2sqrt (example 3.1)

Percentage Accurate: 53.8% → 99.6%

Time: 7.0s

Alternatives: 10

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% 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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. flip-- 58.3%

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

   2. div-inv 58.3%

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

   3. add-sqr-sqrt 58.7%

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

   4. add-sqr-sqrt 58.6%

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

   5. associate--l+ 58.6%

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

3. Applied egg-rr 58.6%

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

   1. +-commutative 58.6%

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

   2. associate-+l- 99.8%

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

   3. +-inverses 99.8%

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

   4. metadata-eval 99.8%

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

   5. associate-*r/ 99.8%

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

   6. metadata-eval 99.8%

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

   7. +-commutative 99.8%

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

5. Simplified 99.8%

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

   1. +-commutative 99.8%

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

   2. add-sqr-sqrt 99.7%

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

   3. pow2 99.7%

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

   4. pow1/2 99.7%

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

   5. +-commutative 99.7%

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

   6. sqrt-pow1 99.7%

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

   7. metadata-eval 99.7%

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

7. Applied egg-rr 99.7%

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

   1. flip-+ 61.5%

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

   2. associate-/r/ 61.5%

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

9. Applied egg-rr 99.8%

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

    1. sqrt-pow1 99.8%

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

    2. metadata-eval 99.8%

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

    3. metadata-eval 99.8%

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

    4. pow-prod-up 99.6%

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

    5. pow-prod-down 99.8%

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

    6. pow2 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{1}{0.5 \cdot \sqrt{\frac{1}{x}}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 5e-8) (/ 1.0 (/ 1.0 (* 0.5 (sqrt (/ 1.0 x))))) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 5e-8) {
		tmp = 1.0 / (1.0 / (0.5 * sqrt((1.0 / x))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 5d-8) then
        tmp = 1.0d0 / (1.0d0 / (0.5d0 * sqrt((1.0d0 / x))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 5e-8) {
		tmp = 1.0 / (1.0 / (0.5 * Math.sqrt((1.0 / x))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 5e-8:
		tmp = 1.0 / (1.0 / (0.5 * math.sqrt((1.0 / x))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 5e-8)
		tmp = Float64(1.0 / Float64(1.0 / Float64(0.5 * sqrt(Float64(1.0 / x)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 5e-8)
		tmp = 1.0 / (1.0 / (0.5 * sqrt((1.0 / x))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-8], N[(1.0 / N[(1.0 / N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 4.9999999999999998e-8

   1. Initial program 4.0%

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

      1. flip-- 4.0%

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

      2. div-inv 4.0%

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

      3. add-sqr-sqrt 4.9%

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

      4. add-sqr-sqrt 4.7%

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

      5. associate--l+ 4.7%

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

   3. Applied egg-rr 4.7%

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

      1. +-commutative 4.7%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   7. Applied egg-rr 47.0%

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

      1. *-commutative 47.0%

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

      2. associate-/r/ 47.0%

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

      3. *-lft-identity 47.0%

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

      4. metadata-eval 47.0%

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

      5. +-inverses 47.0%

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

      6. +-inverses 47.0%

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

      7. metadata-eval 47.0%

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

      8. *-lft-identity 47.0%

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

      9. +-inverses 47.0%

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

      10. metadata-eval 47.0%

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

      11. *-commutative 47.0%

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

   9. Simplified 47.0%

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

   10. Taylor expanded in x around inf 99.6%

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

   if 4.9999999999999998e-8 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{1}{0.5 \cdot \sqrt{\frac{1}{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. flip-- 58.3%

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

   2. div-inv 58.3%

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

   3. add-sqr-sqrt 58.7%

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

   4. add-sqr-sqrt 58.6%

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

   5. associate--l+ 58.6%

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

3. Applied egg-rr 58.6%

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

   1. +-commutative 58.6%

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

   2. associate-+l- 99.8%

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

   3. +-inverses 99.8%

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

   4. metadata-eval 99.8%

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

   5. associate-*r/ 99.8%

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

   6. metadata-eval 99.8%

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

   7. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. associate-*r* 99.7%

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

      4. distribute-rgt-out 99.7%

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

      5. *-commutative 99.7%

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

   4. Simplified 99.7%

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

   if 1.25 < x

   1. Initial program 7.1%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 8.0%

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

      5. associate--l+ 8.0%

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

   3. Applied egg-rr 8.0%

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

      1. +-commutative 8.0%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   7. Applied egg-rr 46.3%

