.0 / (t_0 + math.sqrt(x))) / (x + 0.5)
	else:
		tmp = math.pow(x, -0.5) - math.sqrt((1.0 / (1.0 + x)))
	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-11)
		tmp = Float64(Float64(1.0 / Float64(t_0 + sqrt(x))) / Float64(x + 0.5));
	else
		tmp = Float64((x ^ -0.5) - sqrt(Float64(1.0 / Float64(1.0 + x))));
	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-11)
		tmp = (1.0 / (t_0 + sqrt(x))) / (x + 0.5);
	else
		tmp = (x ^ -0.5) - sqrt((1.0 / (1.0 + x)));
	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-11], N[(N[(1.0 / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Sqrt[N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.00000000000000018e-11

   1. Initial program 38.5%

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

      1. frac-sub 38.6%

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

      2. clear-num 38.6%

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

      3. sqrt-unprod 38.6%

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

      4. +-commutative 38.6%

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

      5. *-un-lft-identity 38.6%

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

      6. *-rgt-identity 38.6%

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

      7. +-commutative 38.6%

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

   3. Applied egg-rr 38.6%

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

      1. associate-/r/ 38.6%

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

      2. associate-*l/ 38.6%

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

      3. *-lft-identity 38.6%

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

      4. distribute-rgt-in 38.6%

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

      5. *-lft-identity 38.6%

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

   5. Simplified 38.6%

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

      1. flip-- 38.8%

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

      2. add-sqr-sqrt 39.5%

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

      3. add-sqr-sqrt 40.2%

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

   7. Applied egg-rr 40.2%

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

      1. associate--l+ 79.6%

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

      2. +-inverses 79.6%

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

      3. metadata-eval 79.6%

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

   9. Simplified 79.6%

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

   10. Taylor expanded in x around inf 99.5%

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

       1. +-commutative 7.4%

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

   12. Simplified 99.5%

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

   if 5.00000000000000018e-11 < (-.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. 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. inv-pow 99.5%

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

      9. sqrt-pow2 99.9%

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

      10. metadata-eval 99.9%

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

      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%

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

   3. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. distribute-lft1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. mul0-lft 99.9%

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

      5. +-rgt-identity 99.9%

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

   5. Simplified 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. sqrt-unprod 99.9%

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

      3. metadata-eval 99.9%

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

      4. sqrt-pow2 99.9%

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

      5. inv-pow 99.9%

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

      6. metadata-eval 99.9%

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

      7. sqrt-pow2 99.9%

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

      8. inv-pow 99.9%

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

      9. frac-times 99.9%

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

      10. metadata-eval 99.9%

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

      11. add-sqr-sqrt 99.9%

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

      12. +-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{1 + x}}\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1}{t_0 + \sqrt{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 4e-14)
     (/ (/ 1.0 (+ t_0 (sqrt x))) x)
     (- (pow x -0.5) (sqrt (/ 1.0 (+ 1.0 x)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 4e-14) {
		tmp = (1.0 / (t_0 + sqrt(x))) / x;
	} else {
		tmp = pow(x, -0.5) - sqrt((1.0 / (1.0 + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 4d-14) then
        tmp = (1.0d0 / (t_0 + sqrt(x))) / x
    else
        tmp = (x ** (-0.5d0)) - sqrt((1.0d0 / (1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 4e-14) {
		tmp = (1.0 / (t_0 + Math.sqrt(x))) / x;
	} else {
		tmp = Math.pow(x, -0.5) - Math.sqrt((1.0 / (1.0 + x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 4e-14:
		tmp = (1.0 / (t_0 + math.sqrt(x))) / x
	else:
		tmp = math.pow(x, -0.5) - math.sqrt((1.0 / (1.0 + x)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 4e-14)
		tmp = Float64(Float64(1.0 / Float64(t_0 + sqrt(x))) / x);
	else
		tmp = Float64((x ^ -0.5) - sqrt(Float64(1.0 / Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 4e-14)
		tmp = (1.0 / (t_0 + sqrt(x))) / x;
	else
		tmp = (x ^ -0.5) - sqrt((1.0 / (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 4e-14], N[(N[(1.0 / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Sqrt[N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 4e-14

