2sqrt (example 3.1)

Percentage Accurate: 53.3% → 99.7%

Time: 6.0s

Alternatives: 8

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: 53.3% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. flip-- 51.9%

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

   2. div-inv 51.9%

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

   3. add-sqr-sqrt 52.1%

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

   4. add-sqr-sqrt 52.6%

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

3. Applied egg-rr 52.6%

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

   1. associate-*r/ 52.6%

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

   2. *-rgt-identity 52.6%

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

   3. remove-double-neg 52.6%

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

   4. sub-neg 52.6%

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

   5. div-sub 51.8%

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

   6. rem-square-sqrt 51.6%

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

   7. sqr-neg 51.6%

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

   8. div-sub 52.1%

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

   9. sqr-neg 52.1%

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

   10. +-commutative 52.1%

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

   11. rem-square-sqrt 52.6%

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

   12. associate--l+ 99.7%

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

   13. +-inverses 99.7%

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

   14. metadata-eval 99.7%

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

   15. sub-neg 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 4e-6) (* (pow x -0.5) 0.5) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 4e-6) {
		tmp = pow(x, -0.5) * 0.5;
	} 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 <= 4d-6) then
        tmp = (x ** (-0.5d0)) * 0.5d0
    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 <= 4e-6) {
		tmp = Math.pow(x, -0.5) * 0.5;
	} 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 <= 4e-6:
		tmp = math.pow(x, -0.5) * 0.5
	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 <= 4e-6)
		tmp = Float64((x ^ -0.5) * 0.5);
	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 <= 4e-6)
		tmp = (x ^ -0.5) * 0.5;
	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, 4e-6], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 3.99999999999999982e-6

   1. Initial program 4.8%

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

      1. flip-- 5.1%

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

      2. div-inv 5.1%

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

      3. add-sqr-sqrt 5.5%

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

      4. add-sqr-sqrt 6.1%

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

   3. Applied egg-rr 6.1%

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

      1. associate-*r/ 6.1%

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

      2. *-rgt-identity 6.1%

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

      3. remove-double-neg 6.1%

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

      4. sub-neg 6.1%

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

      5. div-sub 4.8%

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

      6. rem-square-sqrt 4.6%

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

      7. sqr-neg 4.6%

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

      8. div-sub 5.5%

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

      9. sqr-neg 5.5%

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

      10. +-commutative 5.5%

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

      11. rem-square-sqrt 6.1%

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

      12. associate--l+ 99.5%

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

      13. +-inverses 99.5%

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

      14. metadata-eval 99.5%

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

      15. sub-neg 99.5%

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

   5. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. add-sqr-sqrt 99.3%

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

      3. pow2 99.3%

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

      4. pow1/2 99.3%

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

      5. sqrt-pow1 99.3%

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

      6. metadata-eval 99.3%

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in x around inf 99.1%

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

      1. inv-pow 99.1%

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

      2. count-2 99.1%

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

      3. unpow-prod-down 99.1%

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

      4. metadata-eval 99.1%

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

      5. metadata-eval 99.1%

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

      6. inv-pow 99.1%

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

      7. metadata-eval 99.1%

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

      8. sqrt-div 99.4%

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

      9. *-commutative 99.4%

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

      10. inv-pow 99.4%

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

      11. sqrt-pow1 99.6%

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

      12. metadata-eval 99.6%

          \[\leadsto {x}^{\color{blue}{-0.5}} \cdot \sqrt{0.25} \]

      13. metadata-eval 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 3.99999999999999982e-6 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. associate--l+ 99.4%