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

      1. *-commutative 46.3%

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

      2. associate-/r/ 46.4%

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

      3. *-lft-identity 46.4%

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

      4. metadata-eval 46.4%

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

      5. +-inverses 46.4%

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

      6. +-inverses 46.4%

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

      7. metadata-eval 46.4%

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

      8. *-lft-identity 46.4%

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

      9. +-inverses 46.4%

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

      10. metadata-eval 46.4%

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

      11. *-commutative 46.4%

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

   9. Simplified 46.4%

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

   10. Taylor expanded in x around inf 97.3%

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

4. Final simplification 98.6%

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

Alternative 5: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. *-commutative 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+l- 99.6%

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

   6. Applied egg-rr 99.6%

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

   if 1 < x

   1. Initial program 7.1%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 8.0%

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

      5. associate--l+ 8.0%

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

   3. Applied egg-rr 8.0%

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

      1. +-commutative 8.0%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   7. Applied egg-rr 46.3%

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

      1. *-commutative 46.3%

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

      2. associate-/r/ 46.4%

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

      3. *-lft-identity 46.4%

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

      4. metadata-eval 46.4%

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

      5. +-inverses 46.4%

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

      6. +-inverses 46.4%

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

      7. metadata-eval 46.4%

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

      8. *-lft-identity 46.4%

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

      9. +-inverses 46.4%

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

      10. metadata-eval 46.4%

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

      11. *-commutative 46.4%

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

   9. Simplified 46.4%

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

   10. Taylor expanded in x around inf 97.3%

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

4. Final simplification 98.6%

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

Alternative 6: 98.4% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (+ (* x 0.5) (- 1.0 (sqrt x))) (/ 1.0 (* (sqrt x) 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x * 0.5) + (1.0 - sqrt(x));
	} else {
		tmp = 1.0 / (sqrt(x) * 2.0);
	}
	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 = 1.0d0 / (sqrt(x) * 2.0d0)
    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 = 1.0 / (Math.sqrt(x) * 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = (x * 0.5) + (1.0 - math.sqrt(x))
	else:
		tmp = 1.0 / (math.sqrt(x) * 2.0)
	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(1.0 / Float64(sqrt(x) * 2.0));
	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 = 1.0 / (sqrt(x) * 2.0);
	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[(1.0 / N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $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 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. *-commutative 99.6%

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

   4. Applied egg-rr 99.6%

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

   if 1 < x

   1. Initial program 7.1%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 8.0%

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

      5. associate--l+ 8.0%

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

   3. Applied egg-rr 8.0%

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

      1. +-commutative 8.0%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   7. Applied egg-rr 46.3%

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

      1. *-commutative 46.3%

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

      2. associate-/r/ 46.4%

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

      3. *-lft-identity 46.4%

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

      4. metadata-eval 46.4%

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

      5. +-inverses 46.4%

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

      6. +-inverses 46.4%

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

      7. metadata-eval 46.4%

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

      8. *-lft-identity 46.4%

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

      9. +-inverses 46.4%

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

      10. metadata-eval 46.4%

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

      11. *-commutative 46.4%

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

   9. Simplified 46.4%

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

   10. Taylor expanded in x around inf 97.2%

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

4. Final simplification 98.6%

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

Alternative 7: 98.4% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (- 1.0 (- (sqrt x) (* x 0.5))) (/ 1.0 (* (sqrt x) 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 - (sqrt(x) - (x * 0.5));
	} else {
		tmp = 1.0 / (sqrt(x) * 2.0);
	}
	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) - (x * 0.5d0))
    else
        tmp = 1.0d0 / (sqrt(x) * 2.0d0)
    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) - (x * 0.5));
	} else {
		tmp = 1.0 / (Math.sqrt(x) * 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 - (math.sqrt(x) - (x * 0.5))
	else:
		tmp = 1.0 / (math.sqrt(x) * 2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 - Float64(sqrt(x) - Float64(x * 0.5)));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 - (sqrt(x) - (x * 0.5));
	else
		tmp = 1.0 / (sqrt(x) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $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 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. *-commutative 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+l- 99.6%