   1. Initial program 38.0%

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

      1. frac-sub 38.1%

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

      2. clear-num 38.1%

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

      3. sqrt-unprod 38.1%

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

      4. +-commutative 38.1%

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

      5. *-un-lft-identity 38.1%

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

      6. *-rgt-identity 38.1%

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

      7. +-commutative 38.1%

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

   3. Applied egg-rr 38.1%

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

      1. associate-/r/ 38.1%

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

      2. associate-*l/ 38.1%

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

      3. *-lft-identity 38.1%

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

      4. distribute-rgt-in 38.1%

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

      5. *-lft-identity 38.1%

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

   5. Simplified 38.1%

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

      1. flip-- 38.0%

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

      2. add-sqr-sqrt 38.4%

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

      3. add-sqr-sqrt 39.1%

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

   7. Applied egg-rr 39.1%

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

      1. associate--l+ 79.3%

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

      2. +-inverses 79.3%

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

      3. metadata-eval 79.3%

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

   9. Simplified 79.3%

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

   10. Taylor expanded in x around inf 99.2%

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

   if 4e-14 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.0%

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

      1. *-un-lft-identity 99.0%

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

      2. clear-num 99.0%

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

      3. associate-/r/ 99.0%

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

      4. prod-diff 99.0%

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

      5. *-un-lft-identity 99.0%

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

      6. fma-neg 99.0%

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

      7. *-un-lft-identity 99.0%

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

      8. inv-pow 99.1%

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

      9. sqrt-pow2 99.4%

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

      10. metadata-eval 99.4%

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

      11. pow1/2 99.4%

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

      12. pow-flip 99.4%

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

      13. +-commutative 99.4%

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

      14. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-udef 99.4%

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

      2. distribute-lft1-in 99.4%

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

      3. metadata-eval 99.4%

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

      4. mul0-lft 99.4%

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

      5. +-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

      1. add-sqr-sqrt 99.4%

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

      2. sqrt-unprod 99.4%

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

      3. metadata-eval 99.4%

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

      4. sqrt-pow2 99.4%

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

      5. inv-pow 99.4%

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

      6. metadata-eval 99.4%

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

      7. sqrt-pow2 99.4%

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

      8. inv-pow 99.4%

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

      9. frac-times 99.4%

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

      10. metadata-eval 99.4%

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

      11. add-sqr-sqrt 99.4%

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

      12. +-commutative 99.4%

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

   7. Applied egg-rr 99.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{1 + x}}\\ \end{array} \]

Alternative 5: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

   1. frac-sub 72.6%

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

   2. clear-num 72.6%

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

   3. sqrt-unprod 72.6%

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

   4. +-commutative 72.6%

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

   5. *-un-lft-identity 72.6%

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

   6. *-rgt-identity 72.6%

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

   7. +-commutative 72.6%

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

3. Applied egg-rr 72.6%

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

   1. associate-/r/ 72.6%

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

   2. associate-*l/ 72.6%

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

   3. *-lft-identity 72.6%

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

   4. distribute-rgt-in 72.6%

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

   5. *-lft-identity 72.6%

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

5. Simplified 72.6%

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

   1. flip-- 72.7%

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

   2. add-sqr-sqrt 73.0%

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

   3. add-sqr-sqrt 73.3%

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

7. Applied egg-rr 73.3%

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

   1. associate--l+ 90.7%

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

   2. +-inverses 90.7%

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

   3. metadata-eval 90.7%

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

9. Simplified 90.7%

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

    1. frac-2neg 90.7%

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

    2. div-inv 90.7%

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

    3. distribute-neg-frac 90.7%

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

    4. metadata-eval 90.7%

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

    5. +-commutative 90.7%

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

    6. add-sqr-sqrt 90.7%

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

