Main:bigenough3 from C

Percentage Accurate: 53.7% → 99.7%

Time: 8.1s

Alternatives: 9

Speedup: 2.0×

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: 53.7% 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

\[\begin{array}{l} \\ {\left({\left(\sqrt{x} + \sqrt{x + 1}\right)}^{2}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (pow (+ (sqrt x) (sqrt (+ x 1.0))) 2.0) -0.5))

C

double code(double x) {
	return pow(pow((sqrt(x) + sqrt((x + 1.0))), 2.0), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((sqrt(x) + sqrt((x + 1.0d0))) ** 2.0d0) ** (-0.5d0)
end function

Java

public static double code(double x) {
	return Math.pow(Math.pow((Math.sqrt(x) + Math.sqrt((x + 1.0))), 2.0), -0.5);
}

Python

def code(x):
	return math.pow(math.pow((math.sqrt(x) + math.sqrt((x + 1.0))), 2.0), -0.5)

Julia

function code(x)
	return (Float64(sqrt(x) + sqrt(Float64(x + 1.0))) ^ 2.0) ^ -0.5
end

MATLAB

function tmp = code(x)
	tmp = ((sqrt(x) + sqrt((x + 1.0))) ^ 2.0) ^ -0.5;
end

Wolfram

code[x_] := N[Power[N[Power[N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 51.1%

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

   1. flip-- 51.3%

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

   2. div-inv 51.3%

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

   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 52.7%

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

   5. associate--l+ 52.7%

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

4. Applied egg-rr 52.7%

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

   1. associate-*r/ 52.7%

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

   2. *-rgt-identity 52.7%

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

   3. +-commutative 52.7%

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

   4. associate-+l- 99.8%

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

   5. +-inverses 99.8%

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

   6. metadata-eval 99.8%

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

   7. +-commutative 99.8%

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

6. Simplified 99.8%

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

   1. +-commutative 99.8%

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

   2. add-sqr-sqrt 99.6%

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

   3. fma-def 99.8%

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

   4. pow1/2 99.8%

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

   5. sqrt-pow1 99.8%

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

   6. metadata-eval 99.8%

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

   7. pow1/2 99.8%

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

   8. sqrt-pow1 99.8%

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

   9. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

   1. inv-pow 99.8%

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

   2. add-sqr-sqrt 99.5%

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

   3. unpow-prod-down 99.4%

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

10. Applied egg-rr 99.8%

    \[\leadsto \color{blue}{{\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}\right)}^{2}\right)}^{-0.5}} \]
11. Step-by-step derivation

    1. +-commutative 99.8%

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

    2. unpow1/2 99.8%

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

    3. metadata-eval 99.8%

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

    4. pow-sqr 99.6%

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

    5. hypot-1-def 99.6%

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

    6. pow-sqr 99.7%

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

    7. metadata-eval 99.7%

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

    8. metadata-eval 99.7%

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

    9. pow-sqr 99.8%

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

    10. metadata-eval 99.8%

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

    11. metadata-eval 99.8%

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

    12. sqr-pow 99.8%

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

    13. unpow1 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

    \[\leadsto {\left({\left(\sqrt{x} + \sqrt{x + 1}\right)}^{2}\right)}^{-0.5} \]
14. Add Preprocessing

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;0.5 \cdot {\left(\sqrt{x}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 1e-5) (* 0.5 (pow (sqrt x) -1.0)) t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = 0.5 * pow(sqrt(x), -1.0);
	} 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 <= 1d-5) then
        tmp = 0.5d0 * (sqrt(x) ** (-1.0d0))
    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 <= 1e-5) {
		tmp = 0.5 * Math.pow(Math.sqrt(x), -1.0);
	} 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 <= 1e-5:
		tmp = 0.5 * math.pow(math.sqrt(x), -1.0)
	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 <= 1e-5)
		tmp = Float64(0.5 * (sqrt(x) ^ -1.0));
	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 <= 1e-5)
		tmp = 0.5 * (sqrt(x) ^ -1.0);
	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, 1e-5], N[(0.5 * N[Power[N[Sqrt[x], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 1.00000000000000008e-5

