Main:bigenough3 from C

Percentage Accurate: 54.0% → 99.7%

Time: 7.5s

Alternatives: 11

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: 54.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

   1. flip-- 49.9%

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

   2. div-inv 49.9%

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

   3. add-sqr-sqrt 50.4%

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

   4. add-sqr-sqrt 51.1%

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

   5. associate--l+ 51.1%

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

4. Applied egg-rr 51.1%

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

   1. associate-*r/ 51.1%

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

   2. *-rgt-identity 51.1%

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

   3. +-commutative 51.1%

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

   4. associate-+l- 99.7%

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

   5. div-sub 99.7%

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

   6. +-inverses 99.7%

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

   7. div0 99.7%

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

   8. --rgt-identity 99.7%

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

   9. +-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 99.6%

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

   2. sqrt-unprod 99.7%

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

   3. inv-pow 99.7%

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

   4. inv-pow 99.7%

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

   5. pow-prod-up 99.8%

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

   6. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{0.25 - \frac{0.125}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 5e-5) (sqrt (/ (- 0.25 (/ 0.125 x)) x)) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = sqrt(((0.25 - (0.125 / x)) / 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((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 5d-5) then
        tmp = sqrt(((0.25d0 - (0.125d0 / x)) / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = Math.sqrt(((0.25 - (0.125 / x)) / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 5e-5:
		tmp = math.sqrt(((0.25 - (0.125 / x)) / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 5e-5)
		tmp = sqrt(Float64(Float64(0.25 - Float64(0.125 / x)) / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 5e-5)
		tmp = sqrt(((0.25 - (0.125 / x)) / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-5], N[Sqrt[N[(N[(0.25 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 5.00000000000000024e-5

   1. Initial program 5.0%

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

      1. flip-- 5.2%

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

      2. div-inv 5.2%

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

      3. add-sqr-sqrt 6.2%

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

      4. add-sqr-sqrt 7.4%

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

      5. associate--l+ 7.4%

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

   4. Applied egg-rr 7.4%

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

      1. associate-*r/ 7.4%

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

      2. *-rgt-identity 7.4%

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

      3. +-commutative 7.4%

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

      4. associate-+l- 99.6%

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

      5. div-sub 99.6%

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

      6. +-inverses 99.6%

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

      7. div0 99.6%

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

      8. --rgt-identity 99.6%

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

      9. +-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 99.2%

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

      2. sqrt-unprod 99.6%

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

      3. inv-pow 99.6%

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

      4. inv-pow 99.6%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in x around inf 59.4%

      \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot x - 0.125}{{x}^{2}}}} \]

   10. Taylor expanded in x around inf 99.8%

       \[\leadsto \sqrt{\color{blue}{\frac{0.25 - 0.125 \cdot \frac{1}{x}}{x}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 99.8%

          \[\leadsto \sqrt{\frac{0.25 - \color{blue}{\frac{0.125 \cdot 1}{x}}}{x}} \]

       2. metadata-eval 99.8%

          \[\leadsto \sqrt{\frac{0.25 - \frac{\color{blue}{0.125}}{x}}{x}} \]

   12. Simplified 99.8%

       \[\leadsto \sqrt{\color{blue}{\frac{0.25 - \frac{0.125}{x}}{x}}} \]

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{0.25 - \frac{0.125}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

   1. flip-- 49.9%

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

   2. div-inv 49.9%

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

   3. add-sqr-sqrt 50.4%

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

   4. add-sqr-sqrt 51.1%

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

   5. associate--l+ 51.1%

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

4. Applied egg-rr 51.1%

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

   1. associate-*r/ 51.1%

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

   2. *-rgt-identity 51.1%

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

   3. +-commutative 51.1%

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

   4. associate-+l- 99.7%

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

   5. div-sub 99.7%

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

   6. +-inverses 99.7%

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

   7. div0 99.7%

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

   8. --rgt-identity 99.7%

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

   9. +-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. Final simplification 99.7%

