2isqrt (example 3.6)

Percentage Accurate: 68.7% → 99.8%

Time: 11.9s

Alternatives: 13

Speedup: 1.8×

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: 68.7% 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: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. *-un-lft-identity 67.9%

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

   2. clear-num 67.9%

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

   3. associate-/r/ 67.9%

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

   4. prod-diff 67.9%

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

   5. *-un-lft-identity 67.9%

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

   6. fma-neg 67.9%

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

   7. *-un-lft-identity 67.9%

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

   8. pow1/2 67.9%

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

   9. pow-flip 63.8%

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

   10. metadata-eval 63.8%

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

   11. pow1/2 63.8%

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

   12. pow-flip 68.0%

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

   13. +-commutative 68.0%

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

   14. metadata-eval 68.0%

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

3. Applied egg-rr 68.0%

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

   1. +-commutative 68.0%

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

   2. sub-neg 68.0%

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

   3. fma-udef 68.0%

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

   4. distribute-lft1-in 68.0%

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

   5. metadata-eval 68.0%

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

   6. mul0-lft 68.0%

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

   7. +-commutative 68.0%

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

   8. associate-+r+ 68.0%

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

   9. sub-neg 68.0%

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

   10. neg-sub0 68.0%

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

   11. +-commutative 68.0%

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

   12. sub-neg 68.0%

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

5. Simplified 68.0%

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

   1. metadata-eval 68.0%

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

   2. pow-flip 63.8%

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

   3. +-commutative 63.8%

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

   4. pow1/2 63.8%

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

   5. flip-- 63.7%

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

   6. pow-prod-up 58.2%

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

   7. metadata-eval 58.2%

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

   8. inv-pow 58.2%

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

   9. +-commutative 58.2%

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

   10. +-commutative 58.2%

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

   11. frac-times 62.3%

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

   12. metadata-eval 62.3%

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

   13. add-sqr-sqrt 67.8%

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

   14. +-commutative 67.8%

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

   15. pow1/2 67.8%

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

7. Applied egg-rr 67.8%

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

   1. frac-sub 68.6%

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

   2. *-un-lft-identity 68.6%

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

   3. +-commutative 68.6%

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

   4. +-commutative 68.6%

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

9. Applied egg-rr 68.6%

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

    1. *-rgt-identity 68.6%

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

    2. associate--l+ 88.6%

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

    3. +-inverses 88.6%

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

    4. metadata-eval 88.6%

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

    5. associate-/r* 89.4%

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

11. Simplified 89.4%

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

    1. associate-/l/ 99.4%

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

    2. inv-pow 99.4%

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

    3. metadata-eval 99.4%

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

    4. pow-prod-up 99.7%

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

    5. times-frac 99.8%

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

    6. +-commutative 99.8%

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

13. Applied egg-rr 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 92.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{x + 1}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ x 1.0)))) 4e-13)
   (/ (/ (/ 1.0 x) (+ x 1.0)) (* 2.0 (sqrt (/ 1.0 x))))
   (- (pow x -0.5) (pow (+ x 1.0) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 4e-13) {
		tmp = ((1.0 / x) / (x + 1.0)) / (2.0 * sqrt((1.0 / x)));
	} else {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((x + 1.0d0)))) <= 4d-13) then
        tmp = ((1.0d0 / x) / (x + 1.0d0)) / (2.0d0 * sqrt((1.0d0 / x)))
    else
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((x + 1.0)))) <= 4e-13) {
		tmp = ((1.0 / x) / (x + 1.0)) / (2.0 * Math.sqrt((1.0 / x)));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((x + 1.0)))) <= 4e-13:
		tmp = ((1.0 / x) / (x + 1.0)) / (2.0 * math.sqrt((1.0 / x)))
	else:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(x + 1.0)))) <= 4e-13)
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(x + 1.0)) / Float64(2.0 * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 4e-13)
		tmp = ((1.0 / x) / (x + 1.0)) / (2.0 * sqrt((1.0 / x)));
	else
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-13], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 4.0000000000000001e-13

   1. Initial program 39.7%

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

      1. *-un-lft-identity 39.7%

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

      2. clear-num 39.7%

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

      3. associate-/r/ 39.7%

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

      4. prod-diff 39.7%

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

      5. *-un-lft-identity 39.7%

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

      6. fma-neg 39.7%

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

      7. *-un-lft-identity 39.7%

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

      8. pow1/2 39.7%

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

      9. pow-flip 31.7%

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

      10. metadata-eval 31.7%

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

      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 39.7%

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

      13. +-commutative 39.7%

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

      14. metadata-eval 39.7%

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

   3. Applied egg-rr 39.7%

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

      1. +-commutative 39.7%

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

      2. sub-neg 39.7%

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

      3. fma-udef 39.7%

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

      4. distribute-lft1-in 39.7%

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

      5. metadata-eval 39.7%

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

      6. mul0-lft 39.7%

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

      7. +-commutative 39.7%

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

      8. associate-+r+ 39.7%

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

      9. sub-neg 39.7%

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

      10. neg-sub0 39.7%

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

      11. +-commutative 39.7%

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

      12. sub-neg 39.7%

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

   5. Simplified 39.7%

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

      1. metadata-eval 39.7%

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

      2. pow-flip 31.7%

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

      3. +-commutative 31.7%

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

      4. pow1/2 31.7%

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

      5. flip-- 31.7%

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

      6. pow-prod-up 21.7%

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

      7. metadata-eval 21.7%

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

      8. inv-pow 21.7%

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

      9. +-commutative 21.7%

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

      10. +-commutative 21.7%

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

      11. frac-times 29.5%

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

      12. metadata-eval 29.5%

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

      13. add-sqr-sqrt 39.8%

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

      14. +-commutative 39.8%

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

      15. pow1/2 39.8%

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

   7. Applied egg-rr 39.8%

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

      1. frac-sub 41.0%

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

      2. *-un-lft-identity 41.0%

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

      3. +-commutative 41.0%

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

      4. +-commutative 41.0%

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

   9. Applied egg-rr 41.0%

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

       1. *-rgt-identity 41.0%

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

       2. associate--l+ 79.0%

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

       3. +-inverses 79.0%

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

       4. metadata-eval 79.0%

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

       5. associate-/r* 80.6%

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

   11. Simplified 80.6%

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

   12. Taylor expanded in x around inf 79.8%

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

   if 4.0000000000000001e-13 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.3%

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

      1. *-un-lft-identity 99.3%

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

      2. clear-num 99.3%

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

      3. associate-/r/ 99.3%

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

      4. prod-diff 99.3%

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

      5. *-un-lft-identity 99.3%

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

      6. fma-neg 99.3%

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

      7. *-un-lft-identity 99.3%

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

      8. pow1/2 99.3%

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

      9. pow-flip 99.6%

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

      10. metadata-eval 99.6%

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

      11. pow1/2 99.6%

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

      12. pow-flip 99.6%

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

      13. +-commutative 99.6%

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

      14. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. fma-udef 99.6%

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

      4. distribute-lft1-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. mul0-lft 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. sub-neg 99.6%

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

      10. neg-sub0 99.6%

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

      11. +-commutative 99.6%

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

      12. sub-neg 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{x + 1}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. *-un-lft-identity 67.9%