   1. Initial program 5.2%

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

      1. flip-- 5.7%

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

      2. div-inv 5.7%

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

      3. add-sqr-sqrt 6.1%

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

      4. add-sqr-sqrt 7.5%

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

      5. associate--l+ 7.5%

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

   4. Applied egg-rr 7.5%

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

      1. associate-*r/ 7.5%

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

      2. *-rgt-identity 7.5%

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

      3. +-commutative 7.5%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   6. Simplified 99.6%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

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

      4. sqrt-pow1 99.3%

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

      5. metadata-eval 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in x around inf 99.0%

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

       1. inv-pow 99.0%

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

       2. count-2 99.0%

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

       3. unpow-prod-down 99.0%

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

       4. metadata-eval 99.0%

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

   11. Applied egg-rr 99.0%

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

   if 1.00000000000000008e-5 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.2%

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

4. Final simplification 99.1%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

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 * pow(sqrt(x), -1.0);
	}
	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(x) ** (-1.0d0))
    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.pow(Math.sqrt(x), -1.0);
	}
	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.pow(math.sqrt(x), -1.0)
	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(x) ^ -1.0));
	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(x) ^ -1.0);
	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[Power[N[Sqrt[x], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. associate-*r* 99.6%

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

      4. distribute-rgt-out 99.6%

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

      5. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   if 1.25 < x

   1. Initial program 7.3%

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

      1. flip-- 7.8%

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

      2. div-inv 7.8%

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

      3. add-sqr-sqrt 8.5%

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

      4. add-sqr-sqrt 10.3%

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

      5. associate--l+ 10.3%

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

   4. Applied egg-rr 10.3%

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

      1. associate-*r/ 10.3%

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

      2. *-rgt-identity 10.3%

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

      3. +-commutative 10.3%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   6. Simplified 99.6%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

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

      4. sqrt-pow1 99.3%

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

      5. metadata-eval 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in x around inf 97.4%

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

       1. inv-pow 97.4%

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

       2. count-2 97.4%

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

       3. unpow-prod-down 97.4%

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

       4. metadata-eval 97.4%

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

   11. Applied egg-rr 97.4%

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

4. Final simplification 98.4%

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

Alternative 4: 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 51.1%

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

   1. flip-- 51.3%

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

   2. div-inv 51.3%

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

   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 52.7%

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

   5. associate--l+ 52.7%

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

4. Applied egg-rr 52.7%

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

   1. associate-*r/ 52.7%

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

   2. *-rgt-identity 52.7%

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

   3. +-commutative 52.7%

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

   4. associate-+l- 99.8%

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

   5. +-inverses 99.8%

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

   6. metadata-eval 99.8%

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

   7. +-commutative 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 5: 85.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + \left(x \cdot 0.5 - \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+205}:\\ \;\;\;\;x \cdot \frac{1}{2 \cdot {x}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (+ 1.0 (- (* x 0.5) (sqrt x)))
   (if (<= x 2e+205) (* x (/ 1.0 (* 2.0 (pow x 1.5)))) (/ (sqrt x) x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + ((x * 0.5) - sqrt(x));
	} else if (x <= 2e+205) {
		tmp = x * (1.0 / (2.0 * pow(x, 1.5)));
	} else {
		tmp = sqrt(x) / 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 if (x <= 2d+205) then
        tmp = x * (1.0d0 / (2.0d0 * (x ** 1.5d0)))
    else
        tmp = sqrt(x) / 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 if (x <= 2e+205) {
		tmp = x * (1.0 / (2.0 * Math.pow(x, 1.5)));
	} else {
		tmp = Math.sqrt(x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 + ((x * 0.5) - math.sqrt(x))
	elif x <= 2e+205:
		tmp = x * (1.0 / (2.0 * math.pow(x, 1.5)))
	else:
		tmp = math.sqrt(x) / 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)));
	elseif (x <= 2e+205)
		tmp = Float64(x * Float64(1.0 / Float64(2.0 * (x ^ 1.5))));
	else
		tmp = Float64(sqrt(x) / 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));
	elseif (x <= 2e+205)
		tmp = x * (1.0 / (2.0 * (x ^ 1.5)));
	else
		tmp = sqrt(x) / 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], If[LessEqual[x, 2e+205], N[(x * N[(1.0 / N[(2.0 * N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.5%