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

Alternative 4: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25 - \frac{0.125}{x}}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.05)
   (+ 1.0 (- (* x (+ 0.5 (* x -0.125))) (sqrt x)))
   (sqrt (/ (- 0.25 (/ 0.125 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.05) {
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x));
	} else {
		tmp = sqrt(((0.25 - (0.125 / x)) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.05d0) then
        tmp = 1.0d0 + ((x * (0.5d0 + (x * (-0.125d0)))) - sqrt(x))
    else
        tmp = sqrt(((0.25d0 - (0.125d0 / x)) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.05) {
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - Math.sqrt(x));
	} else {
		tmp = Math.sqrt(((0.25 - (0.125 / x)) / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.05:
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - math.sqrt(x))
	else:
		tmp = math.sqrt(((0.25 - (0.125 / x)) / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.05)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) - sqrt(x)));
	else
		tmp = sqrt(Float64(Float64(0.25 - Float64(0.125 / x)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.05)
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x));
	else
		tmp = sqrt(((0.25 - (0.125 / x)) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.05], N[(1.0 + N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(0.25 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) - \sqrt{x}} \]
   4. Step-by-step derivation

      1. associate--l+ 100.0%

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

      2. fma-neg 100.0%

         \[\leadsto 1 + \left(x \cdot \left(0.5 + x \cdot \color{blue}{\mathsf{fma}\left(0.0625, x, -0.125\right)}\right) - \sqrt{x}\right) \]

      3. metadata-eval 100.0%

         \[\leadsto 1 + \left(x \cdot \left(0.5 + x \cdot \mathsf{fma}\left(0.0625, x, \color{blue}{-0.125}\right)\right) - \sqrt{x}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \left(x \cdot \left(0.5 + x \cdot \mathsf{fma}\left(0.0625, x, -0.125\right)\right) - \sqrt{x}\right)} \]

   6. Taylor expanded in x around 0 100.0%

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

   if 1.05000000000000004 < x

   1. Initial program 6.7%

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

      1. flip-- 7.1%

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

      2. div-inv 7.1%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 9.4%

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

      5. associate--l+ 9.4%

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

   4. Applied egg-rr 9.4%

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

      1. associate-*r/ 9.4%

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

      2. *-rgt-identity 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l- 99.6%

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

      5. div-sub 99.6%

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

      6. +-inverses 99.6%

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

      7. div0 99.6%

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

      8. --rgt-identity 99.6%

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

      9. +-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 99.2%

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

      2. sqrt-unprod 99.6%

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

      3. inv-pow 99.6%

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

      4. inv-pow 99.6%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in x around inf 59.3%

      \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot x - 0.125}{{x}^{2}}}} \]

   10. Taylor expanded in x around inf 98.8%

       \[\leadsto \sqrt{\color{blue}{\frac{0.25 - 0.125 \cdot \frac{1}{x}}{x}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 98.8%

          \[\leadsto \sqrt{\frac{0.25 - \color{blue}{\frac{0.125 \cdot 1}{x}}}{x}} \]

       2. metadata-eval 98.8%

          \[\leadsto \sqrt{\frac{0.25 - \frac{\color{blue}{0.125}}{x}}{x}} \]

   12. Simplified 98.8%

       \[\leadsto \sqrt{\color{blue}{\frac{0.25 - \frac{0.125}{x}}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25 - \frac{0.125}{x}}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25 - \frac{0.125}{x}}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.58) (- 1.0 (sqrt x)) (sqrt (/ (- 0.25 (/ 0.125 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.58) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = sqrt(((0.25 - (0.125 / x)) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.58d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = sqrt(((0.25d0 - (0.125d0 / x)) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.58) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = Math.sqrt(((0.25 - (0.125 / x)) / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.58:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = math.sqrt(((0.25 - (0.125 / x)) / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.58)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = sqrt(Float64(Float64(0.25 - Float64(0.125 / x)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.58)
		tmp = 1.0 - sqrt(x);
	else
		tmp = sqrt(((0.25 - (0.125 / x)) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.58], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(0.25 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.57999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.1%

