2sqrt (example 3.1)

Percentage Accurate: 52.9% → 99.7%

Time: 7.6s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\left(\sqrt{x} + \sqrt{x + 1}\right)}^{-2}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (sqrt (pow (+ (sqrt x) (sqrt (+ x 1.0))) -2.0)))

C

double code(double x) {
	return sqrt(pow((sqrt(x) + sqrt((x + 1.0))), -2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((sqrt(x) + sqrt((x + 1.0d0))) ** (-2.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt(Math.pow((Math.sqrt(x) + Math.sqrt((x + 1.0))), -2.0));
}

Python

def code(x):
	return math.sqrt(math.pow((math.sqrt(x) + math.sqrt((x + 1.0))), -2.0))

Julia

function code(x)
	return sqrt((Float64(sqrt(x) + sqrt(Float64(x + 1.0))) ^ -2.0))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((sqrt(x) + sqrt((x + 1.0))) ^ -2.0));
end

Wolfram

code[x_] := N[Sqrt[N[Power[N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.7%

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

   1. flip-- 52.0%

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

   2. div-inv 52.0%

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

   3. add-sqr-sqrt 51.7%

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

   4. add-sqr-sqrt 52.6%

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

   5. associate--l+ 52.6%

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

3. Applied egg-rr 52.6%

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

   1. +-commutative 52.6%

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

   2. associate-+l- 99.7%

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

   3. +-inverses 99.7%

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

   4. metadata-eval 99.7%

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

   5. associate-*r/ 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

5. Simplified 99.7%

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

   1. +-commutative 99.7%

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

   2. add-sqr-sqrt 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.7%

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

   5. pow1/2 99.7%

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

   6. sqrt-pow1 99.7%

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

   7. metadata-eval 99.7%

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

   8. pow1/2 99.7%

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

   9. sqrt-pow1 99.7%

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

   10. metadata-eval 99.7%

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

   11. +-commutative 99.7%

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

7. Applied egg-rr 99.7%

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

   1. fma-udef 99.6%

      \[\leadsto \frac{1}{\color{blue}{{x}^{0.25} \cdot {x}^{0.25} + \sqrt{1 + x}}} \]

   2. pow-prod-up 99.7%

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

   3. metadata-eval 99.7%

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

   4. pow1/2 99.7%

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

   5. pow1/2 99.7%

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

   6. metadata-eval 99.7%

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

   7. pow-pow 96.2%

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

   8. pow1/3 99.5%

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

   9. +-commutative 99.5%

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

   10. add-sqr-sqrt 99.4%

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

   11. sqrt-unprod 99.5%

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

   12. inv-pow 99.5%

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

   13. inv-pow 99.5%

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

   14. pow-prod-up 99.5%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 2e-7) (* 0.5 (sqrt (/ 1.0 x))) t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 2e-7) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0)) - sqrt(x)
    if (t_0 <= 2d-7) then
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 2e-7) {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 2e-7:
		tmp = 0.5 * math.sqrt((1.0 / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 2e-7)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 2e-7)
		tmp = 0.5 * sqrt((1.0 / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-7], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 1.9999999999999999e-7

   1. Initial program 4.2%

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

      1. flip-- 4.8%

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

      2. div-inv 4.8%

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

      3. add-sqr-sqrt 3.8%

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

      4. add-sqr-sqrt 5.4%

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

      5. associate--l+ 5.4%

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

   3. Applied egg-rr 5.4%

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

      1. +-commutative 5.4%

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

      2. associate-+l- 99.5%

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

      3. +-inverses 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate-*r/ 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   5. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. add-sqr-sqrt 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.5%

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

      5. pow1/2 99.5%

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

      6. sqrt-pow1 99.5%

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

      7. metadata-eval 99.5%

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

      8. pow1/2 99.5%

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

      9. sqrt-pow1 99.4%

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

      10. metadata-eval 99.4%

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

      11. +-commutative 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. fma-udef 99.4%

         \[\leadsto \frac{1}{\color{blue}{{x}^{0.25} \cdot {x}^{0.25} + \sqrt{1 + x}}} \]