    7. hypot-def 99.5%

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

11. Applied egg-rr 99.5%

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

    1. associate-*l/ 99.5%

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

    2. associate-*r/ 99.5%

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

    3. metadata-eval 99.5%

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

    4. neg-mul-1 99.5%

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

    5. associate-/r* 99.5%

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

    6. metadata-eval 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 6: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 110000000:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 110000000.0)
   (- (pow x -0.5) (sqrt (/ 1.0 (+ 1.0 x))))
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 110000000.0) {
		tmp = pow(x, -0.5) - sqrt((1.0 / (1.0 + x)));
	} else {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 110000000.0d0) then
        tmp = (x ** (-0.5d0)) - sqrt((1.0d0 / (1.0d0 + x)))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 110000000.0) {
		tmp = Math.pow(x, -0.5) - Math.sqrt((1.0 / (1.0 + x)));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 110000000.0:
		tmp = math.pow(x, -0.5) - math.sqrt((1.0 / (1.0 + x)))
	else:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 110000000.0)
		tmp = Float64((x ^ -0.5) - sqrt(Float64(1.0 / Float64(1.0 + x))));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 110000000.0)
		tmp = (x ^ -0.5) - sqrt((1.0 / (1.0 + x)));
	else
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 110000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Sqrt[N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1e8

   1. Initial program 99.0%

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

      1. *-un-lft-identity 99.0%

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

      2. clear-num 99.0%

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

      3. associate-/r/ 99.0%

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

      4. prod-diff 99.0%

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

      5. *-un-lft-identity 99.0%

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

      6. fma-neg 99.0%

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

      7. *-un-lft-identity 99.0%

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

      8. inv-pow 99.1%

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

      9. sqrt-pow2 99.4%

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

      10. metadata-eval 99.4%

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

      11. pow1/2 99.4%

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

      12. pow-flip 99.4%

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

      13. +-commutative 99.4%

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

      14. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-udef 99.4%

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

      2. distribute-lft1-in 99.4%

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

      3. metadata-eval 99.4%

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

      4. mul0-lft 99.4%

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

      5. +-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

      1. add-sqr-sqrt 99.4%

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

      2. sqrt-unprod 99.4%

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

      3. metadata-eval 99.4%

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

      4. sqrt-pow2 99.4%

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

      5. inv-pow 99.4%

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

      6. metadata-eval 99.4%

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

      7. sqrt-pow2 99.4%

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

      8. inv-pow 99.4%

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

      9. frac-times 99.4%

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

      10. metadata-eval 99.4%

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

      11. add-sqr-sqrt 99.4%

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

      12. +-commutative 99.4%

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

   7. Applied egg-rr 99.4%

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

   if 1.1e8 < x

   1. Initial program 38.0%

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

      1. *-un-lft-identity 38.0%

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

      2. clear-num 38.0%

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

      3. associate-/r/ 38.0%

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

      4. prod-diff 38.0%

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

      5. *-un-lft-identity 38.0%

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

      6. fma-neg 38.0%

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

      7. *-un-lft-identity 38.0%

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

      8. inv-pow 38.0%

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

      9. sqrt-pow2 31.7%

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

      10. metadata-eval 31.7%

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

      11. pow1/2 31.7%

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

      12. pow-flip 38.1%

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

      13. +-commutative 38.1%

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

      14. metadata-eval 38.1%

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

   3. Applied egg-rr 38.1%

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

      1. fma-udef 38.1%

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

      2. distribute-lft1-in 38.1%

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

      3. metadata-eval 38.1%

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

      4. mul0-lft 38.1%

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

      5. +-rgt-identity 38.1%

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

   5. Simplified 38.1%

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

   6. Taylor expanded in x around inf 62.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 110000000:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 7: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 90000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 90000000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 90000000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 90000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 90000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 90000000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 90000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 90000000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 90000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9e7