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

   if 0.57999999999999996 < x

   1. Initial program 6.7%

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

      1. flip-- 7.1%

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

      2. div-inv 7.1%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 9.4%

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

      5. associate--l+ 9.4%

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

   4. Applied egg-rr 9.4%

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

      1. associate-*r/ 9.4%

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

      2. *-rgt-identity 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l- 99.6%

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

      5. div-sub 99.6%

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

      6. +-inverses 99.6%

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

      7. div0 99.6%

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

      8. --rgt-identity 99.6%

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

      9. +-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 99.2%

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

      2. sqrt-unprod 99.6%

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

      3. inv-pow 99.6%

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

      4. inv-pow 99.6%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in x around inf 59.3%

      \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot x - 0.125}{{x}^{2}}}} \]

   10. Taylor expanded in x around inf 98.8%

       \[\leadsto \sqrt{\color{blue}{\frac{0.25 - 0.125 \cdot \frac{1}{x}}{x}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 98.8%

          \[\leadsto \sqrt{\frac{0.25 - \color{blue}{\frac{0.125 \cdot 1}{x}}}{x}} \]

       2. metadata-eval 98.8%

          \[\leadsto \sqrt{\frac{0.25 - \frac{\color{blue}{0.125}}{x}}{x}} \]

   12. Simplified 98.8%

       \[\leadsto \sqrt{\color{blue}{\frac{0.25 - \frac{0.125}{x}}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25 - \frac{0.125}{x}}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.92:\\ \;\;\;\;1 + \left(x \cdot 0.5 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25 - \frac{0.125}{x}}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.92)
   (+ 1.0 (- (* x 0.5) (sqrt x)))
   (sqrt (/ (- 0.25 (/ 0.125 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.92) {
		tmp = 1.0 + ((x * 0.5) - sqrt(x));
	} else {
		tmp = sqrt(((0.25 - (0.125 / x)) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.92d0) then
        tmp = 1.0d0 + ((x * 0.5d0) - sqrt(x))
    else
        tmp = sqrt(((0.25d0 - (0.125d0 / x)) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.92) {
		tmp = 1.0 + ((x * 0.5) - Math.sqrt(x));
	} else {
		tmp = Math.sqrt(((0.25 - (0.125 / x)) / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.92:
		tmp = 1.0 + ((x * 0.5) - math.sqrt(x))
	else:
		tmp = math.sqrt(((0.25 - (0.125 / x)) / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.92)
		tmp = Float64(1.0 + Float64(Float64(x * 0.5) - sqrt(x)));
	else
		tmp = sqrt(Float64(Float64(0.25 - Float64(0.125 / x)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.92)
		tmp = 1.0 + ((x * 0.5) - sqrt(x));
	else
		tmp = sqrt(((0.25 - (0.125 / x)) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.92], N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(0.25 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.92000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.9%

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

      1. associate--l+ 99.9%

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

      2. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if 0.92000000000000004 < x

   1. Initial program 6.7%

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

      1. flip-- 7.1%

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

      2. div-inv 7.1%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 9.4%

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

      5. associate--l+ 9.4%

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

   4. Applied egg-rr 9.4%

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

      1. associate-*r/ 9.4%

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

      2. *-rgt-identity 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l- 99.6%

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

      5. div-sub 99.6%

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

      6. +-inverses 99.6%

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

      7. div0 99.6%

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

      8. --rgt-identity 99.6%

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

      9. +-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 99.2%

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

      2. sqrt-unprod 99.6%

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

      3. inv-pow 99.6%

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

      4. inv-pow 99.6%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in x around inf 59.3%

      \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot x - 0.125}{{x}^{2}}}} \]

