2sqrt (example 3.1)

Percentage Accurate: 53.4% → 99.7%

Time: 6.3s

Alternatives: 7

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: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

   1. flip-- 59.2%

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

   2. div-inv 59.2%

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

   3. add-sqr-sqrt 59.5%

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

   4. add-sqr-sqrt 60.5%

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

3. Applied egg-rr 60.5%

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

   1. *-commutative 60.5%

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

   2. associate-/r/ 60.5%

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

   3. +-commutative 60.5%

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

   4. associate--l+ 99.8%

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

   5. +-inverses 99.8%

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

   6. metadata-eval 99.8%

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

   7. /-rgt-identity 99.8%

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

   8. +-commutative 99.8%

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

5. Simplified 99.8%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 99.8%

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

   3. inv-pow 99.8%

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

   4. +-commutative 99.8%

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

   5. inv-pow 99.8%

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

   6. +-commutative 99.8%

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

   7. pow-prod-up 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

   10. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 0.0001:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 0.0001) (* 0.5 (pow x -0.5)) t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = 0.5 * pow(x, -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((x + 1.0d0)) - sqrt(x)
    if (t_0 <= 0.0001d0) then
        tmp = 0.5d0 * (x ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = 0.5 * Math.pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 0.0001:
		tmp = 0.5 * math.pow(x, -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 0.0001)
		tmp = Float64(0.5 * (x ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 0.0001)
		tmp = 0.5 * (x ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0001], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 1.00000000000000005e-4

   1. Initial program 6.5%

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

      1. flip-- 6.8%

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

      2. div-inv 6.8%

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

      3. add-sqr-sqrt 7.4%

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

      4. add-sqr-sqrt 9.8%

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

   3. Applied egg-rr 9.8%

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

      1. *-commutative 9.8%

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

      2. associate-/r/ 9.8%

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

      3. +-commutative 9.8%

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

      4. associate--l+ 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. /-rgt-identity 99.6%

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

      8. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. flip3-+ 68.0%

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

      3. associate-/r/ 67.9%

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

      4. +-commutative 67.9%

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

      5. sqrt-pow2 67.9%

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

      6. metadata-eval 67.9%

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

      7. sqrt-pow2 67.7%

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

      8. +-commutative 67.7%

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

      9. metadata-eval 67.7%

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

      10. add-sqr-sqrt 68.0%

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

      11. add-sqr-sqrt 67.7%

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

   7. Applied egg-rr 52.5%

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

   8. Taylor expanded in x around inf 68.1%

      \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \color{blue}{\left(0.5 + \left(0.125 \cdot \frac{1}{x} + x\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 68.1%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \color{blue}{\left(\left(0.125 \cdot \frac{1}{x} + x\right) + 0.5\right)} \]

      2. +-commutative 68.1%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \left(\color{blue}{\left(x + 0.125 \cdot \frac{1}{x}\right)} + 0.5\right) \]

      3. associate-+l+ 68.1%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \color{blue}{\left(x + \left(0.125 \cdot \frac{1}{x} + 0.5\right)\right)} \]

      4. associate-*r/ 68.1%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \left(x + \left(\color{blue}{\frac{0.125 \cdot 1}{x}} + 0.5\right)\right) \]

      5. metadata-eval 68.1%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \left(x + \left(\frac{\color{blue}{0.125}}{x} + 0.5\right)\right) \]

   10. Simplified 68.1%

       \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \color{blue}{\left(x + \left(\frac{0.125}{x} + 0.5\right)\right)} \]

   11. Taylor expanded in x around inf 98.4%

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

       1. rem-exp-log 91.2%

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

       2. exp-neg 91.2%

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

       3. unpow1/2 91.2%

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

       4. exp-prod 91.2%

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

       5. distribute-lft-neg-out 91.2%

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

       6. distribute-rgt-neg-in 91.2%

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

       7. metadata-eval 91.2%

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

       8. exp-to-pow 98.5%

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

   13. Simplified 98.5%

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

   if 1.00000000000000005e-4 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.0001:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

   1. flip-- 59.2%

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

   2. div-inv 59.2%

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

   3. add-sqr-sqrt 59.5%

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

   4. add-sqr-sqrt 60.5%

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

3. Applied egg-rr 60.5%

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

   1. *-commutative 60.5%

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

   2. associate-/r/ 60.5%

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

   3. +-commutative 60.5%

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

   4. associate--l+ 99.8%

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

   5. +-inverses 99.8%

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

   6. metadata-eval 99.8%

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

   7. /-rgt-identity 99.8%

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

   8. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $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. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-inverses 99.9%

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

      6. metadata-eval 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. +-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. inv-pow 99.9%

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

      2. +-commutative 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. unpow-prod-down 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. pow-sqr 99.8%

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

      2. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around 0 97.9%

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

   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 8.2%

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

      4. add-sqr-sqrt 10.6%

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

   3. Applied egg-rr 10.6%

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

      1. *-commutative 10.6%

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

      2. associate-/r/ 10.6%

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

      3. +-commutative 10.6%

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

      4. associate--l+ 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. /-rgt-identity 99.6%

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

      8. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. flip3-+ 68.2%

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

      3. associate-/r/ 68.2%

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

      4. +-commutative 68.2%

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

      5. sqrt-pow2 68.2%

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

      6. metadata-eval 68.2%

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

      7. sqrt-pow2 68.0%

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

      8. +-commutative 68.0%

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

      9. metadata-eval 68.0%

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

      10. add-sqr-sqrt 68.3%

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

      11. add-sqr-sqrt 68.0%

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

   7. Applied egg-rr 52.9%

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

   8. Taylor expanded in x around inf 68.3%

      \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \color{blue}{\left(0.5 + \left(0.125 \cdot \frac{1}{x} + x\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 68.3%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \color{blue}{\left(\left(0.125 \cdot \frac{1}{x} + x\right) + 0.5\right)} \]