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

      1. associate--l+ 99.5%

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

      2. +-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   if 1 < x < 2.00000000000000003e205

   1. Initial program 8.6%

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

      1. flip-- 9.4%

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

      2. div-inv 9.4%

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

      3. add-sqr-sqrt 10.9%

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

      4. add-sqr-sqrt 13.0%

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

      5. associate--l+ 13.0%

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

   4. Applied egg-rr 13.0%

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

      1. associate-*r/ 13.0%

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

      2. *-rgt-identity 13.0%

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

      3. +-commutative 13.0%

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

      4. associate-+l- 99.7%

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

      5. +-inverses 99.7%

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

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. add-sqr-sqrt 99.4%

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

      2. pow2 99.4%

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

      3. pow1/2 99.4%

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

      4. sqrt-pow1 99.4%

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

      5. metadata-eval 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in x around inf 96.5%

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

       1. flip3-+ 96.3%

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

       2. associate-/r/ 96.3%

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

       3. count-2 96.3%

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

       4. sqrt-pow2 96.0%

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

       5. metadata-eval 96.0%

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

       6. add-sqr-sqrt 96.3%

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

       7. add-sqr-sqrt 96.0%

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

       8. add-sqr-sqrt 96.3%

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

       9. +-inverses 96.3%

          \[\leadsto \frac{1}{2 \cdot {x}^{1.5}} \cdot \left(x + \color{blue}{0}\right) \]

   11. Applied egg-rr 96.3%

       \[\leadsto \color{blue}{\frac{1}{2 \cdot {x}^{1.5}} \cdot \left(x + 0\right)} \]

   if 2.00000000000000003e205 < x

   1. Initial program 4.4%

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

      1. flip-- 4.4%

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

      2. div-inv 4.4%

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

      3. add-sqr-sqrt 3.3%

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

      4. add-sqr-sqrt 4.4%

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

      5. associate--l+ 4.4%

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

   4. Applied egg-rr 4.4%

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

      1. associate-*r/ 4.4%

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

      2. *-rgt-identity 4.4%

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

      3. +-commutative 4.4%

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

      4. associate-+l- 99.5%

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

      5. +-inverses 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

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

      4. sqrt-pow1 99.3%

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

      5. metadata-eval 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in x around inf 99.5%

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

       1. flip-+ 0.0%

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

       2. add-sqr-sqrt 2.7%

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

       3. add-sqr-sqrt 0.0%

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

       4. +-inverses 0.0%

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

       5. +-inverses 0.0%

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

       6. +-inverses 0.0%

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

       7. +-inverses 0.0%

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

       8. add-sqr-sqrt 1.0%

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

       9. add-sqr-sqrt 0.0%

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

       10. clear-num 0.0%

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

       11. flip-+ 3.5%

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

       12. flip3-+ 2.3%

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

       13. div-inv 2.3%

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

       14. count-2 2.3%

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

       15. sqrt-pow2 2.3%

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

       16. metadata-eval 2.3%

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

   11. Applied egg-rr 2.3%

       \[\leadsto \color{blue}{\left(2 \cdot {x}^{1.5}\right) \cdot \frac{1}{x + 0}} \]