      2. pow-prod-up 99.5%

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

      3. metadata-eval 99.5%

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

      4. pow1/2 99.5%

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

      5. pow1/2 99.5%

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

      6. metadata-eval 99.5%

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

      7. pow-pow 92.5%

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

      8. pow1/3 99.2%

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

      9. +-commutative 99.2%

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

      10. flip3-+ 62.9%

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

      11. pow1/3 57.9%

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

      12. pow-pow 62.5%

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

      13. metadata-eval 62.5%

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

      14. pow-pow 62.5%

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

      15. metadata-eval 62.5%

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

      16. sqrt-pow2 62.5%

          \[\leadsto \frac{1}{\frac{{\left(1 + x\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}}{{\left(\sqrt[3]{1 + x}\right)}^{1.5} \cdot {\left(\sqrt[3]{1 + x}\right)}^{1.5} + \left(\sqrt{x} \cdot \sqrt{x} - {\left(\sqrt[3]{1 + x}\right)}^{1.5} \cdot \sqrt{x}\right)}} \]

      17. metadata-eval 62.5%

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

   9. Applied egg-rr 46.0%

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

   10. Taylor expanded in x around inf 99.7%

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

   if 1.9999999999999999e-7 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.1%

      \[\sqrt{x + 1} - \sqrt{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. flip-- 52.0%

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

   2. div-inv 52.0%

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

   3. add-sqr-sqrt 51.7%

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

   4. add-sqr-sqrt 52.6%

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

   5. associate--l+ 52.6%

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

3. Applied egg-rr 52.6%

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

   1. +-commutative 52.6%

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

   2. associate-+l- 99.7%

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

   3. +-inverses 99.7%

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

   4. metadata-eval 99.7%

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

   5. associate-*r/ 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25)
   (- (+ 1.0 (* x (+ 0.5 (* x -0.125)))) (sqrt x))
   (* 0.5 (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - sqrt(x);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.25d0) then
        tmp = (1.0d0 + (x * (0.5d0 + (x * (-0.125d0))))) - sqrt(x)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - math.sqrt(x)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * -0.125)))) - sqrt(x));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.25)
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - sqrt(x);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{\left(1 + \left(-0.125 \cdot {x}^{2} + 0.5 \cdot x\right)\right)} - \sqrt{x} \]
   3. Step-by-step derivation

      1. +-commutative 99.0%

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

      2. unpow2 99.0%

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

      3. associate-*r* 99.0%

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

      4. distribute-rgt-out 99.0%

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

      5. *-commutative 99.0%

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

   4. Simplified 99.0%

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

   if 1.25 < x

   1. Initial program 6.3%

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

      1. flip-- 7.0%

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

      2. div-inv 7.0%

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

      3. add-sqr-sqrt 6.5%

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

      4. add-sqr-sqrt 8.2%

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

      5. associate--l+ 8.2%

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

   3. Applied egg-rr 8.2%

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

      1. +-commutative 8.2%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. add-sqr-sqrt 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.5%

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

      5. pow1/2 99.5%

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

      6. sqrt-pow1 99.5%

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

      7. metadata-eval 99.5%

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

      8. pow1/2 99.5%

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

      9. sqrt-pow1 99.4%

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

      10. metadata-eval 99.4%

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

      11. +-commutative 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. fma-udef 99.4%

         \[\leadsto \frac{1}{\color{blue}{{x}^{0.25} \cdot {x}^{0.25} + \sqrt{1 + x}}} \]

      2. pow-prod-up 99.6%

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

      3. metadata-eval 99.6%

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

      4. pow1/2 99.6%

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

      5. pow1/2 99.6%

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

      6. metadata-eval 99.6%

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

      7. pow-pow 92.7%

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

      8. pow1/3 99.2%

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

      9. +-commutative 99.2%

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

      10. flip3-+ 64.0%

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

      11. pow1/3 59.1%

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

      12. pow-pow 63.6%

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

      13. metadata-eval 63.6%

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

      14. pow-pow 63.6%

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

      15. metadata-eval 63.6%

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

      16. sqrt-pow2 63.7%

          \[\leadsto \frac{1}{\frac{{\left(1 + x\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}}{{\left(\sqrt[3]{1 + x}\right)}^{1.5} \cdot {\left(\sqrt[3]{1 + x}\right)}^{1.5} + \left(\sqrt{x} \cdot \sqrt{x} - {\left(\sqrt[3]{1 + x}\right)}^{1.5} \cdot \sqrt{x}\right)}} \]

      17. metadata-eval 63.7%

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

   9. Applied egg-rr 47.6%

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

   10. Taylor expanded in x around inf 98.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]

Alternative 5: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x * 0.5) + (1.0 - sqrt(x));
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	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 - sqrt(x))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. associate--l+ 98.5%

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

      3. *-commutative 98.5%

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

   4. Applied egg-rr 98.5%

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

   if 1 < x

   1. Initial program 6.3%

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

      1. flip-- 7.0%

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

      2. div-inv 7.0%

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

      3. add-sqr-sqrt 6.5%

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

      4. add-sqr-sqrt 8.2%

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

      5. associate--l+ 8.2%

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

   3. Applied egg-rr 8.2%

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

      1. +-commutative 8.2%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. add-sqr-sqrt 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.5%

