2sqrt (example 3.1)

Percentage Accurate: 53.2% → 99.7%

Time: 1.7min

Alternatives: 7

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: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

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 51.3%

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

   1. flip-- 51.9%

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

   2. div-inv 51.9%

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

   3. add-sqr-sqrt 51.5%

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

   4. add-sqr-sqrt 51.9%

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

   5. associate--l+ 51.8%

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

3. Applied egg-rr 51.8%

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

   1. associate-*r/ 51.8%

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

   2. *-rgt-identity 51.8%

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

   3. +-commutative 51.8%

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

   4. associate-+l- 99.7%

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

   5. +-inverses 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 1e-5) (* 0.5 (pow x -0.5)) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = 0.5 * pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 1d-5) then
        tmp = 0.5d0 * (x ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = 0.5 * Math.pow(x, -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 <= 1e-5:
		tmp = 0.5 * math.pow(x, -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 <= 1e-5)
		tmp = Float64(0.5 * (x ^ -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 <= 1e-5)
		tmp = 0.5 * (x ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-5], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 1.00000000000000008e-5

   1. Initial program 4.4%

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

      1. flip-- 5.3%

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

      2. div-inv 5.3%

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

      3. add-sqr-sqrt 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 5.3%

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

      5. associate--l+ 5.3%

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

   3. Applied egg-rr 5.3%

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

      1. associate-*r/ 5.3%

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

      2. *-rgt-identity 5.3%

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

      3. +-commutative 5.3%

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

      4. associate-+l- 99.5%

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

      5. +-inverses 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   5. Simplified 99.5%

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

      1. flip3-+ 69.4%

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

      2. associate-/r/ 69.4%

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

      3. sqrt-pow2 69.3%

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

      4. metadata-eval 69.3%

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

      5. +-commutative 69.3%

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

      6. sqrt-pow2 69.2%

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

      7. metadata-eval 69.2%

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

      8. add-sqr-sqrt 69.5%

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

      9. add-sqr-sqrt 69.3%

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

      10. associate-+r- 69.2%

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

      11. +-commutative 69.2%

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

      12. sqrt-unprod 51.4%

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

   7. Applied egg-rr 51.4%

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

   8. Taylor expanded in x around inf 69.2%

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

   9. Taylor expanded in x around inf 99.5%

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

       1. rem-exp-log 92.1%

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

       2. exp-neg 92.1%

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

       3. unpow1/2 92.1%

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

       4. exp-prod 92.1%

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

       5. distribute-lft-neg-out 92.1%

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

       6. *-commutative 92.1%

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

       7. distribute-lft-neg-in 92.1%

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

       8. metadata-eval 92.1%

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

       9. log-pow 92.1%

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

       10. rem-exp-log 99.7%

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

   11. Simplified 99.7%

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

   if 1.00000000000000008e-5 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.6%

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

4. Final simplification 99.7%

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

Alternative 3: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\frac{1}{1 + \left(\sqrt{x} + x \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4) (/ 1.0 (+ 1.0 (+ (sqrt x) (* x 0.5)))) (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0 / (1.0 + (sqrt(x) + (x * 0.5)));
	} else {
		tmp = 0.5 * pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = 1.0d0 / (1.0d0 + (sqrt(x) + (x * 0.5d0)))
    else
        tmp = 0.5d0 * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0 / (1.0 + (Math.sqrt(x) + (x * 0.5)));
	} else {
		tmp = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = 1.0 / (1.0 + (math.sqrt(x) + (x * 0.5)))
	else:
		tmp = 0.5 * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = Float64(1.0 / Float64(1.0 + Float64(sqrt(x) + Float64(x * 0.5))));
	else
		tmp = Float64(0.5 * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = 1.0 / (1.0 + (sqrt(x) + (x * 0.5)));
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], N[(1.0 / N[(1.0 + N[(N[Sqrt[x], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   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+ 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l- 99.8%

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

      5. +-inverses 99.8%

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

      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%

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

      2. add-cube-cbrt 99.8%

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

      3. fma-def 99.8%

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

      4. cbrt-prod 99.9%

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

      5. add-sqr-sqrt 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{\color{blue}{x}}, \sqrt[3]{\sqrt{x}}, \sqrt{1 + x}\right)} \]