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

   2. clear-num 67.9%

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

   3. associate-/r/ 67.9%

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

   4. prod-diff 67.9%

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

   5. *-un-lft-identity 67.9%

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

   6. fma-neg 67.9%

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

   7. *-un-lft-identity 67.9%

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

   8. pow1/2 67.9%

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

   9. pow-flip 63.8%

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

   10. metadata-eval 63.8%

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

   11. pow1/2 63.8%

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

   12. pow-flip 68.0%

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

   13. +-commutative 68.0%

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

   14. metadata-eval 68.0%

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

3. Applied egg-rr 68.0%

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

   1. +-commutative 68.0%

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

   2. sub-neg 68.0%

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

   3. fma-udef 68.0%

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

   4. distribute-lft1-in 68.0%

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

   5. metadata-eval 68.0%

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

   6. mul0-lft 68.0%

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

   7. +-commutative 68.0%

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

   8. associate-+r+ 68.0%

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

   9. sub-neg 68.0%

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

   10. neg-sub0 68.0%

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

   11. +-commutative 68.0%

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

   12. sub-neg 68.0%

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

5. Simplified 68.0%

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

   1. metadata-eval 68.0%

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

   2. pow-flip 63.8%

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

   3. +-commutative 63.8%

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

   4. pow1/2 63.8%

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

   5. flip-- 63.7%

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

   6. pow-prod-up 58.2%

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

   7. metadata-eval 58.2%

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

   8. inv-pow 58.2%

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

   9. +-commutative 58.2%

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

   10. +-commutative 58.2%

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

   11. frac-times 62.3%

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

   12. metadata-eval 62.3%

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

   13. add-sqr-sqrt 67.8%

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

   14. +-commutative 67.8%

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

   15. pow1/2 67.8%

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

7. Applied egg-rr 67.8%

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

   1. frac-sub 68.6%

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

   2. *-un-lft-identity 68.6%

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

   3. +-commutative 68.6%

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

   4. +-commutative 68.6%

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

9. Applied egg-rr 68.6%

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

    1. *-rgt-identity 68.6%

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

    2. associate--l+ 88.6%

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

    3. +-inverses 88.6%

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

    4. metadata-eval 88.6%

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

    5. associate-/r* 89.4%

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

11. Simplified 89.4%

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

    1. associate-/l/ 99.4%

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

    2. div-inv 99.4%

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

    3. +-commutative 99.4%

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

13. Applied egg-rr 99.4%

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

14. Final simplification 99.4%

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

Alternative 4: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. *-un-lft-identity 67.9%

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

   2. clear-num 67.9%

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

   3. associate-/r/ 67.9%

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

   4. prod-diff 67.9%

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

   5. *-un-lft-identity 67.9%

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

   6. fma-neg 67.9%

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

   7. *-un-lft-identity 67.9%

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

   8. pow1/2 67.9%

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

   9. pow-flip 63.8%

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

   10. metadata-eval 63.8%

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

   11. pow1/2 63.8%

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

   12. pow-flip 68.0%

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

   13. +-commutative 68.0%

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

   14. metadata-eval 68.0%

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

3. Applied egg-rr 68.0%

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

   1. +-commutative 68.0%

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

   2. sub-neg 68.0%

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

   3. fma-udef 68.0%

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

   4. distribute-lft1-in 68.0%

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

   5. metadata-eval 68.0%

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

   6. mul0-lft 68.0%

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

   7. +-commutative 68.0%

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

   8. associate-+r+ 68.0%

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

   9. sub-neg 68.0%

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

   10. neg-sub0 68.0%

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

   11. +-commutative 68.0%

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

   12. sub-neg 68.0%

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

5. Simplified 68.0%

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

   1. metadata-eval 68.0%

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

   2. pow-flip 63.8%

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

   3. +-commutative 63.8%

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

   4. pow1/2 63.8%

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

   5. flip-- 63.7%

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

   6. pow-prod-up 58.2%

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

   7. metadata-eval 58.2%

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

   8. inv-pow 58.2%

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

   9. +-commutative 58.2%

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

   10. +-commutative 58.2%

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

   11. frac-times 62.3%

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

   12. metadata-eval 62.3%

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

   13. add-sqr-sqrt 67.8%

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

   14. +-commutative 67.8%

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

   15. pow1/2 67.8%

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

7. Applied egg-rr 67.8%

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

   1. frac-sub 68.6%

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

   2. *-un-lft-identity 68.6%

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

   3. +-commutative 68.6%

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

   4. +-commutative 68.6%

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

9. Applied egg-rr 68.6%

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

10. Taylor expanded in x around 0 88.6%

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

11. Final simplification 88.6%

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

Alternative 5: 92.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. *-un-lft-identity 67.9%