   1. Initial program 99.0%

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

      1. *-un-lft-identity 99.0%

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

      2. clear-num 99.0%

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

      3. associate-/r/ 99.0%

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

      4. prod-diff 99.0%

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

      5. *-un-lft-identity 99.0%

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

      6. fma-neg 99.0%

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

      7. *-un-lft-identity 99.0%

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

      8. inv-pow 99.1%

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

      9. sqrt-pow2 99.4%

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

      10. metadata-eval 99.4%

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

      11. pow1/2 99.4%

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

      12. pow-flip 99.4%

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

      13. +-commutative 99.4%

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

      14. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-udef 99.4%

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

      2. distribute-lft1-in 99.4%

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

      3. metadata-eval 99.4%

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

      4. mul0-lft 99.4%

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

      5. +-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   if 9e7 < x

   1. Initial program 38.0%

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

      1. *-un-lft-identity 38.0%

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

      2. clear-num 38.0%

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

      3. associate-/r/ 38.0%

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

      4. prod-diff 38.0%

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

      5. *-un-lft-identity 38.0%

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

      6. fma-neg 38.0%

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

      7. *-un-lft-identity 38.0%

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

      8. inv-pow 38.0%

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

      9. sqrt-pow2 31.7%

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

      10. metadata-eval 31.7%

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

      11. pow1/2 31.7%

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

      12. pow-flip 38.1%

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

      13. +-commutative 38.1%

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

      14. metadata-eval 38.1%

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

   3. Applied egg-rr 38.1%

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

      1. fma-udef 38.1%

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

      2. distribute-lft1-in 38.1%

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

      3. metadata-eval 38.1%

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

      4. mul0-lft 38.1%

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

      5. +-rgt-identity 38.1%

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

   5. Simplified 38.1%

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

   6. Taylor expanded in x around inf 62.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 90000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 8: 82.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around 0 99.2%

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

      1. add-log-exp 4.2%

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

      2. *-un-lft-identity 4.2%

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

      3. log-prod 4.2%

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

      4. metadata-eval 4.2%

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

      5. add-log-exp 99.2%

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

      6. pow1/2 99.2%

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

      7. pow-flip 99.6%

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

      8. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-lft-identity 99.6%

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

   6. Simplified 99.6%

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

   if 1.69999999999999996 < x

   1. Initial program 39.0%

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

      1. *-un-lft-identity 39.0%

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

      2. clear-num 39.0%

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

      3. associate-/r/ 39.0%

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

      4. prod-diff 39.0%

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

      5. *-un-lft-identity 39.0%

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

      6. fma-neg 39.0%

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

      7. *-un-lft-identity 39.0%

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

      8. inv-pow 39.0%

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

      9. sqrt-pow2 32.8%

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

      10. metadata-eval 32.8%

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

      11. pow1/2 32.8%

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

      12. pow-flip 39.0%

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

      13. +-commutative 39.0%

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

      14. metadata-eval 39.0%

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

   3. Applied egg-rr 39.0%

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

      1. fma-udef 39.0%

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

      2. distribute-lft1-in 39.0%

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

      3. metadata-eval 39.0%

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

      4. mul0-lft 39.0%

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

      5. +-rgt-identity 39.0%

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

   5. Simplified 39.0%

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

   6. Taylor expanded in x around inf 61.6%

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

4. Final simplification 82.7%

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

Alternative 9: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(x, x, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.8)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (sqrt (/ 1.0 (fma x x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 5.8) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = sqrt((1.0 / fma(x, x, x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.8)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = sqrt(Float64(1.0 / fma(x, x, x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.8], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(1.0 / N[(x * x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.79999999999999982

