2sqrt (example 3.1)

Percentage Accurate: 54.5% → 99.7%

Time: 7.1s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

   1. flip-- 54.1%

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

   2. div-inv 54.1%

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

   3. add-sqr-sqrt 54.1%

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

   4. add-sqr-sqrt 54.5%

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

   5. associate--l+ 54.5%

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

3. Applied egg-rr 54.5%

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

   1. +-commutative 54.5%

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

   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. 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. inv-pow 99.8%

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

   5. pow-prod-up 99.8%

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

   6. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 2e-6) (* (pow x -0.5) 0.5) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 2e-6) {
		tmp = pow(x, -0.5) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 2d-6) then
        tmp = (x ** (-0.5d0)) * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 2e-6) {
		tmp = Math.pow(x, -0.5) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 2e-6:
		tmp = math.pow(x, -0.5) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 2e-6)
		tmp = Float64((x ^ -0.5) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 2e-6)
		tmp = (x ^ -0.5) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-6], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 1.99999999999999991e-6

   1. Initial program 4.2%

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

      1. flip-- 4.6%

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

      2. div-inv 4.6%

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

      3. add-sqr-sqrt 4.3%

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

      4. add-sqr-sqrt 4.6%

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

      5. associate--l+ 4.6%

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

   3. Applied egg-rr 4.6%

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

      1. +-commutative 4.6%

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

      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%

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

      1. add-sqr-sqrt 99.0%

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

      2. pow2 99.0%

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

      3. add-sqr-sqrt 99.0%

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

      4. add-sqr-sqrt 98.9%

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

      5. hypot-def 98.9%

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

      6. pow1/2 98.9%

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

      7. sqrt-pow1 99.0%

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

      8. metadata-eval 99.0%

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

      9. pow1/2 99.0%

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

      10. sqrt-pow1 99.0%

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

      11. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around inf 98.0%

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

      1. associate-/r* 98.1%

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

      2. div-inv 98.1%

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

      3. pow1/2 98.1%

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

      4. pow-flip 98.0%

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

      5. metadata-eval 98.0%

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

      6. sqrt-pow2 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

   10. Applied egg-rr 99.8%

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

   if 1.99999999999999991e-6 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

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 53.9%

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

   1. flip-- 54.1%

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

   2. div-inv 54.1%

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

   3. add-sqr-sqrt 54.1%

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

   4. add-sqr-sqrt 54.5%

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

   5. associate--l+ 54.5%

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

3. Applied egg-rr 54.5%

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

   1. +-commutative 54.5%

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

   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: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) * 0.5;
	}
	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 = math.pow(x, -0.5) * 0.5
	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((x ^ -0.5) * 0.5);
	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 = (x ^ -0.5) * 0.5;
	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[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $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 99.0%

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

      1. +-commutative 99.0%

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

      2. unpow2 99.0%

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

      3. associate-*r* 99.0%

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

      4. distribute-rgt-out 99.0%

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

      5. *-commutative 99.0%

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

   4. Simplified 99.0%

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

   if 1.25 < x

   1. Initial program 7.1%

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

      1. flip-- 7.6%

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

      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 7.6%

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

      4. add-sqr-sqrt 8.4%

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

      5. associate--l+ 8.4%

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

   3. Applied egg-rr 8.4%

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

      1. +-commutative 8.4%

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

      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%

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

      1. add-sqr-sqrt 99.0%

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

      2. pow2 99.0%

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

      3. add-sqr-sqrt 98.9%

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

      4. add-sqr-sqrt 98.9%

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

      5. hypot-def 98.9%

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

      6. pow1/2 98.9%

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

      7. sqrt-pow1 99.0%

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

      8. metadata-eval 99.0%

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

      9. pow1/2 99.0%

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

      10. sqrt-pow1 99.0%

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

      11. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around inf 95.8%

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

      1. associate-/r* 95.9%

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

      2. div-inv 95.9%

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

      3. pow1/2 95.9%

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

      4. pow-flip 95.8%

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

      5. metadata-eval 95.8%

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

      6. sqrt-pow2 97.6%

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

      7. metadata-eval 97.6%

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

      8. metadata-eval 97.6%

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

      9. metadata-eval 97.6%

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

   10. Applied egg-rr 97.6%

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

4. Final simplification 98.3%

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

Alternative 5: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. associate--l+ 98.7%