   10. Taylor expanded in x around inf 98.8%

       \[\leadsto \sqrt{\color{blue}{\frac{0.25 - 0.125 \cdot \frac{1}{x}}{x}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 98.8%

          \[\leadsto \sqrt{\frac{0.25 - \color{blue}{\frac{0.125 \cdot 1}{x}}}{x}} \]

       2. metadata-eval 98.8%

          \[\leadsto \sqrt{\frac{0.25 - \frac{\color{blue}{0.125}}{x}}{x}} \]

   12. Simplified 98.8%

       \[\leadsto \sqrt{\color{blue}{\frac{0.25 - \frac{0.125}{x}}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.92:\\ \;\;\;\;1 + \left(x \cdot 0.5 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25 - \frac{0.125}{x}}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.36:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.36) (- 1.0 (sqrt x)) (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = 0.5 * pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.36d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = 0.5d0 * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.36:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = 0.5 * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.36)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = Float64(0.5 * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.36)
		tmp = 1.0 - sqrt(x);
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.36], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.35999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.1%

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

   if 0.35999999999999999 < x

   1. Initial program 6.7%

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

      1. flip-- 7.1%

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

      2. div-inv 7.1%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 9.4%

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

      5. associate--l+ 9.4%

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

   4. Applied egg-rr 9.4%

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

      1. associate-*r/ 9.4%

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

      2. *-rgt-identity 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l- 99.6%

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

      5. div-sub 99.6%

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

      6. +-inverses 99.6%

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

      7. div0 99.6%

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

      8. --rgt-identity 99.6%

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

      9. +-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. Taylor expanded in x around inf 97.9%

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

      1. rem-exp-log 90.7%

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

      2. exp-neg 90.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{-\log x}}} \]

      3. unpow1/2 90.7%

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

      4. exp-prod 90.7%

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

      5. distribute-lft-neg-out 90.7%

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

      6. distribute-rgt-neg-in 90.7%

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

      7. metadata-eval 90.7%

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

      8. exp-to-pow 98.0%

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

   9. Simplified 98.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.36:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.13:\\ \;\;\;\;1 + \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.13) (+ 1.0 (sqrt x)) (sqrt (/ 0.25 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.13) {
		tmp = 1.0 + sqrt(x);
	} else {
		tmp = sqrt((0.25 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.13d0) then
        tmp = 1.0d0 + sqrt(x)
    else
        tmp = sqrt((0.25d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.13) {
		tmp = 1.0 + Math.sqrt(x);
	} else {
		tmp = Math.sqrt((0.25 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.13:
		tmp = 1.0 + math.sqrt(x)
	else:
		tmp = math.sqrt((0.25 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.13)
		tmp = Float64(1.0 + sqrt(x));
	else
		tmp = sqrt(Float64(0.25 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.13)
		tmp = 1.0 + sqrt(x);
	else
		tmp = sqrt((0.25 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.13], N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.25 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.13

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.1%

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

      1. sub-neg 99.1%

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

      2. rem-square-sqrt 0.0%

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

      3. fabs-sqr 0.0%

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

      4. rem-square-sqrt 97.5%

         \[\leadsto 1 + \left|\color{blue}{-\sqrt{x}}\right| \]

      5. rem-sqrt-square 97.5%

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

      6. sqr-neg 97.5%

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

      7. rem-square-sqrt 97.5%

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

   5. Simplified 97.5%

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

   if 0.13 < x

   1. Initial program 6.7%

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

      1. flip-- 7.1%

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

      2. div-inv 7.1%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 9.4%

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

      5. associate--l+ 9.4%

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

   4. Applied egg-rr 9.4%

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

      1. associate-*r/ 9.4%

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

      2. *-rgt-identity 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l- 99.6%

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

      5. div-sub 99.6%

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

      6. +-inverses 99.6%

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

      7. div0 99.6%

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

      8. --rgt-identity 99.6%

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

      9. +-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 99.2%

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

      2. sqrt-unprod 99.6%

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

      3. inv-pow 99.6%

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

      4. inv-pow 99.6%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in x around inf 97.9%