      2. +-commutative 68.3%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \left(\color{blue}{\left(x + 0.125 \cdot \frac{1}{x}\right)} + 0.5\right) \]

      3. associate-+l+ 68.3%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \color{blue}{\left(x + \left(0.125 \cdot \frac{1}{x} + 0.5\right)\right)} \]

      4. associate-*r/ 68.3%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \left(x + \left(\color{blue}{\frac{0.125 \cdot 1}{x}} + 0.5\right)\right) \]

      5. metadata-eval 68.3%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \left(x + \left(\frac{\color{blue}{0.125}}{x} + 0.5\right)\right) \]

   10. Simplified 68.3%

       \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \color{blue}{\left(x + \left(\frac{0.125}{x} + 0.5\right)\right)} \]

   11. Taylor expanded in x around inf 97.9%

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

       1. rem-exp-log 90.8%

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

       2. exp-neg 90.8%

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

       3. unpow1/2 90.8%

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

       4. exp-prod 90.8%

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

       5. distribute-lft-neg-out 90.8%

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

       6. distribute-rgt-neg-in 90.8%

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

       7. metadata-eval 90.8%

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

       8. exp-to-pow 98.1%

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

   13. Simplified 98.1%

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

4. Final simplification 98.0%

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

Alternative 5: 96.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} 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 <= 0.25d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 0.25], 1.0, N[(0.5 * N[Power[x, -0.5], $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 96.1%

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

   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 8.2%

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

      4. add-sqr-sqrt 10.6%

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

   3. Applied egg-rr 10.6%

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

      1. *-commutative 10.6%

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

      2. associate-/r/ 10.6%

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

      3. +-commutative 10.6%

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

      4. associate--l+ 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. /-rgt-identity 99.6%

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

      8. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. flip3-+ 68.2%

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

      3. associate-/r/ 68.2%

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

      4. +-commutative 68.2%

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

      5. sqrt-pow2 68.2%

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

      6. metadata-eval 68.2%

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

      7. sqrt-pow2 68.0%

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

      8. +-commutative 68.0%

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

      9. metadata-eval 68.0%

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

      10. add-sqr-sqrt 68.3%

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

      11. add-sqr-sqrt 68.0%

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

   7. Applied egg-rr 52.9%

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

   8. Taylor expanded in x around inf 68.3%

      \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \color{blue}{\left(0.5 + \left(0.125 \cdot \frac{1}{x} + x\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 68.3%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \color{blue}{\left(\left(0.125 \cdot \frac{1}{x} + x\right) + 0.5\right)} \]

      2. +-commutative 68.3%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \left(\color{blue}{\left(x + 0.125 \cdot \frac{1}{x}\right)} + 0.5\right) \]

      3. associate-+l+ 68.3%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \color{blue}{\left(x + \left(0.125 \cdot \frac{1}{x} + 0.5\right)\right)} \]

      4. associate-*r/ 68.3%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \left(x + \left(\color{blue}{\frac{0.125 \cdot 1}{x}} + 0.5\right)\right) \]

      5. metadata-eval 68.3%

         \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \left(x + \left(\frac{\color{blue}{0.125}}{x} + 0.5\right)\right) \]

   10. Simplified 68.3%

       \[\leadsto \frac{1}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \cdot \color{blue}{\left(x + \left(\frac{0.125}{x} + 0.5\right)\right)} \]

   11. Taylor expanded in x around inf 97.9%

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

       1. rem-exp-log 90.8%

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

       2. exp-neg 90.8%

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

       3. unpow1/2 90.8%

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

       4. exp-prod 90.8%

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

       5. distribute-lft-neg-out 90.8%

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

       6. distribute-rgt-neg-in 90.8%

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

       7. metadata-eval 90.8%

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

       8. exp-to-pow 98.1%

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

   13. Simplified 98.1%

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

4. Final simplification 96.9%

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

Alternative 6: 96.7% 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 96.1%

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

   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 8.2%

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

      4. add-sqr-sqrt 10.6%

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

   3. Applied egg-rr 10.6%

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

      1. *-commutative 10.6%

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

      2. associate-/r/ 10.6%

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

      3. +-commutative 10.6%

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

      4. associate--l+ 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. /-rgt-identity 99.6%

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

      8. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. inv-pow 99.6%

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

      2. +-commutative 99.6%

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

      3. add-sqr-sqrt 99.1%

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

      4. unpow-prod-down 98.9%

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

   7. Applied egg-rr 98.8%

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

      1. pow-sqr 99.0%

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

      2. metadata-eval 99.0%

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

   9. Simplified 99.0%

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

   10. Taylor expanded in x around inf 97.1%

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

       1. expm1-log1p-u 97.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left({x}^{0.25}, {x}^{0.25}\right)\right)}^{-2}\right)\right)} \]

       2. expm1-udef 9.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left({x}^{0.25}, {x}^{0.25}\right)\right)}^{-2}\right)} - 1} \]

   12. Applied egg-rr 9.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\sqrt{x} + \sqrt{x}\right)}^{-1}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 97.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt{x} + \sqrt{x}\right)}^{-1}\right)\right)} \]

       2. expm1-log1p 97.7%

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

       3. unpow-1 97.7%

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

       4. count-2 97.7%

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

       5. associate-/r* 97.7%

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

       6. metadata-eval 97.7%

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

   14. Simplified 97.7%

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

4. Final simplification 96.8%

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

Alternative 7: 51.3% 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 59.0%

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

2. Taylor expanded in x around 0 56.7%

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

3. Final simplification 56.7%

   \[\leadsto 1 \]

Developer target: 99.8% 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 2023261 
(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)))