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around 0 99.2%

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

      1. add-log-exp 4.2%

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

      2. *-un-lft-identity 4.2%

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

      3. log-prod 4.2%

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

      4. metadata-eval 4.2%

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

      5. add-log-exp 99.2%

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

      6. pow1/2 99.2%

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

      7. pow-flip 99.6%

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

      8. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-lft-identity 99.6%

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

   6. Simplified 99.6%

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

   if 5.79999999999999982 < x

   1. Initial program 39.0%

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

      1. frac-sub 39.0%

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

      2. clear-num 39.0%

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

      3. sqrt-unprod 39.0%

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

      4. +-commutative 39.0%

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

      5. *-un-lft-identity 39.0%

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

      6. *-rgt-identity 39.0%

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

      7. +-commutative 39.0%

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

   3. Applied egg-rr 39.0%

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

      1. associate-/r/ 39.0%

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

      2. associate-*l/ 39.0%

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

      3. *-lft-identity 39.0%

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

      4. distribute-rgt-in 39.0%

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

      5. *-lft-identity 39.0%

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

   5. Simplified 39.0%

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

   6. Taylor expanded in x around 0 38.1%

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

      1. add-sqr-sqrt 38.1%

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

      2. sqrt-unprod 38.1%

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

      3. frac-times 38.1%

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

      4. metadata-eval 38.1%

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

      5. add-sqr-sqrt 38.1%

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

      6. +-commutative 38.1%

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

      7. fma-def 38.1%

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

   8. Applied egg-rr 38.1%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(x, x, x\right)}}\\ \end{array} \]

Alternative 10: 68.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.8)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (/ 1.0 (sqrt (+ x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.8) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 1.0 / sqrt((x + (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.8d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = 1.0d0 / sqrt((x + (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.8) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 1.0 / Math.sqrt((x + (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.8:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = 1.0 / math.sqrt((x + (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.8)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(1.0 / sqrt(Float64(x + Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.8)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = 1.0 / sqrt((x + (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.8], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.79999999999999982

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around 0 99.2%

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

      1. add-log-exp 4.2%

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

      2. *-un-lft-identity 4.2%

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

      3. log-prod 4.2%

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

      4. metadata-eval 4.2%

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

      5. add-log-exp 99.2%

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

      6. pow1/2 99.2%

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

      7. pow-flip 99.6%

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

      8. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-lft-identity 99.6%

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

   6. Simplified 99.6%

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

   if 5.79999999999999982 < x

   1. Initial program 39.0%

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

      1. frac-sub 39.0%

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

      2. clear-num 39.0%

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

      3. sqrt-unprod 39.0%

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

      4. +-commutative 39.0%

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

      5. *-un-lft-identity 39.0%

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

      6. *-rgt-identity 39.0%

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

      7. +-commutative 39.0%

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

   3. Applied egg-rr 39.0%

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

      1. associate-/r/ 39.0%

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

      2. associate-*l/ 39.0%

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

      3. *-lft-identity 39.0%

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

      4. distribute-rgt-in 39.0%

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

      5. *-lft-identity 39.0%

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

   5. Simplified 39.0%

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

   6. Taylor expanded in x around 0 38.1%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x}}\\ \end{array} \]

Alternative 11: 68.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.6%

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

   if 1.3999999999999999 < x

   1. Initial program 39.0%

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

      1. frac-sub 39.0%

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

      2. clear-num 39.0%

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

      3. sqrt-unprod 39.0%

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

      4. +-commutative 39.0%

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

      5. *-un-lft-identity 39.0%

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

      6. *-rgt-identity 39.0%

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

      7. +-commutative 39.0%

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

   3. Applied egg-rr 39.0%

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

      1. associate-/r/ 39.0%

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

      2. associate-*l/ 39.0%

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

      3. *-lft-identity 39.0%

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

      4. distribute-rgt-in 39.0%

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

      5. *-lft-identity 39.0%

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

   5. Simplified 39.0%

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

   6. Taylor expanded in x around 0 38.1%

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

4. Final simplification 72.2%

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

Alternative 12: 68.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.0%

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

   if 0.69999999999999996 < x

   1. Initial program 39.0%

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

      1. frac-sub 39.0%

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

      2. clear-num 39.0%

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

      3. sqrt-unprod 39.0%

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

      4. +-commutative 39.0%

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

      5. *-un-lft-identity 39.0%

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

      6. *-rgt-identity 39.0%

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

      7. +-commutative 39.0%

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

   3. Applied egg-rr 39.0%

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

      1. associate-/r/ 39.0%

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

      2. associate-*l/ 39.0%

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

      3. *-lft-identity 39.0%

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

      4. distribute-rgt-in 39.0%

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

      5. *-lft-identity 39.0%

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

   5. Simplified 39.0%

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

   6. Taylor expanded in x around 0 38.1%

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

4. Final simplification 71.9%

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

Alternative 13: 68.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.8) (+ -1.0 (pow x -0.5)) (/ 1.0 (sqrt (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.8) {
		tmp = -1.0 + pow(x, -0.5);
	} else {
		tmp = 1.0 / sqrt((x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.8d0) then
        tmp = (-1.0d0) + (x ** (-0.5d0))
    else
        tmp = 1.0d0 / sqrt((x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.8) {
		tmp = -1.0 + Math.pow(x, -0.5);
	} else {
		tmp = 1.0 / Math.sqrt((x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.8:
		tmp = -1.0 + math.pow(x, -0.5)
	else:
		tmp = 1.0 / math.sqrt((x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.8)
		tmp = Float64(-1.0 + (x ^ -0.5));
	else
		tmp = Float64(1.0 / sqrt(Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.8)
		tmp = -1.0 + (x ^ -0.5);
	else
		tmp = 1.0 / sqrt((x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.8], N[(-1.0 + N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.0%