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

   2. clear-num 67.9%

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

   3. associate-/r/ 67.9%

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

   4. prod-diff 67.9%

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

   5. *-un-lft-identity 67.9%

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

   6. fma-neg 67.9%

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

   7. *-un-lft-identity 67.9%

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

   8. pow1/2 67.9%

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

   9. pow-flip 63.8%

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

   10. metadata-eval 63.8%

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

   11. pow1/2 63.8%

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

   12. pow-flip 68.0%

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

   13. +-commutative 68.0%

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

   14. metadata-eval 68.0%

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

3. Applied egg-rr 68.0%

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

   1. +-commutative 68.0%

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

   2. sub-neg 68.0%

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

   3. fma-udef 68.0%

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

   4. distribute-lft1-in 68.0%

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

   5. metadata-eval 68.0%

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

   6. mul0-lft 68.0%

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

   7. +-commutative 68.0%

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

   8. associate-+r+ 68.0%

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

   9. sub-neg 68.0%

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

   10. neg-sub0 68.0%

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

   11. +-commutative 68.0%

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

   12. sub-neg 68.0%

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

5. Simplified 68.0%

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

   1. metadata-eval 68.0%

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

   2. pow-flip 63.8%

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

   3. +-commutative 63.8%

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

   4. pow1/2 63.8%

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

   5. flip-- 63.7%

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

   6. pow-prod-up 58.2%

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

   7. metadata-eval 58.2%

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

   8. inv-pow 58.2%

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

   9. +-commutative 58.2%

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

   10. +-commutative 58.2%

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

   11. frac-times 62.3%

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

   12. metadata-eval 62.3%

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

   13. add-sqr-sqrt 67.8%

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

   14. +-commutative 67.8%

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

   15. pow1/2 67.8%

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

7. Applied egg-rr 67.8%

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

   1. frac-sub 68.6%

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

   2. *-un-lft-identity 68.6%

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

   3. +-commutative 68.6%

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

   4. +-commutative 68.6%

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

9. Applied egg-rr 68.6%

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

    1. *-rgt-identity 68.6%

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

    2. associate--l+ 88.6%

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

    3. +-inverses 88.6%

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

    4. metadata-eval 88.6%

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

    5. associate-/r* 89.4%

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

11. Simplified 89.4%

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

12. Final simplification 89.4%

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

Alternative 6: 91.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;\frac{1}{\sqrt{x}} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{x + 1}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.72)
   (- (/ 1.0 (sqrt x)) (+ 1.0 (* x (+ -0.5 (* x 0.375)))))
   (/ (/ (/ 1.0 x) (+ x 1.0)) (* 2.0 (sqrt (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.72) {
		tmp = (1.0 / sqrt(x)) - (1.0 + (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = ((1.0 / x) / (x + 1.0)) / (2.0 * sqrt((1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.72d0) then
        tmp = (1.0d0 / sqrt(x)) - (1.0d0 + (x * ((-0.5d0) + (x * 0.375d0))))
    else
        tmp = ((1.0d0 / x) / (x + 1.0d0)) / (2.0d0 * sqrt((1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.72) {
		tmp = (1.0 / Math.sqrt(x)) - (1.0 + (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = ((1.0 / x) / (x + 1.0)) / (2.0 * Math.sqrt((1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.72:
		tmp = (1.0 / math.sqrt(x)) - (1.0 + (x * (-0.5 + (x * 0.375))))
	else:
		tmp = ((1.0 / x) / (x + 1.0)) / (2.0 * math.sqrt((1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.72)
		tmp = Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 + Float64(x * Float64(-0.5 + Float64(x * 0.375)))));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(x + 1.0)) / Float64(2.0 * sqrt(Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.72)
		tmp = (1.0 / sqrt(x)) - (1.0 + (x * (-0.5 + (x * 0.375))));
	else
		tmp = ((1.0 / x) / (x + 1.0)) / (2.0 * sqrt((1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.72], N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 + N[(x * N[(-0.5 + N[(x * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.2%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(1 + \left(-0.5 \cdot x + 0.375 \cdot {x}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(\color{blue}{x \cdot -0.5} + 0.375 \cdot {x}^{2}\right)\right) \]

      2. *-commutative 99.2%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{{x}^{2} \cdot 0.375}\right)\right) \]

      3. unpow2 99.2%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{\left(x \cdot x\right)} \cdot 0.375\right)\right) \]

      4. associate-*l* 99.2%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{x \cdot \left(x \cdot 0.375\right)}\right)\right) \]

      5. distribute-lft-out 99.2%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \color{blue}{x \cdot \left(-0.5 + x \cdot 0.375\right)}\right) \]

   4. Simplified 99.2%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right)} \]

   if 0.71999999999999997 < x

   1. Initial program 40.6%

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

      1. *-un-lft-identity 40.6%

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

      2. clear-num 40.6%

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

      3. associate-/r/ 40.6%

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

      4. prod-diff 40.6%

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

      5. *-un-lft-identity 40.6%

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

      6. fma-neg 40.6%

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

      7. *-un-lft-identity 40.6%

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

      8. pow1/2 40.6%

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

      9. pow-flip 32.8%

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

      10. metadata-eval 32.8%

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

      11. pow1/2 32.8%

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

      12. pow-flip 40.6%

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

      13. +-commutative 40.6%

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

      14. metadata-eval 40.6%

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

   3. Applied egg-rr 40.6%

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

      1. +-commutative 40.6%

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

      2. sub-neg 40.6%

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

      3. fma-udef 40.6%

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

      4. distribute-lft1-in 40.6%

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

      5. metadata-eval 40.6%

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

      6. mul0-lft 40.6%

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

      7. +-commutative 40.6%

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

      8. associate-+r+ 40.6%

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

      9. sub-neg 40.6%

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

      10. neg-sub0 40.6%

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

      11. +-commutative 40.6%

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

      12. sub-neg 40.6%

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

   5. Simplified 40.6%

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

      1. metadata-eval 40.6%

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

      2. pow-flip 32.8%

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

      3. +-commutative 32.8%

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

      4. pow1/2 32.8%

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

      5. flip-- 32.8%

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

      6. pow-prod-up 23.2%

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

      7. metadata-eval 23.2%

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

      8. inv-pow 23.2%

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

      9. +-commutative 23.2%

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

      10. +-commutative 23.2%

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

      11. frac-times 30.8%

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

      12. metadata-eval 30.8%

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

      13. add-sqr-sqrt 40.8%

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

      14. +-commutative 40.8%

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

      15. pow1/2 40.8%

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

   7. Applied egg-rr 40.8%

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

      1. frac-sub 42.3%

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

      2. *-un-lft-identity 42.3%

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

      3. +-commutative 42.3%

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

      4. +-commutative 42.3%

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

   9. Applied egg-rr 42.3%

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

       1. *-rgt-identity 42.3%

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

       2. associate--l+ 79.4%

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

       3. +-inverses 79.4%

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

       4. metadata-eval 79.4%

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

       5. associate-/r* 81.0%

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

   11. Simplified 81.0%

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

   12. Taylor expanded in x around inf 78.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;\frac{1}{\sqrt{x}} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{x + 1}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]

Alternative 7: 67.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. *-un-lft-identity 67.9%

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

   2. clear-num 67.9%

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

   3. associate-/r/ 67.9%

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

   4. prod-diff 67.9%

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

   5. *-un-lft-identity 67.9%

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

   6. fma-neg 67.9%

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

   7. *-un-lft-identity 67.9%

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

   8. pow1/2 67.9%

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

   9. pow-flip 63.8%

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

   10. metadata-eval 63.8%

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

   11. pow1/2 63.8%

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

   12. pow-flip 68.0%

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

   13. +-commutative 68.0%

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

   14. metadata-eval 68.0%

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

3. Applied egg-rr 68.0%

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

   1. +-commutative 68.0%

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

   2. sub-neg 68.0%

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

   3. fma-udef 68.0%

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

   4. distribute-lft1-in 68.0%

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

   5. metadata-eval 68.0%

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

   6. mul0-lft 68.0%

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

   7. +-commutative 68.0%

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

   8. associate-+r+ 68.0%

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

   9. sub-neg 68.0%

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

   10. neg-sub0 68.0%

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

   11. +-commutative 68.0%

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

   12. sub-neg 68.0%

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

5. Simplified 68.0%

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

   1. metadata-eval 68.0%

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

   2. pow-flip 63.8%

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

   3. +-commutative 63.8%

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

   4. pow1/2 63.8%

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

   5. flip-- 63.7%

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

   6. pow-prod-up 58.2%

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

   7. metadata-eval 58.2%

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

   8. inv-pow 58.2%

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

   9. +-commutative 58.2%

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

   10. +-commutative 58.2%

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

   11. frac-times 62.3%

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

   12. metadata-eval 62.3%

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

   13. add-sqr-sqrt 67.8%

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

   14. +-commutative 67.8%

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

   15. pow1/2 67.8%

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

7. Applied egg-rr 67.8%

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

   1. frac-sub 68.6%

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

   2. *-un-lft-identity 68.6%

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

   3. +-commutative 68.6%

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

   4. +-commutative 68.6%

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

9. Applied egg-rr 68.6%

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

    1. *-rgt-identity 68.6%

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

    2. associate--l+ 88.6%

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

    3. +-inverses 88.6%

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

    4. metadata-eval 88.6%

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

    5. associate-/r* 89.4%

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

11. Simplified 89.4%

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

12. Taylor expanded in x around 0 66.1%

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

13. Final simplification 66.1%

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

Alternative 8: 67.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+76}:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 8e+76) (+ (+ (pow x -0.5) (* x 0.5)) -1.0) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 8e+76) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8d+76) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8e+76) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8e+76:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8e+76)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8e+76)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8e+76], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.0000000000000004e76