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

      3. *-commutative 98.7%

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

   4. Applied egg-rr 98.7%

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

   if 1 < x

   1. Initial program 7.1%

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

      1. flip-- 7.6%

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

      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 7.6%

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

      4. add-sqr-sqrt 8.4%

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

      5. associate--l+ 8.4%

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

   3. Applied egg-rr 8.4%

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

      1. +-commutative 8.4%

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

      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%

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

      1. add-sqr-sqrt 99.0%

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

      2. pow2 99.0%

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

      3. add-sqr-sqrt 98.9%

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

      4. add-sqr-sqrt 98.9%

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

      5. hypot-def 98.9%

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

      6. pow1/2 98.9%

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

      7. sqrt-pow1 99.0%

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

      8. metadata-eval 99.0%

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

      9. pow1/2 99.0%

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

      10. sqrt-pow1 99.0%

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

      11. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around inf 95.8%

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

      1. associate-/r* 95.9%

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

      2. div-inv 95.9%

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

      3. pow1/2 95.9%

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

      4. pow-flip 95.8%

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

      5. metadata-eval 95.8%

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

      6. sqrt-pow2 97.6%

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

      7. metadata-eval 97.6%

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

      8. metadata-eval 97.6%

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

      9. metadata-eval 97.6%

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

   10. Applied egg-rr 97.6%

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

4. Final simplification 98.1%

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

Alternative 6: 97.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

      1. 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. add-sqr-sqrt 99.9%

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

      2. pow2 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. hypot-def 99.9%

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

      6. pow1/2 99.9%

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

      7. sqrt-pow1 99.9%

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

      8. metadata-eval 99.9%

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

      9. pow1/2 99.9%

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

      10. sqrt-pow1 99.9%

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

      11. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around 0 97.4%

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

   if 1 < x

   1. Initial program 7.1%

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

      1. flip-- 7.6%

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

      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 7.6%

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

      4. add-sqr-sqrt 8.4%

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

      5. associate--l+ 8.4%

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

   3. Applied egg-rr 8.4%

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

      1. +-commutative 8.4%

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

      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%

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

      1. add-sqr-sqrt 99.0%

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

      2. pow2 99.0%

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

      3. add-sqr-sqrt 98.9%

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

      4. add-sqr-sqrt 98.9%

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

      5. hypot-def 98.9%

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

      6. pow1/2 98.9%

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

      7. sqrt-pow1 99.0%

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

      8. metadata-eval 99.0%

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

      9. pow1/2 99.0%

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

      10. sqrt-pow1 99.0%

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

      11. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around inf 95.8%

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

      1. associate-/r* 95.9%

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

      2. div-inv 95.9%

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

      3. pow1/2 95.9%

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

      4. pow-flip 95.8%

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

      5. metadata-eval 95.8%

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

      6. sqrt-pow2 97.6%

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

      7. metadata-eval 97.6%

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

      8. metadata-eval 97.6%

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

      9. metadata-eval 97.6%

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

   10. Applied egg-rr 97.6%

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

4. Final simplification 97.5%

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

Alternative 7: 96.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 96.0%

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

   if 0.25 < x

   1. Initial program 7.8%

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

      1. flip-- 8.4%

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

      2. div-inv 8.4%

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

      3. add-sqr-sqrt 8.3%

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

      4. add-sqr-sqrt 9.1%

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

      5. associate--l+ 9.1%

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

   3. Applied egg-rr 9.1%

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

      1. +-commutative 9.1%

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

      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%

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

      1. add-sqr-sqrt 99.0%

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

      2. pow2 99.0%

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

      3. add-sqr-sqrt 99.0%

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

      4. add-sqr-sqrt 98.9%

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

      5. hypot-def 98.9%

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

      6. pow1/2 98.9%

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

      7. sqrt-pow1 99.0%

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

      8. metadata-eval 99.0%

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

      9. pow1/2 99.0%

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

      10. sqrt-pow1 99.0%

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

      11. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around inf 95.2%

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

      1. associate-/r* 95.3%

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

      2. div-inv 95.3%

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

      3. pow1/2 95.3%

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

      4. pow-flip 95.2%

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

      5. metadata-eval 95.2%

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

      6. sqrt-pow2 96.9%

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

      7. metadata-eval 96.9%

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

      8. metadata-eval 96.9%

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

      9. metadata-eval 96.9%

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

   10. Applied egg-rr 96.9%

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

4. Final simplification 96.5%

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

Alternative 8: 96.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.25d0) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.25d0 / 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 = Math.sqrt((0.25 / x));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 0.25], 1.0, N[Sqrt[N[(0.25 / 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 96.0%

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

   if 0.25 < x

   1. Initial program 7.8%

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

      1. flip-- 8.4%

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

      2. div-inv 8.4%

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

      3. add-sqr-sqrt 8.3%

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

      4. add-sqr-sqrt 9.1%

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

      5. associate--l+ 9.1%

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

   3. Applied egg-rr 9.1%

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

      1. +-commutative 9.1%

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

      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%

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

      1. add-sqr-sqrt 99.0%

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

      2. pow2 99.0%

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

      3. add-sqr-sqrt 99.0%

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

      4. add-sqr-sqrt 98.9%

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

      5. hypot-def 98.9%

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

      6. pow1/2 98.9%

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

      7. sqrt-pow1 99.0%

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

      8. metadata-eval 99.0%

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

      9. pow1/2 99.0%

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

      10. sqrt-pow1 99.0%

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

      11. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around inf 95.2%

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

      1. add-sqr-sqrt 95.4%

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

      2. sqrt-unprod 95.2%

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

      3. frac-times 95.3%

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

      4. metadata-eval 95.3%

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

      5. swap-sqr 95.1%

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

      6. add-sqr-sqrt 95.2%

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

      7. sqrt-pow2 96.0%

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

      8. metadata-eval 96.0%

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

      9. metadata-eval 96.0%

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

      10. sqrt-pow2 96.8%

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

      11. metadata-eval 96.8%

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

      12. metadata-eval 96.8%

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

      13. metadata-eval 96.8%

          \[\leadsto \sqrt{\frac{1}{x \cdot \color{blue}{4}}} \]

   10. Applied egg-rr 96.8%

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

       1. *-commutative 96.8%

          \[\leadsto \sqrt{\frac{1}{\color{blue}{4 \cdot x}}} \]

       2. associate-/r* 96.8%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{4}}{x}}} \]

       3. metadata-eval 96.8%

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

   12. Simplified 96.8%

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

4. Final simplification 96.4%

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

Alternative 9: 52.2% 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 53.9%

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

2. Taylor expanded in x around 0 51.5%

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

3. Final simplification 51.5%

   \[\leadsto 1 \]

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 2023314 
(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)))