      \[\leadsto \sqrt{\color{blue}{\frac{0.25}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.13:\\ \;\;\;\;1 + \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.36:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.36) (- 1.0 (sqrt x)) (sqrt (/ 0.25 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = sqrt((0.25 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.36d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = sqrt((0.25d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = Math.sqrt((0.25 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.36:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = math.sqrt((0.25 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.36)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = sqrt(Float64(0.25 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.36)
		tmp = 1.0 - sqrt(x);
	else
		tmp = sqrt((0.25 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.36], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.25 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.35999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.1%

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

   if 0.35999999999999999 < x

   1. Initial program 6.7%

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

      1. flip-- 7.1%

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

      2. div-inv 7.1%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 9.4%

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

      5. associate--l+ 9.4%

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

   4. Applied egg-rr 9.4%

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

      1. associate-*r/ 9.4%

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

      2. *-rgt-identity 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l- 99.6%

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

      5. div-sub 99.6%

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

      6. +-inverses 99.6%

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

      7. div0 99.6%

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

      8. --rgt-identity 99.6%

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

      9. +-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 99.2%

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

      2. sqrt-unprod 99.6%

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

      3. inv-pow 99.6%

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

      4. inv-pow 99.6%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in x around inf 97.9%

      \[\leadsto \sqrt{\color{blue}{\frac{0.25}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.36:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.25}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (/ 0.25 x)))

C

double code(double x) {
	return sqrt((0.25 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((0.25d0 / x))
end function

Java

public static double code(double x) {
	return Math.sqrt((0.25 / x));
}

Python

def code(x):
	return math.sqrt((0.25 / x))

Julia

function code(x)
	return sqrt(Float64(0.25 / x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((0.25 / x));
end

Wolfram

code[x_] := N[Sqrt[N[(0.25 / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.7%

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

   1. flip-- 49.9%

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

   2. div-inv 49.9%

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

   3. add-sqr-sqrt 50.4%

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

   4. add-sqr-sqrt 51.1%

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

   5. associate--l+ 51.1%

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

4. Applied egg-rr 51.1%

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

   1. associate-*r/ 51.1%

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

   2. *-rgt-identity 51.1%

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

   3. +-commutative 51.1%

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

   4. associate-+l- 99.7%

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

   5. div-sub 99.7%

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

   6. +-inverses 99.7%

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

   7. div0 99.7%

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

   8. --rgt-identity 99.7%

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

   9. +-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 99.6%

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

   2. sqrt-unprod 99.7%

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

   3. inv-pow 99.7%

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

   4. inv-pow 99.7%

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

   5. pow-prod-up 99.8%

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

   6. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in x around inf 55.9%

   \[\leadsto \sqrt{\color{blue}{\frac{0.25}{x}}} \]

10. Final simplification 55.9%

    \[\leadsto \sqrt{\frac{0.25}{x}} \]
11. Add Preprocessing

Alternative 11: 6.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt x))

C

double code(double x) {
	return sqrt(x);
}

Fortran

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

Java

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

Python

def code(x):
	return math.sqrt(x)

Julia

function code(x)
	return sqrt(x)
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[x], $MachinePrecision]

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in x around 0 46.5%

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

   1. sub-neg 46.5%

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

   2. rem-square-sqrt 0.0%

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

   3. fabs-sqr 0.0%

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

   4. rem-square-sqrt 47.9%

      \[\leadsto 1 + \left|\color{blue}{-\sqrt{x}}\right| \]

   5. rem-sqrt-square 47.9%

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

   6. sqr-neg 47.9%

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

   7. rem-square-sqrt 47.9%

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

5. Simplified 47.9%

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

6. Taylor expanded in x around inf 6.1%

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

7. Final simplification 6.1%

   \[\leadsto \sqrt{x} \]
8. 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 2024078 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :alt
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

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