   1. Initial program 79.6%

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

      1. *-un-lft-identity 79.6%

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

      2. clear-num 79.6%

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

      3. associate-/r/ 79.6%

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

      4. prod-diff 79.6%

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

      5. *-un-lft-identity 79.6%

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

      6. fma-neg 79.6%

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

      7. *-un-lft-identity 79.6%

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

      8. pow1/2 79.6%

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

      9. pow-flip 80.0%

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

      10. metadata-eval 80.0%

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

      11. pow1/2 80.0%

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

      12. pow-flip 79.8%

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

      13. +-commutative 79.8%

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

      14. metadata-eval 79.8%

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

   3. Applied egg-rr 79.8%

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

      1. +-commutative 79.8%

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

      2. sub-neg 79.8%

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

      3. fma-udef 79.8%

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

      4. distribute-lft1-in 79.8%

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

      5. metadata-eval 79.8%

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

      6. mul0-lft 79.8%

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

      7. +-commutative 79.8%

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

      8. associate-+r+ 79.8%

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

      9. sub-neg 79.8%

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

      10. neg-sub0 79.8%

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

      11. +-commutative 79.8%

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

      12. sub-neg 79.8%

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

   5. Simplified 79.8%

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

   6. Taylor expanded in x around 0 76.9%

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

   if 8.0000000000000004e76 < x

   1. Initial program 49.8%

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

      1. *-un-lft-identity 49.8%

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

      2. clear-num 49.8%

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

      3. associate-/r/ 49.8%

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

      4. prod-diff 49.8%

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

      5. *-un-lft-identity 49.8%

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

      6. fma-neg 49.8%

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

      7. *-un-lft-identity 49.8%

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

      8. pow1/2 49.8%

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

      9. pow-flip 38.9%

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

      10. metadata-eval 38.9%

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

      11. pow1/2 38.9%

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

      12. pow-flip 49.8%

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

      13. +-commutative 49.8%

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

      14. metadata-eval 49.8%

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

   3. Applied egg-rr 49.8%

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

      1. +-commutative 49.8%

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

      2. sub-neg 49.8%

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

      3. fma-udef 49.8%

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

      4. distribute-lft1-in 49.8%

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

      5. metadata-eval 49.8%

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

      6. mul0-lft 49.8%

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

      7. +-commutative 49.8%

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

      8. associate-+r+ 49.8%

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

      9. sub-neg 49.8%

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

      10. neg-sub0 49.8%

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

      11. +-commutative 49.8%

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

      12. sub-neg 49.8%

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

   5. Simplified 49.8%

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

      1. sqr-pow 27.0%

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

      2. metadata-eval 27.0%

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

      3. pow-flip 30.7%

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

      4. +-commutative 30.7%

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

      5. pow1/2 30.7%

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

      6. fma-neg 4.3%

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

      7. metadata-eval 4.3%

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

      8. metadata-eval 4.3%

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

      9. pow1/2 4.3%

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

      10. +-commutative 4.3%

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

      11. pow-flip 4.4%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -\color{blue}{{\left(1 + x\right)}^{\left(-0.5\right)}}\right) \]

      12. +-commutative 4.4%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\color{blue}{\left(x + 1\right)}}^{\left(-0.5\right)}\right) \]

      13. metadata-eval 4.4%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\left(x + 1\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 4.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\left(x + 1\right)}^{-0.5}\right)} \]

   8. Taylor expanded in x around inf 49.8%

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

      1. unpow1/2 49.8%

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

      2. +-inverses 49.8%

         \[\leadsto \color{blue}{0} \]

   10. Simplified 49.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+76}:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 68.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. pow1/2 99.7%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul0-lft 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+r+ 100.0%

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

      9. sub-neg 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.0%

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

   if 1 < x

   1. Initial program 40.6%

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

      1. *-un-lft-identity 40.6%

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

      2. clear-num 40.6%

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

      3. associate-/r/ 40.6%

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

      4. prod-diff 40.6%

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

      5. *-un-lft-identity 40.6%

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

      6. fma-neg 40.6%

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

      7. *-un-lft-identity 40.6%

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

      8. pow1/2 40.6%

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

      9. pow-flip 32.8%

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

      10. metadata-eval 32.8%

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

      11. pow1/2 32.8%

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

      12. pow-flip 40.6%

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

      13. +-commutative 40.6%

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

      14. metadata-eval 40.6%

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

   3. Applied egg-rr 40.6%

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

      1. +-commutative 40.6%

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

      2. sub-neg 40.6%

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

      3. fma-udef 40.6%

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

      4. distribute-lft1-in 40.6%

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

      5. metadata-eval 40.6%

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

      6. mul0-lft 40.6%

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

      7. +-commutative 40.6%

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

      8. associate-+r+ 40.6%

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

      9. sub-neg 40.6%

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

      10. neg-sub0 40.6%

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

      11. +-commutative 40.6%

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

      12. sub-neg 40.6%

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

   5. Simplified 40.6%

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

      1. metadata-eval 40.6%

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

      2. pow-flip 32.8%

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

      3. +-commutative 32.8%

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

      4. pow1/2 32.8%

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

      5. flip-- 32.8%

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

      6. pow-prod-up 23.2%

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

      7. metadata-eval 23.2%

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

      8. inv-pow 23.2%

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

      9. +-commutative 23.2%

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

      10. +-commutative 23.2%

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

      11. frac-times 30.8%

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

      12. metadata-eval 30.8%

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

      13. add-sqr-sqrt 40.8%

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

      14. +-commutative 40.8%

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

      15. pow1/2 40.8%

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

   7. Applied egg-rr 40.8%

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

   8. Taylor expanded in x around 0 38.0%

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

   9. Taylor expanded in x around inf 39.5%

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

       1. unpow2 39.5%

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

       2. associate-*l* 39.5%

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

       3. unpow1/2 39.5%

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

       4. distribute-rgt-in 39.5%

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

       5. *-lft-identity 39.5%

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

       6. unpow1/2 39.5%

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

       7. rem-exp-log 39.5%

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

       8. exp-neg 39.5%

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

       9. exp-prod 39.5%

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

       10. distribute-lft-neg-out 39.5%

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

       11. distribute-rgt-neg-in 39.5%

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

       12. metadata-eval 39.5%

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

       13. exp-to-pow 39.5%

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

       14. pow-plus 39.5%

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

       15. metadata-eval 39.5%

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

   11. Simplified 39.5%

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

4. Final simplification 66.9%

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

Alternative 10: 67.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.6e+153) (/ 1.0 (+ x (sqrt x))) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / (x + sqrt(x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.6d+153) then
        tmp = 1.0d0 / (x + sqrt(x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / (x + Math.sqrt(x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.6e+153:
		tmp = 1.0 / (x + math.sqrt(x))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.6e+153)
		tmp = Float64(1.0 / Float64(x + sqrt(x)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.6e+153)
		tmp = 1.0 / (x + sqrt(x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.6e+153], N[(1.0 / N[(x + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.6000000000000003e153