      6. add-cube-cbrt 99.9%

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

      7. sqrt-prod 99.9%

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

      8. sqrt-unprod 99.9%

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

      9. add-cbrt-cube 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\sqrt{\sqrt[3]{x}}}, \sqrt{1 + x}\right)} \]

      10. pow1/3 99.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt{\color{blue}{{x}^{0.3333333333333333}}}, \sqrt{1 + x}\right)} \]

      11. sqrt-pow1 99.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{{x}^{\left(\frac{0.3333333333333333}{2}\right)}}, \sqrt{1 + x}\right)} \]

      12. metadata-eval 99.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, {x}^{\color{blue}{0.16666666666666666}}, \sqrt{1 + x}\right)} \]

      13. +-commutative 99.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, {x}^{0.16666666666666666}, \sqrt{\color{blue}{x + 1}}\right)} \]

   7. Applied egg-rr 99.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, {x}^{0.16666666666666666}, \sqrt{x + 1}\right)}} \]
   8. Step-by-step derivation

      1. fma-udef 99.9%

         \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x} \cdot {x}^{0.16666666666666666} + \sqrt{x + 1}}} \]

      2. +-commutative 99.9%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{x + 1} + \sqrt[3]{x} \cdot {x}^{0.16666666666666666}}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt[3]{x} \cdot {x}^{0.16666666666666666}} \]

      4. unpow1/3 99.8%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \color{blue}{{x}^{0.3333333333333333}} \cdot {x}^{0.16666666666666666}} \]

      5. metadata-eval 99.8%

         \[\leadsto \frac{1}{\sqrt{1 + x} + {x}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}} \cdot {x}^{0.16666666666666666}} \]

      6. pow-sqr 99.8%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \color{blue}{\left({x}^{0.16666666666666666} \cdot {x}^{0.16666666666666666}\right)} \cdot {x}^{0.16666666666666666}} \]

      7. unpow3 99.8%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{3}}} \]

   9. Simplified 99.8%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 + x} + {\left({x}^{0.16666666666666666}\right)}^{3}}} \]

   10. Taylor expanded in x around 0 98.4%

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

       1. *-commutative 98.4%

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

   12. Simplified 98.4%

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

   if 2.39999999999999991 < x

   1. Initial program 5.5%

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

      1. flip-- 6.7%

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

      2. div-inv 6.7%

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

      3. add-sqr-sqrt 6.0%

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

      4. add-sqr-sqrt 6.7%

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

      5. associate--l+ 6.7%

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

   3. Applied egg-rr 6.7%

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

      1. associate-*r/ 6.7%

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

      2. *-rgt-identity 6.7%

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

      3. +-commutative 6.7%

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

      4. associate-+l- 99.5%

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

      5. +-inverses 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   5. Simplified 99.5%

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

      1. flip3-+ 69.8%

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

      2. associate-/r/ 69.8%

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

      3. sqrt-pow2 69.8%

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

      4. metadata-eval 69.8%

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

      5. +-commutative 69.8%

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

      6. sqrt-pow2 69.6%

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

      7. metadata-eval 69.6%

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

      8. add-sqr-sqrt 69.9%

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

      9. add-sqr-sqrt 69.7%

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

      10. associate-+r- 69.7%

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

      11. +-commutative 69.7%

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

      12. sqrt-unprod 52.1%

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

   7. Applied egg-rr 52.1%

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

   8. Taylor expanded in x around inf 68.9%

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

   9. Taylor expanded in x around inf 98.7%

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

       1. rem-exp-log 91.5%

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

       2. exp-neg 91.4%

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

       3. unpow1/2 91.4%

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

       4. exp-prod 91.5%

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

       5. distribute-lft-neg-out 91.5%

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

       6. *-commutative 91.5%

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

       7. distribute-lft-neg-in 91.5%

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

       8. metadata-eval 91.5%

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

       9. log-pow 91.5%

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

       10. rem-exp-log 99.0%

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

   11. Simplified 99.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\frac{1}{1 + \left(\sqrt{x} + x \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 4: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 / (1.0 + sqrt(x));
	else
		tmp = 0.5 * (x ^ -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[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l- 99.8%

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

      5. +-inverses 99.8%

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

      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%

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

      2. add-cube-cbrt 99.8%

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

      3. fma-def 99.8%

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

      4. cbrt-prod 99.9%

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

      5. add-sqr-sqrt 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{\color{blue}{x}}, \sqrt[3]{\sqrt{x}}, \sqrt{1 + x}\right)} \]