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

      2. +-commutative 99.4%

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

      3. *-commutative 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 1 < x

   1. Initial program 7.1%

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

      1. flip-- 7.6%

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

      2. div-inv 7.6%

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

      3. add-sqr-sqrt 8.0%

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

      4. add-sqr-sqrt 8.9%

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

   3. Applied egg-rr 8.9%

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

      1. associate-*r/ 8.9%

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

      2. *-rgt-identity 8.9%

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

      3. remove-double-neg 8.9%

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

      4. sub-neg 8.9%

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

      5. div-sub 7.3%

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

      6. rem-square-sqrt 7.0%

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

      7. sqr-neg 7.0%

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

      8. div-sub 8.0%

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

      9. sqr-neg 8.0%

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

      10. +-commutative 8.0%

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

      11. rem-square-sqrt 8.9%

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

      12. associate--l+ 99.5%

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

      13. +-inverses 99.5%

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

      14. metadata-eval 99.5%

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

      15. sub-neg 99.5%

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

   5. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. add-sqr-sqrt 99.3%

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

      3. pow2 99.3%

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

      4. pow1/2 99.3%

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

      5. sqrt-pow1 99.3%

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

      6. metadata-eval 99.3%

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in x around inf 97.4%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{x}} \]
   9. 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%

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

      5. metadata-eval 97.4%

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

      6. inv-pow 97.4%

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

      7. metadata-eval 97.4%

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

      8. sqrt-div 97.6%

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

      9. *-commutative 97.6%

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

      10. inv-pow 97.6%

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

      11. sqrt-pow1 97.8%

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

      12. metadata-eval 97.8%

          \[\leadsto {x}^{\color{blue}{-0.5}} \cdot \sqrt{0.25} \]

      13. metadata-eval 97.8%

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

   10. Applied egg-rr 97.8%

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

4. Final simplification 98.6%

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

Alternative 4: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.4%

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

   if 1 < x

   1. Initial program 7.1%

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

      1. flip-- 7.6%

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

      2. div-inv 7.6%

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

      3. add-sqr-sqrt 8.0%

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

      4. add-sqr-sqrt 8.9%

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

   3. Applied egg-rr 8.9%

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

      1. associate-*r/ 8.9%

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

      2. *-rgt-identity 8.9%

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

      3. remove-double-neg 8.9%

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

      4. sub-neg 8.9%

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

      5. div-sub 7.3%

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

      6. rem-square-sqrt 7.0%

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

      7. sqr-neg 7.0%

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

      8. div-sub 8.0%

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

      9. sqr-neg 8.0%

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

      10. +-commutative 8.0%

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

      11. rem-square-sqrt 8.9%

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

      12. associate--l+ 99.5%

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

      13. +-inverses 99.5%

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

      14. metadata-eval 99.5%

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

      15. sub-neg 99.5%

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

   5. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. add-sqr-sqrt 99.3%

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

      3. pow2 99.3%

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

      4. pow1/2 99.3%

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

      5. sqrt-pow1 99.3%

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

      6. metadata-eval 99.3%

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in x around inf 97.4%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{x}} \]
   9. 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%

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

      5. metadata-eval 97.4%

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

      6. inv-pow 97.4%

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

      7. metadata-eval 97.4%

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

      8. sqrt-div 97.6%

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

      9. *-commutative 97.6%

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

      10. inv-pow 97.6%

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

      11. sqrt-pow1 97.8%

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

      12. metadata-eval 97.8%

          \[\leadsto {x}^{\color{blue}{-0.5}} \cdot \sqrt{0.25} \]

      13. metadata-eval 97.8%

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

   10. Applied egg-rr 97.8%

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

4. Final simplification 98.6%

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

Alternative 5: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

      1. add-log-exp 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-log-exp 100.0%

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

      2. flip-- 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. add-cube-cbrt 99.8%

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

      6. add-sqr-sqrt 99.7%

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

      7. times-frac 99.8%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{x + \left(1 - x\right)}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)}} \]
   6. Step-by-step derivation

      1. *-lft-identity 99.7%

         \[\leadsto \frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{\color{blue}{1 \cdot \left(x + \left(1 - x\right)\right)}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{\color{blue}{\left(1 - 0\right)} \cdot \left(x + \left(1 - x\right)\right)}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \]

      3. +-inverses 99.7%

         \[\leadsto \frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{\left(1 - \color{blue}{\left(x - x\right)}\right) \cdot \left(x + \left(1 - x\right)\right)}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \]