   1. Initial program 69.6%

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

      1. *-un-lft-identity 69.6%

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

      2. clear-num 69.6%

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

      3. associate-/r/ 69.6%

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

      4. prod-diff 69.6%

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

      5. *-un-lft-identity 69.6%

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

      6. fma-neg 69.6%

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

      7. *-un-lft-identity 69.6%

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

      8. pow1/2 69.6%

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

      9. pow-flip 69.9%

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

      10. metadata-eval 69.9%

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

      11. pow1/2 69.9%

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

      12. pow-flip 69.8%

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

      13. +-commutative 69.8%

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

      14. metadata-eval 69.8%

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

   3. Applied egg-rr 69.8%

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

      1. +-commutative 69.8%

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

      2. sub-neg 69.8%

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

      3. fma-udef 69.8%

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

      4. distribute-lft1-in 69.8%

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

      5. metadata-eval 69.8%

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

      6. mul0-lft 69.8%

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

      7. +-commutative 69.8%

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

      8. associate-+r+ 69.8%

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

      9. sub-neg 69.8%

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

      10. neg-sub0 69.8%

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

      11. +-commutative 69.8%

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

      12. sub-neg 69.8%

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

   5. Simplified 69.8%

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

      1. metadata-eval 69.8%

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

      2. pow-flip 69.9%

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

      3. +-commutative 69.9%

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

      4. pow1/2 69.9%

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

      5. flip-- 69.8%

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

      6. pow-prod-up 69.6%

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

      7. metadata-eval 69.6%

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

      8. inv-pow 69.6%

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

      9. +-commutative 69.6%

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

      10. +-commutative 69.6%

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

      11. frac-times 69.5%

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

      12. metadata-eval 69.5%

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

      13. add-sqr-sqrt 69.5%

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

      14. +-commutative 69.5%

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

      15. pow1/2 69.5%

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

   7. Applied egg-rr 69.5%

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

   8. Taylor expanded in x around 0 66.4%

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

      1. +-commutative 66.4%

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

      2. distribute-lft-in 66.4%

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

      3. pow1 66.4%

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

      4. pow-prod-up 66.7%

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

      5. metadata-eval 66.7%

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

      6. pow1/2 66.7%

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

      7. *-rgt-identity 66.7%

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

   10. Applied egg-rr 66.7%

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

   if 4.6000000000000003e153 < x

   1. Initial program 63.8%

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

      1. *-un-lft-identity 63.8%

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

      2. clear-num 63.8%

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

      3. associate-/r/ 63.8%

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

      4. prod-diff 63.8%

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

      5. *-un-lft-identity 63.8%

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

      6. fma-neg 63.8%

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

      7. *-un-lft-identity 63.8%

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

      8. pow1/2 63.8%

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

      9. pow-flip 49.5%

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

      10. metadata-eval 49.5%

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

      11. pow1/2 49.5%

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

      12. pow-flip 63.8%

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

      13. +-commutative 63.8%

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

      14. metadata-eval 63.8%

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

   3. Applied egg-rr 63.8%

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

      1. +-commutative 63.8%

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

      2. sub-neg 63.8%

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

      3. fma-udef 63.8%

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

      4. distribute-lft1-in 63.8%

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

      5. metadata-eval 63.8%

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

      6. mul0-lft 63.8%

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

      7. +-commutative 63.8%

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

      8. associate-+r+ 63.8%

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

      9. sub-neg 63.8%

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

      10. neg-sub0 63.8%

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

      11. +-commutative 63.8%

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

      12. sub-neg 63.8%

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

   5. Simplified 63.8%

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

      1. sqr-pow 33.9%

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

      2. metadata-eval 33.9%

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

      3. pow-flip 38.7%

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

      4. +-commutative 38.7%

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

      5. pow1/2 38.7%

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

      6. fma-neg 4.3%

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

      7. metadata-eval 4.3%

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

      8. metadata-eval 4.3%

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

      9. pow1/2 4.3%

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

      10. +-commutative 4.3%

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

      11. pow-flip 4.4%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -\color{blue}{{\left(1 + x\right)}^{\left(-0.5\right)}}\right) \]

      12. +-commutative 4.4%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\color{blue}{\left(x + 1\right)}}^{\left(-0.5\right)}\right) \]

      13. metadata-eval 4.4%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\left(x + 1\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 4.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\left(x + 1\right)}^{-0.5}\right)} \]

   8. Taylor expanded in x around inf 63.8%

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

      1. unpow1/2 63.8%

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

      2. +-inverses 63.8%

         \[\leadsto \color{blue}{0} \]

   10. Simplified 63.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 66.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) (+ (pow x -0.5) -1.0) 0.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. pow1/2 99.7%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul0-lft 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+r+ 100.0%

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

      9. sub-neg 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 97.8%

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

   if 1 < x

   1. Initial program 40.6%

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

      1. *-un-lft-identity 40.6%

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

      2. clear-num 40.6%

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

      3. associate-/r/ 40.6%

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

      4. prod-diff 40.6%

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

      5. *-un-lft-identity 40.6%

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

      6. fma-neg 40.6%

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

      7. *-un-lft-identity 40.6%

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

      8. pow1/2 40.6%

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

      9. pow-flip 32.8%

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

      10. metadata-eval 32.8%

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

      11. pow1/2 32.8%

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

      12. pow-flip 40.6%

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

      13. +-commutative 40.6%

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

      14. metadata-eval 40.6%

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

   3. Applied egg-rr 40.6%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 40.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right) + \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right)} \]

      2. sub-neg 40.6%

         \[\leadsto \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right)} \]

      3. fma-udef 40.6%

         \[\leadsto \color{blue}{\left(-1 \cdot {\left(1 + x\right)}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      4. distribute-lft1-in 40.6%

         \[\leadsto \color{blue}{\left(-1 + 1\right) \cdot {\left(1 + x\right)}^{-0.5}} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      5. metadata-eval 40.6%

         \[\leadsto \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      6. mul0-lft 40.6%

         \[\leadsto \color{blue}{0} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      7. +-commutative 40.6%

         \[\leadsto 0 + \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}\right)} \]

      8. associate-+r+ 40.6%

         \[\leadsto \color{blue}{\left(0 + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) + {x}^{-0.5}} \]

      9. sub-neg 40.6%

         \[\leadsto \color{blue}{\left(0 - {\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      10. neg-sub0 40.6%

          \[\leadsto \color{blue}{\left(-{\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      11. +-commutative 40.6%

          \[\leadsto \color{blue}{{x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)} \]

      12. sub-neg 40.6%

          \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

   5. Simplified 40.6%

      \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]
   6. Step-by-step derivation

      1. sqr-pow 23.9%

         \[\leadsto \color{blue}{{x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)}} - {\left(1 + x\right)}^{-0.5} \]

      2. metadata-eval 23.9%

         \[\leadsto {x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)} - {\left(1 + x\right)}^{\color{blue}{\left(-0.5\right)}} \]

      3. pow-flip 26.7%

         \[\leadsto {x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)} - \color{blue}{\frac{1}{{\left(1 + x\right)}^{0.5}}} \]

      4. +-commutative 26.7%

         \[\leadsto {x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)} - \frac{1}{{\color{blue}{\left(x + 1\right)}}^{0.5}} \]

      5. pow1/2 26.7%

         \[\leadsto {x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)} - \frac{1}{\color{blue}{\sqrt{x + 1}}} \]

      6. fma-neg 7.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{\left(\frac{-0.5}{2}\right)}, {x}^{\left(\frac{-0.5}{2}\right)}, -\frac{1}{\sqrt{x + 1}}\right)} \]

      7. metadata-eval 7.8%

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{-0.25}}, {x}^{\left(\frac{-0.5}{2}\right)}, -\frac{1}{\sqrt{x + 1}}\right) \]

      8. metadata-eval 7.8%

         \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{\color{blue}{-0.25}}, -\frac{1}{\sqrt{x + 1}}\right) \]

      9. pow1/2 7.8%

         \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -\frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right) \]

      10. +-commutative 7.8%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -\frac{1}{{\color{blue}{\left(1 + x\right)}}^{0.5}}\right) \]

      11. pow-flip 7.6%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -\color{blue}{{\left(1 + x\right)}^{\left(-0.5\right)}}\right) \]