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

      5. pow1/2 99.5%

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

      6. sqrt-pow1 99.5%

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

      7. metadata-eval 99.5%

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

      8. pow1/2 99.5%

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

      9. sqrt-pow1 99.4%

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

      10. metadata-eval 99.4%

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

      11. +-commutative 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. fma-udef 99.4%

         \[\leadsto \frac{1}{\color{blue}{{x}^{0.25} \cdot {x}^{0.25} + \sqrt{1 + x}}} \]

      2. pow-prod-up 99.6%

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

      3. metadata-eval 99.6%

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

      4. pow1/2 99.6%

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

      5. pow1/2 99.6%

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

      6. metadata-eval 99.6%

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

      7. pow-pow 92.7%

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

      8. pow1/3 99.2%

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

      9. +-commutative 99.2%

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

      10. flip3-+ 64.0%

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

      11. pow1/3 59.1%

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

      12. pow-pow 63.6%

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

      13. metadata-eval 63.6%

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

      14. pow-pow 63.6%

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

      15. metadata-eval 63.6%

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

      16. sqrt-pow2 63.7%

          \[\leadsto \frac{1}{\frac{{\left(1 + x\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}}{{\left(\sqrt[3]{1 + x}\right)}^{1.5} \cdot {\left(\sqrt[3]{1 + x}\right)}^{1.5} + \left(\sqrt{x} \cdot \sqrt{x} - {\left(\sqrt[3]{1 + x}\right)}^{1.5} \cdot \sqrt{x}\right)}} \]

      17. metadata-eval 63.7%

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

   9. Applied egg-rr 47.6%

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

   10. Taylor expanded in x around inf 98.2%

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

4. Final simplification 98.3%

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

Alternative 6: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. associate--l+ 98.5%

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

      3. *-commutative 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. +-commutative 98.5%

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

      2. associate-+l- 98.6%

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

   6. Applied egg-rr 98.6%

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

   if 1 < x

   1. Initial program 6.3%

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

      1. flip-- 7.0%

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

      2. div-inv 7.0%

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

      3. add-sqr-sqrt 6.5%

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

      4. add-sqr-sqrt 8.2%

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

      5. associate--l+ 8.2%

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

   3. Applied egg-rr 8.2%

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

      1. +-commutative 8.2%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. add-sqr-sqrt 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.5%

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

      5. pow1/2 99.5%

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

      6. sqrt-pow1 99.5%

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

      7. metadata-eval 99.5%

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

      8. pow1/2 99.5%

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

      9. sqrt-pow1 99.4%

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

      10. metadata-eval 99.4%

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

      11. +-commutative 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. fma-udef 99.4%

         \[\leadsto \frac{1}{\color{blue}{{x}^{0.25} \cdot {x}^{0.25} + \sqrt{1 + x}}} \]

      2. pow-prod-up 99.6%

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

      3. metadata-eval 99.6%

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

      4. pow1/2 99.6%

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

      5. pow1/2 99.6%

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

      6. metadata-eval 99.6%

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

      7. pow-pow 92.7%

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

      8. pow1/3 99.2%

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

      9. +-commutative 99.2%

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

      10. flip3-+ 64.0%

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

      11. pow1/3 59.1%

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

      12. pow-pow 63.6%

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

      13. metadata-eval 63.6%

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

      14. pow-pow 63.6%

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

      15. metadata-eval 63.6%

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

      16. sqrt-pow2 63.7%

          \[\leadsto \frac{1}{\frac{{\left(1 + x\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}}{{\left(\sqrt[3]{1 + x}\right)}^{1.5} \cdot {\left(\sqrt[3]{1 + x}\right)}^{1.5} + \left(\sqrt{x} \cdot \sqrt{x} - {\left(\sqrt[3]{1 + x}\right)}^{1.5} \cdot \sqrt{x}\right)}} \]

      17. metadata-eval 63.7%

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

   9. Applied egg-rr 47.6%

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

   10. Taylor expanded in x around inf 98.2%

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

4. Final simplification 98.4%

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

Alternative 7: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.25) 1.0 (* 0.5 (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.25d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.25:
		tmp = 1.0
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.25)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.25)
		tmp = 1.0;
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.25], 1.0, N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 95.3%

      \[\leadsto \color{blue}{1} \]

   if 0.25 < x

   1. Initial program 6.3%

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

      1. flip-- 7.0%

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

      2. div-inv 7.0%

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

      3. add-sqr-sqrt 6.5%

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

      4. add-sqr-sqrt 8.2%

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

      5. associate--l+ 8.2%

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

   3. Applied egg-rr 8.2%

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

      1. +-commutative 8.2%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. add-sqr-sqrt 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.5%