      4. associate-+l- 99.7%

         \[\leadsto \frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{\color{blue}{\left(\left(1 - x\right) + x\right)} \cdot \left(x + \left(1 - x\right)\right)}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{\color{blue}{\left(x + \left(1 - x\right)\right)} \cdot \left(x + \left(1 - x\right)\right)}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \]

      6. unpow2 99.7%

         \[\leadsto \frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{\color{blue}{{\left(x + \left(1 - x\right)\right)}^{2}}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \]

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 96.9%

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

   if 1 < x

   1. Initial program 7.1%

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

      1. flip-- 7.6%

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

      2. div-inv 7.6%

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

      3. add-sqr-sqrt 8.0%

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

      4. add-sqr-sqrt 8.9%

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

   3. Applied egg-rr 8.9%

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

      1. associate-*r/ 8.9%

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

      2. *-rgt-identity 8.9%

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

      3. remove-double-neg 8.9%

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

      4. sub-neg 8.9%

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

      5. div-sub 7.3%

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

      6. rem-square-sqrt 7.0%

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

      7. sqr-neg 7.0%

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

      8. div-sub 8.0%

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

      9. sqr-neg 8.0%

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

      10. +-commutative 8.0%

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

      11. rem-square-sqrt 8.9%

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

      12. associate--l+ 99.5%

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

      13. +-inverses 99.5%

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

      14. metadata-eval 99.5%

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

      15. sub-neg 99.5%

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

   5. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. add-sqr-sqrt 99.3%

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

      3. pow2 99.3%

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

      4. pow1/2 99.3%

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

      5. sqrt-pow1 99.3%

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

      6. metadata-eval 99.3%

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in x around inf 97.4%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{x}} \]
   9. 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%

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

      5. metadata-eval 97.4%

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

      6. inv-pow 97.4%

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

      7. metadata-eval 97.4%

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

      8. sqrt-div 97.6%

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

      9. *-commutative 97.6%

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

      10. inv-pow 97.6%

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

      11. sqrt-pow1 97.8%

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

      12. metadata-eval 97.8%

          \[\leadsto {x}^{\color{blue}{-0.5}} \cdot \sqrt{0.25} \]

      13. metadata-eval 97.8%

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

   10. Applied egg-rr 97.8%

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

4. Final simplification 97.4%

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

Alternative 6: 97.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.25) 1.0 (* (pow x -0.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 0.25:
		tmp = 1.0
	else:
		tmp = math.pow(x, -0.5) * 0.5
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 94.8%

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

   if 0.25 < x

   1. Initial program 7.1%

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

      1. flip-- 7.6%

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

      2. div-inv 7.6%

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

      3. add-sqr-sqrt 8.0%

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

      4. add-sqr-sqrt 8.9%

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

   3. Applied egg-rr 8.9%

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

      1. associate-*r/ 8.9%

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

      2. *-rgt-identity 8.9%

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

      3. remove-double-neg 8.9%

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

      4. sub-neg 8.9%

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

      5. div-sub 7.3%

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

      6. rem-square-sqrt 7.0%

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

      7. sqr-neg 7.0%

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

      8. div-sub 8.0%

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

      9. sqr-neg 8.0%

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

      10. +-commutative 8.0%

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

      11. rem-square-sqrt 8.9%

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

      12. associate--l+ 99.5%

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

      13. +-inverses 99.5%

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

      14. metadata-eval 99.5%

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

      15. sub-neg 99.5%

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

   5. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. add-sqr-sqrt 99.3%

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

      3. pow2 99.3%

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

      4. pow1/2 99.3%

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

      5. sqrt-pow1 99.3%

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

      6. metadata-eval 99.3%

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in x around inf 97.4%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{x}} \]
   9. 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%