      12. +-commutative 7.6%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\color{blue}{\left(x + 1\right)}}^{\left(-0.5\right)}\right) \]

      13. metadata-eval 7.6%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\left(x + 1\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\left(x + 1\right)}^{-0.5}\right)} \]

   8. Taylor expanded in x around inf 37.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} - {\left(\frac{1}{x}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 37.4%

         \[\leadsto \sqrt{\frac{1}{x}} - \color{blue}{\sqrt{\frac{1}{x}}} \]

      2. +-inverses 37.4%

         \[\leadsto \color{blue}{0} \]

   10. Simplified 37.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 27.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.3e+123) (sqrt (/ 0.1111111111111111 x)) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.3e+123) {
		tmp = sqrt((0.1111111111111111 / x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.3d+123) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.3e+123) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.3e+123:
		tmp = math.sqrt((0.1111111111111111 / x))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.3e+123)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.3e+123)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.3e+123], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999993e123

   1. Initial program 72.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Step-by-step derivation

      1. sub-neg 72.6%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} + \left(-\frac{1}{\sqrt{x + 1}}\right)} \]

      2. flip3-+ 49.1%

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{\sqrt{x}}\right)}^{3} + {\left(-\frac{1}{\sqrt{x + 1}}\right)}^{3}}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} + \left(\left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right) - \frac{1}{\sqrt{x}} \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)\right)}} \]

      3. inv-pow 49.1%

         \[\leadsto \frac{{\color{blue}{\left({\left(\sqrt{x}\right)}^{-1}\right)}}^{3} + {\left(-\frac{1}{\sqrt{x + 1}}\right)}^{3}}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} + \left(\left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right) - \frac{1}{\sqrt{x}} \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)\right)} \]

      4. pow-pow 49.3%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x}\right)}^{\left(-1 \cdot 3\right)}} + {\left(-\frac{1}{\sqrt{x + 1}}\right)}^{3}}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} + \left(\left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right) - \frac{1}{\sqrt{x}} \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)\right)} \]

      5. metadata-eval 49.3%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{\color{blue}{-3}} + {\left(-\frac{1}{\sqrt{x + 1}}\right)}^{3}}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} + \left(\left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right) - \frac{1}{\sqrt{x}} \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)\right)} \]

      6. distribute-neg-frac 49.3%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + {\color{blue}{\left(\frac{-1}{\sqrt{x + 1}}\right)}}^{3}}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} + \left(\left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right) - \frac{1}{\sqrt{x}} \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)\right)} \]

      7. metadata-eval 49.3%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + {\left(\frac{\color{blue}{-1}}{\sqrt{x + 1}}\right)}^{3}}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} + \left(\left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right) - \frac{1}{\sqrt{x}} \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)\right)} \]

      8. +-commutative 49.3%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + {\left(\frac{-1}{\sqrt{\color{blue}{1 + x}}}\right)}^{3}}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} + \left(\left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right) - \frac{1}{\sqrt{x}} \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)\right)} \]

      9. frac-times 49.4%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + {\left(\frac{-1}{\sqrt{1 + x}}\right)}^{3}}{\color{blue}{\frac{1 \cdot 1}{\sqrt{x} \cdot \sqrt{x}}} + \left(\left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right) - \frac{1}{\sqrt{x}} \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)\right)} \]

      10. metadata-eval 49.4%

          \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + {\left(\frac{-1}{\sqrt{1 + x}}\right)}^{3}}{\frac{\color{blue}{1}}{\sqrt{x} \cdot \sqrt{x}} + \left(\left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right) - \frac{1}{\sqrt{x}} \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)\right)} \]

      11. add-sqr-sqrt 49.1%

          \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + {\left(\frac{-1}{\sqrt{1 + x}}\right)}^{3}}{\frac{1}{\color{blue}{x}} + \left(\left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right) - \frac{1}{\sqrt{x}} \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)\right)} \]

   3. Applied egg-rr 49.1%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt{x}\right)}^{-3} + {\left(\frac{-1}{\sqrt{1 + x}}\right)}^{3}}{\frac{1}{x} + \left(\frac{-1}{\sqrt{1 + x}} \cdot \frac{-1}{\sqrt{1 + x}} - {x}^{-0.5} \cdot \frac{-1}{\sqrt{1 + x}}\right)}} \]
   4. Step-by-step derivation

      1. cube-div 49.0%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + \color{blue}{\frac{{-1}^{3}}{{\left(\sqrt{1 + x}\right)}^{3}}}}{\frac{1}{x} + \left(\frac{-1}{\sqrt{1 + x}} \cdot \frac{-1}{\sqrt{1 + x}} - {x}^{-0.5} \cdot \frac{-1}{\sqrt{1 + x}}\right)} \]

      2. metadata-eval 49.0%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + \frac{\color{blue}{-1}}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{-1}{\sqrt{1 + x}} \cdot \frac{-1}{\sqrt{1 + x}} - {x}^{-0.5} \cdot \frac{-1}{\sqrt{1 + x}}\right)} \]

      3. cancel-sign-sub-inv 49.0%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \color{blue}{\left(\frac{-1}{\sqrt{1 + x}} \cdot \frac{-1}{\sqrt{1 + x}} + \left(-{x}^{-0.5}\right) \cdot \frac{-1}{\sqrt{1 + x}}\right)}} \]

      4. associate-*r/ 49.0%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\color{blue}{\frac{\frac{-1}{\sqrt{1 + x}} \cdot -1}{\sqrt{1 + x}}} + \left(-{x}^{-0.5}\right) \cdot \frac{-1}{\sqrt{1 + x}}\right)} \]

      5. associate-*l/ 49.0%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{\color{blue}{\frac{-1 \cdot -1}{\sqrt{1 + x}}}}{\sqrt{1 + x}} + \left(-{x}^{-0.5}\right) \cdot \frac{-1}{\sqrt{1 + x}}\right)} \]

      6. metadata-eval 49.0%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{\frac{\color{blue}{1}}{\sqrt{1 + x}}}{\sqrt{1 + x}} + \left(-{x}^{-0.5}\right) \cdot \frac{-1}{\sqrt{1 + x}}\right)} \]

      7. associate-/l/ 49.0%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\color{blue}{\frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x}}} + \left(-{x}^{-0.5}\right) \cdot \frac{-1}{\sqrt{1 + x}}\right)} \]

      8. rem-square-sqrt 49.0%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{\color{blue}{1 + x}} + \left(-{x}^{-0.5}\right) \cdot \frac{-1}{\sqrt{1 + x}}\right)} \]

      9. associate-*r/ 49.0%

         \[\leadsto \frac{{\left(\sqrt{x}\right)}^{-3} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \color{blue}{\frac{\left(-{x}^{-0.5}\right) \cdot -1}{\sqrt{1 + x}}}\right)} \]

   5. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt{x}\right)}^{-3} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 45.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt{x}\right)}^{-3}\right)\right)} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]

      2. expm1-udef 43.4%

         \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\sqrt{x}\right)}^{-3}\right)} - 1\right)} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]

      3. sqrt-pow2 43.4%

         \[\leadsto \frac{\left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-3}{2}\right)}}\right)} - 1\right) + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]

      4. metadata-eval 43.4%

         \[\leadsto \frac{\left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-1.5}}\right)} - 1\right) + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]

   7. Applied egg-rr 43.4%

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-1.5}\right)} - 1\right)} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]
   8. Step-by-step derivation