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

   if 0.80000000000000004 < x

   1. Initial program 39.0%

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

      1. frac-sub 39.0%

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

      2. clear-num 39.0%

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

      3. sqrt-unprod 39.0%

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

      4. +-commutative 39.0%

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

      5. *-un-lft-identity 39.0%

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

      6. *-rgt-identity 39.0%

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

      7. +-commutative 39.0%

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

   3. Applied egg-rr 39.0%

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

      1. associate-/r/ 39.0%

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

      2. associate-*l/ 39.0%

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

      3. *-lft-identity 39.0%

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

      4. distribute-rgt-in 39.0%

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

      5. *-lft-identity 39.0%

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

   5. Simplified 39.0%

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

   6. Taylor expanded in x around 0 38.1%

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

   7. Taylor expanded in x around inf 38.1%

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

      1. unpow2 38.1%

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

   9. Simplified 38.1%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot x}}\\ \end{array} \]

Alternative 14: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{0.5}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.6) (+ -1.0 (pow x -0.5)) (- (/ 1.0 x) (/ 0.5 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.6) {
		tmp = -1.0 + pow(x, -0.5);
	} else {
		tmp = (1.0 / x) - (0.5 / (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.6d0) then
        tmp = (-1.0d0) + (x ** (-0.5d0))
    else
        tmp = (1.0d0 / x) - (0.5d0 / (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.6) {
		tmp = -1.0 + Math.pow(x, -0.5);
	} else {
		tmp = (1.0 / x) - (0.5 / (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.6:
		tmp = -1.0 + math.pow(x, -0.5)
	else:
		tmp = (1.0 / x) - (0.5 / (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64(-1.0 + (x ^ -0.5));
	else
		tmp = Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.6)
		tmp = -1.0 + (x ^ -0.5);
	else
		tmp = (1.0 / x) - (0.5 / (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.6], N[(-1.0 + N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.599999999999999978

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. inv-pow 99.6%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.0%