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

      5. metadata-eval 97.4%

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

      6. inv-pow 97.4%

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

      7. metadata-eval 97.4%

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

      8. sqrt-div 97.6%

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

      9. *-commutative 97.6%

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

      10. inv-pow 97.6%

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

      11. sqrt-pow1 97.8%

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

      12. metadata-eval 97.8%

          \[\leadsto {x}^{\color{blue}{-0.5}} \cdot \sqrt{0.25} \]

      13. metadata-eval 97.8%

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

   10. Applied egg-rr 97.8%

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

4. Final simplification 96.4%

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

Alternative 7: 97.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 94.8%

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

   if 0.25 < x

   1. Initial program 7.1%

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

      1. add-log-exp 7.1%

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

   3. Applied egg-rr 7.1%

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

      1. add-log-exp 7.1%

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

      2. flip-- 7.6%

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

      3. add-sqr-sqrt 8.0%

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

      4. add-sqr-sqrt 8.9%

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

      5. add-cube-cbrt 8.9%

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

      6. add-sqr-sqrt 8.9%

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

      7. times-frac 8.9%

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

   5. Applied egg-rr 8.9%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{x + \left(1 - x\right)}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)}} \]
   6. Step-by-step derivation

      1. *-lft-identity 8.9%

         \[\leadsto \frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{\color{blue}{1 \cdot \left(x + \left(1 - x\right)\right)}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \]

      2. metadata-eval 8.9%

         \[\leadsto \frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{\color{blue}{\left(1 - 0\right)} \cdot \left(x + \left(1 - x\right)\right)}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \]

      3. +-inverses 8.9%

         \[\leadsto \frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{\left(1 - \color{blue}{\left(x - x\right)}\right) \cdot \left(x + \left(1 - x\right)\right)}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \]

      4. associate-+l- 8.9%

         \[\leadsto \frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{\color{blue}{\left(\left(1 - x\right) + x\right)} \cdot \left(x + \left(1 - x\right)\right)}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \]

      5. +-commutative 8.9%

         \[\leadsto \frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{\color{blue}{\left(x + \left(1 - x\right)\right)} \cdot \left(x + \left(1 - x\right)\right)}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \]

      6. unpow2 8.9%

         \[\leadsto \frac{\sqrt[3]{{\left(x + \left(1 - x\right)\right)}^{2}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \cdot \frac{\sqrt[3]{\color{blue}{{\left(x + \left(1 - x\right)\right)}^{2}}}}{\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)} \]

   7. Simplified 99.0%

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

   8. Taylor expanded in x around inf 96.6%

      \[\leadsto {\color{blue}{\left(\sqrt{2} \cdot {\left(1 \cdot x\right)}^{0.25}\right)}}^{-2} \]
   9. Step-by-step derivation

      1. *-commutative 96.6%

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

      2. *-lft-identity 96.6%

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

   10. Simplified 96.6%

       \[\leadsto {\color{blue}{\left({x}^{0.25} \cdot \sqrt{2}\right)}}^{-2} \]
   11. Step-by-step derivation

       1. *-commutative 96.6%

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

       2. unpow-prod-down 96.3%

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

       3. sqrt-pow2 97.2%

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

       4. metadata-eval 97.2%

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

       5. pow-pow 97.8%

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

       6. metadata-eval 97.8%

          \[\leadsto {2}^{-1} \cdot {x}^{\color{blue}{-0.5}} \]

       7. metadata-eval 97.8%

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

       8. sqrt-pow1 97.6%

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

       9. inv-pow 97.6%

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

       10. sqrt-div 97.4%

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

       11. metadata-eval 97.4%

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

       12. inv-pow 97.4%

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

       13. unpow-prod-down 97.4%

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

       14. count-2 97.4%

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

       15. inv-pow 97.4%

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

       16. count-2 97.4%

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

       17. associate-/r* 97.4%

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

       18. metadata-eval 97.4%

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

   12. Applied egg-rr 97.4%

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

4. Final simplification 96.1%

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

Alternative 8: 51.5% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 51.7%

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

2. Taylor expanded in x around 0 49.1%

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

3. Final simplification 49.1%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023217 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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