      1. expm1-def 45.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-1.5}\right)\right)} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]

      2. expm1-log1p 49.3%

         \[\leadsto \frac{\color{blue}{{x}^{-1.5}} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]

   9. Simplified 49.3%

      \[\leadsto \frac{\color{blue}{{x}^{-1.5}} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{{x}^{-0.5}}{\sqrt{1 + x}}\right)} \]

   10. Taylor expanded in x around inf 14.6%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   11. Step-by-step derivation

       1. *-commutative 14.6%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.3333333333333333} \]

   12. Simplified 14.6%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.3333333333333333} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 14.6%

          \[\leadsto \color{blue}{\sqrt{\sqrt{\frac{1}{x}} \cdot 0.3333333333333333} \cdot \sqrt{\sqrt{\frac{1}{x}} \cdot 0.3333333333333333}} \]

       2. sqrt-unprod 14.6%

          \[\leadsto \color{blue}{\sqrt{\left(\sqrt{\frac{1}{x}} \cdot 0.3333333333333333\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot 0.3333333333333333\right)}} \]

       3. inv-pow 14.6%

          \[\leadsto \sqrt{\left(\sqrt{\color{blue}{{x}^{-1}}} \cdot 0.3333333333333333\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot 0.3333333333333333\right)} \]

       4. sqrt-pow1 14.6%

          \[\leadsto \sqrt{\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot 0.3333333333333333\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot 0.3333333333333333\right)} \]

       5. metadata-eval 14.6%

          \[\leadsto \sqrt{\left({x}^{\color{blue}{-0.5}} \cdot 0.3333333333333333\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot 0.3333333333333333\right)} \]

       6. inv-pow 14.6%

          \[\leadsto \sqrt{\left({x}^{-0.5} \cdot 0.3333333333333333\right) \cdot \left(\sqrt{\color{blue}{{x}^{-1}}} \cdot 0.3333333333333333\right)} \]

       7. sqrt-pow1 14.6%

          \[\leadsto \sqrt{\left({x}^{-0.5} \cdot 0.3333333333333333\right) \cdot \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot 0.3333333333333333\right)} \]

       8. metadata-eval 14.6%

          \[\leadsto \sqrt{\left({x}^{-0.5} \cdot 0.3333333333333333\right) \cdot \left({x}^{\color{blue}{-0.5}} \cdot 0.3333333333333333\right)} \]

       9. swap-sqr 14.6%

          \[\leadsto \sqrt{\color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)}} \]

       10. pow-prod-up 14.6%

           \[\leadsto \sqrt{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}} \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)} \]

       11. metadata-eval 14.6%

           \[\leadsto \sqrt{{x}^{\color{blue}{-1}} \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)} \]

       12. inv-pow 14.6%

           \[\leadsto \sqrt{\color{blue}{\frac{1}{x}} \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)} \]

       13. metadata-eval 14.6%

           \[\leadsto \sqrt{\frac{1}{x} \cdot \color{blue}{0.1111111111111111}} \]

   14. Applied egg-rr 14.6%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{x} \cdot 0.1111111111111111}} \]
   15. Step-by-step derivation

       1. associate-*l/ 14.6%

          \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 0.1111111111111111}{x}}} \]

       2. metadata-eval 14.6%

          \[\leadsto \sqrt{\frac{\color{blue}{0.1111111111111111}}{x}} \]

   16. Simplified 14.6%

       \[\leadsto \color{blue}{\sqrt{\frac{0.1111111111111111}{x}}} \]

   if 1.29999999999999993e123 < x

   1. Initial program 58.3%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Step-by-step derivation

      1. *-un-lft-identity 58.3%

         \[\leadsto \color{blue}{1 \cdot \frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}} \]

      2. clear-num 58.3%

         \[\leadsto 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\frac{\sqrt{x + 1}}{1}}} \]

      3. associate-/r/ 58.3%

         \[\leadsto 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\sqrt{x + 1}} \cdot 1} \]

      4. prod-diff 58.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -1 \cdot \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

      5. *-un-lft-identity 58.3%

         \[\leadsto \mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -\color{blue}{\frac{1}{\sqrt{x + 1}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

      6. fma-neg 58.3%

         \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)} + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

      7. *-un-lft-identity 58.3%

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

      8. pow1/2 58.3%

         \[\leadsto \left(\frac{1}{\color{blue}{{x}^{0.5}}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

      9. pow-flip 45.4%

         \[\leadsto \left(\color{blue}{{x}^{\left(-0.5\right)}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

      10. metadata-eval 45.4%

          \[\leadsto \left({x}^{\color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

      11. pow1/2 45.4%

          \[\leadsto \left({x}^{-0.5} - \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

      12. pow-flip 58.3%

          \[\leadsto \left({x}^{-0.5} - \color{blue}{{\left(x + 1\right)}^{\left(-0.5\right)}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

      13. +-commutative 58.3%

          \[\leadsto \left({x}^{-0.5} - {\color{blue}{\left(1 + x\right)}}^{\left(-0.5\right)}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

      14. metadata-eval 58.3%

          \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{-0.5}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   3. Applied egg-rr 58.3%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 58.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right) + \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right)} \]

      2. sub-neg 58.3%

         \[\leadsto \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right)} \]

      3. fma-udef 58.3%

         \[\leadsto \color{blue}{\left(-1 \cdot {\left(1 + x\right)}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      4. distribute-lft1-in 58.3%

         \[\leadsto \color{blue}{\left(-1 + 1\right) \cdot {\left(1 + x\right)}^{-0.5}} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      5. metadata-eval 58.3%

         \[\leadsto \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      6. mul0-lft 58.3%

         \[\leadsto \color{blue}{0} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      7. +-commutative 58.3%

         \[\leadsto 0 + \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}\right)} \]

      8. associate-+r+ 58.3%

         \[\leadsto \color{blue}{\left(0 + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) + {x}^{-0.5}} \]

      9. sub-neg 58.3%

         \[\leadsto \color{blue}{\left(0 - {\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      10. neg-sub0 58.3%

          \[\leadsto \color{blue}{\left(-{\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      11. +-commutative 58.3%

          \[\leadsto \color{blue}{{x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)} \]

      12. sub-neg 58.3%

          \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

   5. Simplified 58.3%

      \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]
   6. Step-by-step derivation

      1. sqr-pow 31.2%

         \[\leadsto \color{blue}{{x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)}} - {\left(1 + x\right)}^{-0.5} \]

      2. metadata-eval 31.2%

         \[\leadsto {x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)} - {\left(1 + x\right)}^{\color{blue}{\left(-0.5\right)}} \]

      3. pow-flip 35.6%

         \[\leadsto {x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)} - \color{blue}{\frac{1}{{\left(1 + x\right)}^{0.5}}} \]

      4. +-commutative 35.6%

         \[\leadsto {x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)} - \frac{1}{{\color{blue}{\left(x + 1\right)}}^{0.5}} \]

      5. pow1/2 35.6%

         \[\leadsto {x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)} - \frac{1}{\color{blue}{\sqrt{x + 1}}} \]

      6. fma-neg 4.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{\left(\frac{-0.5}{2}\right)}, {x}^{\left(\frac{-0.5}{2}\right)}, -\frac{1}{\sqrt{x + 1}}\right)} \]

      7. metadata-eval 4.2%

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{-0.25}}, {x}^{\left(\frac{-0.5}{2}\right)}, -\frac{1}{\sqrt{x + 1}}\right) \]