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

   if 0.599999999999999978 < x

   1. Initial program 39.0%

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

      1. frac-sub 39.0%

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

      2. clear-num 39.0%

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

      3. sqrt-unprod 39.0%

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

      4. +-commutative 39.0%

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

      5. *-un-lft-identity 39.0%

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

      6. *-rgt-identity 39.0%

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

      7. +-commutative 39.0%

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

   3. Applied egg-rr 39.0%

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

      1. associate-/r/ 39.0%

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

      2. associate-*l/ 39.0%

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

      3. *-lft-identity 39.0%

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

      4. distribute-rgt-in 39.0%

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

      5. *-lft-identity 39.0%

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

   5. Simplified 39.0%

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

   6. Taylor expanded in x around 0 38.1%

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

   7. Taylor expanded in x around inf 7.5%

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

      1. unpow2 7.5%

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

      2. associate-*r/ 7.5%

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

      3. metadata-eval 7.5%

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

   9. Simplified 7.5%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{0.5}{x \cdot x}\\ \end{array} \]

Alternative 15: 50.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

   1. *-un-lft-identity 72.6%

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

   2. clear-num 72.6%

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

   3. associate-/r/ 72.6%

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

   4. prod-diff 72.6%

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

   5. *-un-lft-identity 72.6%

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

   6. fma-neg 72.6%

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

   7. *-un-lft-identity 72.6%

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

   8. inv-pow 72.6%

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

   9. sqrt-pow2 70.1%

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

   10. metadata-eval 70.1%

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

   11. pow1/2 70.1%

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

   12. pow-flip 72.8%

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

   13. +-commutative 72.8%

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

   14. metadata-eval 72.8%

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

3. Applied egg-rr 72.8%

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

   1. fma-udef 72.8%

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

   2. distribute-lft1-in 72.8%

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

   3. metadata-eval 72.8%

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

   4. mul0-lft 72.8%

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

   5. +-rgt-identity 72.8%

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

5. Simplified 72.8%

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

   1. add-sqr-sqrt 65.3%

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

   2. sqrt-unprod 72.8%

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

   3. metadata-eval 72.8%

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

   4. sqrt-pow2 72.1%

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

   5. inv-pow 72.1%

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

   6. metadata-eval 72.1%

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

   7. sqrt-pow2 70.1%

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

   8. inv-pow 70.1%

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

   9. frac-times 70.1%

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

   10. metadata-eval 70.1%

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

   11. add-sqr-sqrt 70.2%

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

   12. +-commutative 70.2%

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

7. Applied egg-rr 70.2%

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

8. Taylor expanded in x around inf 54.6%

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

   1. expm1-log1p-u 51.0%

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

   2. expm1-udef 65.1%

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

   3. sqrt-div 65.1%

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

   4. metadata-eval 65.1%

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

   5. inv-pow 65.1%

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

   6. sqrt-pow2 65.1%

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

   7. metadata-eval 65.1%

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

10. Applied egg-rr 65.1%

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

    1. expm1-def 51.0%

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

    2. expm1-log1p 54.7%

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

12. Simplified 54.7%

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

13. Final simplification 54.7%

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

Alternative 16: 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 72.6%

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

   1. frac-sub 72.6%

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

   2. clear-num 72.6%

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

   3. sqrt-unprod 72.6%

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

   4. +-commutative 72.6%

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

   5. *-un-lft-identity 72.6%

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

   6. *-rgt-identity 72.6%

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

   7. +-commutative 72.6%

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

3. Applied egg-rr 72.6%

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

   1. associate-/r/ 72.6%

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

   2. associate-*l/ 72.6%

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

   3. *-lft-identity 72.6%

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

   4. distribute-rgt-in 72.6%

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

   5. *-lft-identity 72.6%

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

5. Simplified 72.6%

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

6. Taylor expanded in x around 0 69.1%

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

7. Taylor expanded in x around inf 7.3%

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

   1. +-commutative 7.3%

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

9. Simplified 7.3%

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

10. Final simplification 7.3%

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

Alternative 17: 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 72.6%

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

   1. frac-sub 72.6%

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

   2. clear-num 72.6%

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

   3. sqrt-unprod 72.6%

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

   4. +-commutative 72.6%

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

   5. *-un-lft-identity 72.6%

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

   6. *-rgt-identity 72.6%

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

   7. +-commutative 72.6%

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

3. Applied egg-rr 72.6%

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

   1. associate-/r/ 72.6%

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

   2. associate-*l/ 72.6%

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

   3. *-lft-identity 72.6%

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

   4. distribute-rgt-in 72.6%

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

   5. *-lft-identity 72.6%

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

5. Simplified 72.6%

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

6. Taylor expanded in x around 0 69.1%

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

7. Taylor expanded in x around inf 7.2%

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

8. Final simplification 7.2%

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

Alternative 18: 1.9% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

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

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 72.6%

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

   1. add-cube-cbrt 59.8%

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

   2. associate-*l* 59.8%

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

   3. frac-2neg 59.8%

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

   4. metadata-eval 59.8%

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

   5. div-inv 59.8%

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

   6. metadata-eval 59.8%

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

   7. distribute-neg-frac 59.8%

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

   8. metadata-eval 59.8%

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

   9. frac-2neg 59.8%

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

   10. prod-diff 57.1%

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

3. Applied egg-rr 57.6%

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

   1. fma-udef 60.0%

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

   2. distribute-rgt-neg-in 60.0%

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

   3. metadata-eval 60.0%

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

   4. *-rgt-identity 60.0%

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

   5. *-commutative 60.0%

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

   6. associate-+l+ 60.0%

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

   7. *-commutative 60.0%

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

   8. fma-udef 60.0%

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

5. Simplified 57.6%

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

6. Taylor expanded in x around 0 1.9%

   \[\leadsto \color{blue}{-1} \]

7. Final simplification 1.9%

   \[\leadsto -1 \]

Developer target: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023273 
(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)))))