   13. Simplified 18.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + \left(x \cdot 0.5 - \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+205}:\\ \;\;\;\;x \cdot \frac{1}{2 \cdot {x}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+205}:\\ \;\;\;\;x \cdot \frac{1}{2 \cdot {x}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25)
   (- (+ 1.0 (* x (+ 0.5 (* x -0.125)))) (sqrt x))
   (if (<= x 2e+205) (* x (/ 1.0 (* 2.0 (pow x 1.5)))) (/ (sqrt x) x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - sqrt(x);
	} else if (x <= 2e+205) {
		tmp = x * (1.0 / (2.0 * pow(x, 1.5)));
	} else {
		tmp = sqrt(x) / 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 if (x <= 2d+205) then
        tmp = x * (1.0d0 / (2.0d0 * (x ** 1.5d0)))
    else
        tmp = sqrt(x) / 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 if (x <= 2e+205) {
		tmp = x * (1.0 / (2.0 * Math.pow(x, 1.5)));
	} else {
		tmp = Math.sqrt(x) / 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)
	elif x <= 2e+205:
		tmp = x * (1.0 / (2.0 * math.pow(x, 1.5)))
	else:
		tmp = math.sqrt(x) / 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));
	elseif (x <= 2e+205)
		tmp = Float64(x * Float64(1.0 / Float64(2.0 * (x ^ 1.5))));
	else
		tmp = Float64(sqrt(x) / 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);
	elseif (x <= 2e+205)
		tmp = x * (1.0 / (2.0 * (x ^ 1.5)));
	else
		tmp = sqrt(x) / 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], If[LessEqual[x, 2e+205], N[(x * N[(1.0 / N[(2.0 * N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.25

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. associate-*r* 99.6%

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

      4. distribute-rgt-out 99.6%

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

      5. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   if 1.25 < x < 2.00000000000000003e205

   1. Initial program 8.6%

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

      1. flip-- 9.4%

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

      2. div-inv 9.4%

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

      3. add-sqr-sqrt 10.9%

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

      4. add-sqr-sqrt 13.0%

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

      5. associate--l+ 13.0%

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

   4. Applied egg-rr 13.0%

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

      1. associate-*r/ 13.0%

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

      2. *-rgt-identity 13.0%

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

      3. +-commutative 13.0%

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

      4. associate-+l- 99.7%

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

      5. +-inverses 99.7%

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

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. add-sqr-sqrt 99.4%

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

      2. pow2 99.4%

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

      3. pow1/2 99.4%

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

      4. sqrt-pow1 99.4%

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

      5. metadata-eval 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in x around inf 96.5%

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

       1. flip3-+ 96.3%

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

       2. associate-/r/ 96.3%

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

       3. count-2 96.3%

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

       4. sqrt-pow2 96.0%

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

       5. metadata-eval 96.0%

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

       6. add-sqr-sqrt 96.3%

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

       7. add-sqr-sqrt 96.0%

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

       8. add-sqr-sqrt 96.3%

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

       9. +-inverses 96.3%

          \[\leadsto \frac{1}{2 \cdot {x}^{1.5}} \cdot \left(x + \color{blue}{0}\right) \]

   11. Applied egg-rr 96.3%

       \[\leadsto \color{blue}{\frac{1}{2 \cdot {x}^{1.5}} \cdot \left(x + 0\right)} \]

   if 2.00000000000000003e205 < x

   1. Initial program 4.4%

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

      1. flip-- 4.4%

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

      2. div-inv 4.4%

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

      3. add-sqr-sqrt 3.3%

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

      4. add-sqr-sqrt 4.4%

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

      5. associate--l+ 4.4%

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

   4. Applied egg-rr 4.4%

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

      1. associate-*r/ 4.4%

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

      2. *-rgt-identity 4.4%

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

      3. +-commutative 4.4%

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

      4. associate-+l- 99.5%

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

      5. +-inverses 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

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

      4. sqrt-pow1 99.3%

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

      5. metadata-eval 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in x around inf 99.5%

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

       1. flip-+ 0.0%

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

       2. add-sqr-sqrt 2.7%

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

       3. add-sqr-sqrt 0.0%

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

       4. +-inverses 0.0%

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

       5. +-inverses 0.0%

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

       6. +-inverses 0.0%

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

       7. +-inverses 0.0%

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

       8. add-sqr-sqrt 1.0%

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

       9. add-sqr-sqrt 0.0%

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

       10. clear-num 0.0%

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

       11. flip-+ 3.5%

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

       12. flip3-+ 2.3%

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

       13. div-inv 2.3%

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

       14. count-2 2.3%

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

       15. sqrt-pow2 2.3%

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

       16. metadata-eval 2.3%

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

   11. Applied egg-rr 2.3%

       \[\leadsto \color{blue}{\left(2 \cdot {x}^{1.5}\right) \cdot \frac{1}{x + 0}} \]