      8. metadata-eval 4.2%

         \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{\color{blue}{-0.25}}, -\frac{1}{\sqrt{x + 1}}\right) \]

      9. pow1/2 4.2%

         \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -\frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right) \]

      10. +-commutative 4.2%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -\frac{1}{{\color{blue}{\left(1 + x\right)}}^{0.5}}\right) \]

      11. pow-flip 4.3%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -\color{blue}{{\left(1 + x\right)}^{\left(-0.5\right)}}\right) \]

      12. +-commutative 4.3%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\color{blue}{\left(x + 1\right)}}^{\left(-0.5\right)}\right) \]

      13. metadata-eval 4.3%

          \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\left(x + 1\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 4.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\left(x + 1\right)}^{-0.5}\right)} \]

   8. Taylor expanded in x around inf 58.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} - {\left(\frac{1}{x}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. unpow1/2 58.3%

         \[\leadsto \sqrt{\frac{1}{x}} - \color{blue}{\sqrt{\frac{1}{x}}} \]

      2. +-inverses 58.3%

         \[\leadsto \color{blue}{0} \]

   10. Simplified 58.3%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 19.0% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 67.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Step-by-step derivation

   1. *-un-lft-identity 67.9%

      \[\leadsto \color{blue}{1 \cdot \frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}} \]

   2. clear-num 67.9%

      \[\leadsto 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\frac{\sqrt{x + 1}}{1}}} \]

   3. associate-/r/ 67.9%

      \[\leadsto 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\sqrt{x + 1}} \cdot 1} \]

   4. prod-diff 67.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -1 \cdot \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

   5. *-un-lft-identity 67.9%

      \[\leadsto \mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -\color{blue}{\frac{1}{\sqrt{x + 1}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   6. fma-neg 67.9%

      \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)} + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   7. *-un-lft-identity 67.9%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   8. pow1/2 67.9%

      \[\leadsto \left(\frac{1}{\color{blue}{{x}^{0.5}}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   9. pow-flip 63.8%

      \[\leadsto \left(\color{blue}{{x}^{\left(-0.5\right)}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   10. metadata-eval 63.8%

       \[\leadsto \left({x}^{\color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   11. pow1/2 63.8%

       \[\leadsto \left({x}^{-0.5} - \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   12. pow-flip 68.0%

       \[\leadsto \left({x}^{-0.5} - \color{blue}{{\left(x + 1\right)}^{\left(-0.5\right)}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   13. +-commutative 68.0%

       \[\leadsto \left({x}^{-0.5} - {\color{blue}{\left(1 + x\right)}}^{\left(-0.5\right)}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   14. metadata-eval 68.0%

       \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{-0.5}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

3. Applied egg-rr 68.0%

   \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
4. Step-by-step derivation

   1. +-commutative 68.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right) + \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right)} \]

   2. sub-neg 68.0%

      \[\leadsto \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right)} \]

   3. fma-udef 68.0%

      \[\leadsto \color{blue}{\left(-1 \cdot {\left(1 + x\right)}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

   4. distribute-lft1-in 68.0%

      \[\leadsto \color{blue}{\left(-1 + 1\right) \cdot {\left(1 + x\right)}^{-0.5}} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

   5. metadata-eval 68.0%

      \[\leadsto \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

   6. mul0-lft 68.0%

      \[\leadsto \color{blue}{0} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

   7. +-commutative 68.0%

      \[\leadsto 0 + \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}\right)} \]

   8. associate-+r+ 68.0%

      \[\leadsto \color{blue}{\left(0 + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) + {x}^{-0.5}} \]

   9. sub-neg 68.0%

      \[\leadsto \color{blue}{\left(0 - {\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

   10. neg-sub0 68.0%

       \[\leadsto \color{blue}{\left(-{\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

   11. +-commutative 68.0%

       \[\leadsto \color{blue}{{x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)} \]

   12. sub-neg 68.0%

       \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

5. Simplified 68.0%

   \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]
6. Step-by-step derivation

   1. sqr-pow 58.7%

      \[\leadsto \color{blue}{{x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)}} - {\left(1 + x\right)}^{-0.5} \]

   2. metadata-eval 58.7%

      \[\leadsto {x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)} - {\left(1 + x\right)}^{\color{blue}{\left(-0.5\right)}} \]

   3. pow-flip 60.2%

      \[\leadsto {x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)} - \color{blue}{\frac{1}{{\left(1 + x\right)}^{0.5}}} \]

   4. +-commutative 60.2%

      \[\leadsto {x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)} - \frac{1}{{\color{blue}{\left(x + 1\right)}}^{0.5}} \]

   5. pow1/2 60.2%

      \[\leadsto {x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)} - \frac{1}{\color{blue}{\sqrt{x + 1}}} \]

   6. fma-neg 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{\left(\frac{-0.5}{2}\right)}, {x}^{\left(\frac{-0.5}{2}\right)}, -\frac{1}{\sqrt{x + 1}}\right)} \]

   7. metadata-eval 50.0%

      \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{-0.25}}, {x}^{\left(\frac{-0.5}{2}\right)}, -\frac{1}{\sqrt{x + 1}}\right) \]

   8. metadata-eval 50.0%

      \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{\color{blue}{-0.25}}, -\frac{1}{\sqrt{x + 1}}\right) \]

   9. pow1/2 50.0%

      \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -\frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right) \]

   10. +-commutative 50.0%

       \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -\frac{1}{{\color{blue}{\left(1 + x\right)}}^{0.5}}\right) \]

   11. pow-flip 49.8%

       \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -\color{blue}{{\left(1 + x\right)}^{\left(-0.5\right)}}\right) \]

   12. +-commutative 49.8%

       \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\color{blue}{\left(x + 1\right)}}^{\left(-0.5\right)}\right) \]

   13. metadata-eval 49.8%

       \[\leadsto \mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\left(x + 1\right)}^{\color{blue}{-0.5}}\right) \]

7. Applied egg-rr 49.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-0.25}, {x}^{-0.25}, -{\left(x + 1\right)}^{-0.5}\right)} \]

8. Taylor expanded in x around inf 21.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} - {\left(\frac{1}{x}\right)}^{0.5}} \]
9. Step-by-step derivation

   1. unpow1/2 21.4%

      \[\leadsto \sqrt{\frac{1}{x}} - \color{blue}{\sqrt{\frac{1}{x}}} \]

   2. +-inverses 21.4%

      \[\leadsto \color{blue}{0} \]

10. Simplified 21.4%

    \[\leadsto \color{blue}{0} \]

11. Final simplification 21.4%

    \[\leadsto 0 \]

Developer target: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0))))))

C

double code(double x) {
	return 1.0 / (((x + 1.0) * sqrt(x)) + (x * sqrt((x + 1.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (((x + 1.0d0) * sqrt(x)) + (x * sqrt((x + 1.0d0))))
end function

Java

public static double code(double x) {
	return 1.0 / (((x + 1.0) * Math.sqrt(x)) + (x * Math.sqrt((x + 1.0))));
}

Python

def code(x):
	return 1.0 / (((x + 1.0) * math.sqrt(x)) + (x * math.sqrt((x + 1.0))))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(x + 1.0) * sqrt(x)) + Float64(x * sqrt(Float64(x + 1.0)))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (((x + 1.0) * sqrt(x)) + (x * sqrt((x + 1.0))));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(x + 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x * N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023319 
(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)))))