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

   6. Applied egg-rr 99.6%

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

   if 1 < x

   1. Initial program 7.1%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 8.0%

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

      5. associate--l+ 8.0%

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

   3. Applied egg-rr 8.0%

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

      1. +-commutative 8.0%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   7. Applied egg-rr 46.3%

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

      1. *-commutative 46.3%

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

      2. associate-/r/ 46.4%

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

      3. *-lft-identity 46.4%

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

      4. metadata-eval 46.4%

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

      5. +-inverses 46.4%

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

      6. +-inverses 46.4%

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

      7. metadata-eval 46.4%

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

      8. *-lft-identity 46.4%

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

      9. +-inverses 46.4%

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

      10. metadata-eval 46.4%

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

      11. *-commutative 46.4%

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

   9. Simplified 46.4%

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

   10. Taylor expanded in x around inf 97.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 - \left(\sqrt{x} - x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} \cdot 2}\\ \end{array} \]

Alternative 8: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.25) 1.0 (/ 1.0 (* (sqrt x) 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (sqrt(x) * 2.0);
	}
	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 = 1.0d0 / (sqrt(x) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (Math.sqrt(x) * 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.25:
		tmp = 1.0
	else:
		tmp = 1.0 / (math.sqrt(x) * 2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.25)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / Float64(sqrt(x) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.25)
		tmp = 1.0;
	else
		tmp = 1.0 / (sqrt(x) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.25], 1.0, N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] * 2.0), $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 96.8%

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

   if 0.25 < x

   1. Initial program 7.1%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 8.0%

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

      5. associate--l+ 8.0%

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

   3. Applied egg-rr 8.0%

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

      1. +-commutative 8.0%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   7. Applied egg-rr 46.3%

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

      1. *-commutative 46.3%

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

      2. associate-/r/ 46.4%

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

      3. *-lft-identity 46.4%

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

      4. metadata-eval 46.4%

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

      5. +-inverses 46.4%

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

      6. +-inverses 46.4%

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

      7. metadata-eval 46.4%

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

      8. *-lft-identity 46.4%

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

      9. +-inverses 46.4%

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

      10. metadata-eval 46.4%

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

      11. *-commutative 46.4%

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

   9. Simplified 46.4%

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

   10. Taylor expanded in x around inf 97.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} \cdot 2}\\ \end{array} \]

Alternative 9: 97.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 / (1.0 + sqrt(x));
	} else {
		tmp = 1.0 / (sqrt(x) * 2.0);
	}
	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 = 1.0d0 / (sqrt(x) * 2.0d0)
    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 = 1.0 / (Math.sqrt(x) * 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 / (1.0 + math.sqrt(x))
	else:
		tmp = 1.0 / (math.sqrt(x) * 2.0)
	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(1.0 / Float64(sqrt(x) * 2.0));
	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 = 1.0 / (sqrt(x) * 2.0);
	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[(1.0 / N[(N[Sqrt[x], $MachinePrecision] * 2.0), $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%

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

      5. associate--l+ 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l- 99.9%

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

      3. +-inverses 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-*r/ 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. pow2 99.9%

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

      4. pow1/2 99.9%

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

      5. +-commutative 99.9%

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

      6. sqrt-pow1 99.9%

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

      7. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around 0 98.8%

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

   if 1 < x

   1. Initial program 7.1%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 8.0%

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

      5. associate--l+ 8.0%

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

   3. Applied egg-rr 8.0%

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

      1. +-commutative 8.0%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   7. Applied egg-rr 46.3%

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

      1. *-commutative 46.3%

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

      2. associate-/r/ 46.4%

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

      3. *-lft-identity 46.4%

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

      4. metadata-eval 46.4%

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

      5. +-inverses 46.4%

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

      6. +-inverses 46.4%

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

      7. metadata-eval 46.4%

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

      8. *-lft-identity 46.4%

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

      9. +-inverses 46.4%

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

      10. metadata-eval 46.4%

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

      11. *-commutative 46.4%

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

   9. Simplified 46.4%

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

   10. Taylor expanded in x around inf 97.2%

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

4. Final simplification 98.1%

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

Alternative 10: 51.9% 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 58.2%

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

2. Taylor expanded in x around 0 56.5%

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

3. Final simplification 56.5%

   \[\leadsto 1 \]

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 2023306 
(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)))