   13. Simplified 18.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+205}:\\ \;\;\;\;x \cdot \frac{1}{2 \cdot {x}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.4:\\ \;\;\;\;1 + x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.4) (+ 1.0 (* x -2.0)) (/ (sqrt x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.4) {
		tmp = 1.0 + (x * -2.0);
	} else {
		tmp = sqrt(x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.4d0) then
        tmp = 1.0d0 + (x * (-2.0d0))
    else
        tmp = sqrt(x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.4) {
		tmp = 1.0 + (x * -2.0);
	} else {
		tmp = Math.sqrt(x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.4:
		tmp = 1.0 + (x * -2.0)
	else:
		tmp = math.sqrt(x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.4)
		tmp = Float64(1.0 + Float64(x * -2.0));
	else
		tmp = Float64(sqrt(x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.4)
		tmp = 1.0 + (x * -2.0);
	else
		tmp = sqrt(x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.4], N[(1.0 + N[(x * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.40000000000000002

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.5%

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

      1. associate--l+ 99.5%

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

      2. +-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x around inf 97.0%

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

   8. Simplified 97.0%

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

   if 0.40000000000000002 < x

   1. Initial program 7.3%

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

      1. flip-- 7.8%

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

      2. div-inv 7.8%

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

      3. add-sqr-sqrt 8.5%

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

      4. add-sqr-sqrt 10.3%

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

      5. associate--l+ 10.3%

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

   4. Applied egg-rr 10.3%

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

      1. associate-*r/ 10.3%

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

      2. *-rgt-identity 10.3%

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

      3. +-commutative 10.3%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   6. Simplified 99.6%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

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

      4. sqrt-pow1 99.3%

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

      5. metadata-eval 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in x around inf 97.4%

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

       1. flip-+ 0.0%

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

       2. add-sqr-sqrt 1.9%

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

       3. add-sqr-sqrt 0.0%

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

       4. +-inverses 0.0%

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

       5. +-inverses 0.0%

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

       6. +-inverses 0.0%

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

       7. +-inverses 0.0%

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

       8. add-sqr-sqrt 0.9%

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

       9. add-sqr-sqrt 0.0%

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

       10. clear-num 0.0%

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

       11. flip-+ 5.5%

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

       12. flip3-+ 5.1%

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

       13. div-inv 5.1%

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

       14. count-2 5.1%

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

       15. sqrt-pow2 5.1%

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

       16. metadata-eval 5.1%

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

   11. Applied egg-rr 5.1%

       \[\leadsto \color{blue}{\left(2 \cdot {x}^{1.5}\right) \cdot \frac{1}{x + 0}} \]

   13. Simplified 18.7%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.4:\\ \;\;\;\;1 + x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

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

   1. flip-- 51.3%

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

   2. div-inv 51.3%

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

   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 52.7%

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

   5. associate--l+ 52.7%

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

4. Applied egg-rr 52.7%

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

   1. associate-*r/ 52.7%

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

   2. *-rgt-identity 52.7%

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

   3. +-commutative 52.7%

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

   4. associate-+l- 99.8%

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

   5. +-inverses 99.8%

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

   6. metadata-eval 99.8%

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

   7. +-commutative 99.8%

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

6. Simplified 99.8%

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

   1. +-commutative 99.8%

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

   2. add-sqr-sqrt 99.6%

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

   3. fma-def 99.8%

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

   4. pow1/2 99.8%

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

   5. sqrt-pow1 99.8%

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

   6. metadata-eval 99.8%

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

   7. pow1/2 99.8%

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

   8. sqrt-pow1 99.8%

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

   9. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in x around 0 56.5%

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

10. Final simplification 56.5%

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

Alternative 9: 51.4% 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 51.1%

   \[\sqrt{x + 1} - \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 49.6%

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

4. Final simplification 49.6%

   \[\leadsto 1 \]
5. Add Preprocessing

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 2024027 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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