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

      5. pow1/2 99.5%

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

      6. sqrt-pow1 99.5%

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

      7. metadata-eval 99.5%

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

      8. pow1/2 99.5%

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

      9. sqrt-pow1 99.4%

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

      10. metadata-eval 99.4%

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

      11. +-commutative 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. fma-udef 99.4%

         \[\leadsto \frac{1}{\color{blue}{{x}^{0.25} \cdot {x}^{0.25} + \sqrt{1 + x}}} \]

      2. pow-prod-up 99.6%

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

      3. metadata-eval 99.6%

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

      4. pow1/2 99.6%

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

      5. pow1/2 99.6%

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

      6. metadata-eval 99.6%

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

      7. pow-pow 92.7%

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

      8. pow1/3 99.2%

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

      9. +-commutative 99.2%

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

      10. flip3-+ 64.0%

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

      11. pow1/3 59.1%

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

      12. pow-pow 63.6%

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

      13. metadata-eval 63.6%

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

      14. pow-pow 63.6%

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

      15. metadata-eval 63.6%

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

      16. sqrt-pow2 63.7%

          \[\leadsto \frac{1}{\frac{{\left(1 + x\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}}{{\left(\sqrt[3]{1 + x}\right)}^{1.5} \cdot {\left(\sqrt[3]{1 + x}\right)}^{1.5} + \left(\sqrt{x} \cdot \sqrt{x} - {\left(\sqrt[3]{1 + x}\right)}^{1.5} \cdot \sqrt{x}\right)}} \]

      17. metadata-eval 63.7%

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

   9. Applied egg-rr 47.6%

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

   10. Taylor expanded in x around inf 98.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]

Alternative 8: 97.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--l+ 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l- 99.9%

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

      3. +-inverses 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-*r/ 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. pow1/2 99.9%

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

      6. sqrt-pow1 99.9%

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

      7. metadata-eval 99.9%

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

      8. pow1/2 99.9%

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

      9. sqrt-pow1 99.9%

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

      10. metadata-eval 99.9%

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

      11. +-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around 0 97.4%

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

   if 1 < x

   1. Initial program 6.3%

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

      1. flip-- 7.0%

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

      2. div-inv 7.0%

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

      3. add-sqr-sqrt 6.5%

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

      4. add-sqr-sqrt 8.2%

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

      5. associate--l+ 8.2%

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

   3. Applied egg-rr 8.2%

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

      1. +-commutative 8.2%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. add-sqr-sqrt 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.5%

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

      5. pow1/2 99.5%

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

      6. sqrt-pow1 99.5%

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

      7. metadata-eval 99.5%

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

      8. pow1/2 99.5%

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

      9. sqrt-pow1 99.4%

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

      10. metadata-eval 99.4%

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

      11. +-commutative 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. fma-udef 99.4%

         \[\leadsto \frac{1}{\color{blue}{{x}^{0.25} \cdot {x}^{0.25} + \sqrt{1 + x}}} \]

      2. pow-prod-up 99.6%

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

      3. metadata-eval 99.6%

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

      4. pow1/2 99.6%

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

      5. pow1/2 99.6%

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

      6. metadata-eval 99.6%

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

      7. pow-pow 92.7%

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

      8. pow1/3 99.2%

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

      9. +-commutative 99.2%

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

      10. flip3-+ 64.0%

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

      11. pow1/3 59.1%

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

      12. pow-pow 63.6%

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

      13. metadata-eval 63.6%

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

      14. pow-pow 63.6%

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

      15. metadata-eval 63.6%

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

      16. sqrt-pow2 63.7%

          \[\leadsto \frac{1}{\frac{{\left(1 + x\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}}{{\left(\sqrt[3]{1 + x}\right)}^{1.5} \cdot {\left(\sqrt[3]{1 + x}\right)}^{1.5} + \left(\sqrt{x} \cdot \sqrt{x} - {\left(\sqrt[3]{1 + x}\right)}^{1.5} \cdot \sqrt{x}\right)}} \]

      17. metadata-eval 63.7%

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

   9. Applied egg-rr 47.6%

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

   10. Taylor expanded in x around inf 98.2%

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

4. Final simplification 97.8%

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

Alternative 9: 51.0% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 51.7%

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

2. Taylor expanded in x around 0 49.6%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 49.6%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023321 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))