      6. add-cube-cbrt 99.9%

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

      7. sqrt-prod 99.9%

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

      8. sqrt-unprod 99.9%

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

      9. add-cbrt-cube 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\sqrt{\sqrt[3]{x}}}, \sqrt{1 + x}\right)} \]

      10. pow1/3 99.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt{\color{blue}{{x}^{0.3333333333333333}}}, \sqrt{1 + x}\right)} \]

      11. sqrt-pow1 99.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{{x}^{\left(\frac{0.3333333333333333}{2}\right)}}, \sqrt{1 + x}\right)} \]

      12. metadata-eval 99.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, {x}^{\color{blue}{0.16666666666666666}}, \sqrt{1 + x}\right)} \]

      13. +-commutative 99.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, {x}^{0.16666666666666666}, \sqrt{\color{blue}{x + 1}}\right)} \]

   7. Applied egg-rr 99.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, {x}^{0.16666666666666666}, \sqrt{x + 1}\right)}} \]
   8. Step-by-step derivation

      1. fma-udef 99.9%

         \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x} \cdot {x}^{0.16666666666666666} + \sqrt{x + 1}}} \]

      2. +-commutative 99.9%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{x + 1} + \sqrt[3]{x} \cdot {x}^{0.16666666666666666}}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt[3]{x} \cdot {x}^{0.16666666666666666}} \]

      4. unpow1/3 99.8%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \color{blue}{{x}^{0.3333333333333333}} \cdot {x}^{0.16666666666666666}} \]

      5. metadata-eval 99.8%

         \[\leadsto \frac{1}{\sqrt{1 + x} + {x}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}} \cdot {x}^{0.16666666666666666}} \]

      6. pow-sqr 99.8%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \color{blue}{\left({x}^{0.16666666666666666} \cdot {x}^{0.16666666666666666}\right)} \cdot {x}^{0.16666666666666666}} \]

      7. unpow3 99.8%

         \[\leadsto \frac{1}{\sqrt{1 + x} + \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{3}}} \]

   9. Simplified 99.8%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 + x} + {\left({x}^{0.16666666666666666}\right)}^{3}}} \]

   10. Taylor expanded in x around 0 97.2%

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

       1. +-commutative 97.2%

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

   12. Simplified 97.2%

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

   if 1 < x

   1. Initial program 5.5%

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

      1. flip-- 6.7%

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

      2. div-inv 6.7%

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

      3. add-sqr-sqrt 6.0%

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

      4. add-sqr-sqrt 6.7%

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

      5. associate--l+ 6.7%

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

   3. Applied egg-rr 6.7%

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

      1. associate-*r/ 6.7%

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

      2. *-rgt-identity 6.7%

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

      3. +-commutative 6.7%

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

      4. associate-+l- 99.5%

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

      5. +-inverses 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   5. Simplified 99.5%

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

      1. flip3-+ 69.8%

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

      2. associate-/r/ 69.8%

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

      3. sqrt-pow2 69.8%

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

      4. metadata-eval 69.8%

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

      5. +-commutative 69.8%

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

      6. sqrt-pow2 69.6%

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

      7. metadata-eval 69.6%

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

      8. add-sqr-sqrt 69.9%

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

      9. add-sqr-sqrt 69.7%

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

      10. associate-+r- 69.7%

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

      11. +-commutative 69.7%

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

      12. sqrt-unprod 52.1%

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

   7. Applied egg-rr 52.1%

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

   8. Taylor expanded in x around inf 68.9%

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

   9. Taylor expanded in x around inf 98.7%

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

       1. rem-exp-log 91.5%

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

       2. exp-neg 91.4%

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

       3. unpow1/2 91.4%

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

       4. exp-prod 91.5%

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

       5. distribute-lft-neg-out 91.5%

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

       6. *-commutative 91.5%

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

       7. distribute-lft-neg-in 91.5%

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

       8. metadata-eval 91.5%

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

       9. log-pow 91.5%

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

       10. rem-exp-log 99.0%

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

   11. Simplified 99.0%

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

4. Final simplification 98.1%

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

Alternative 5: 96.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 95.4%

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

   if 0.25 < x

   1. Initial program 6.2%

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

      1. flip-- 7.5%

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

      2. div-inv 7.5%

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

      3. add-sqr-sqrt 6.8%

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

      4. add-sqr-sqrt 7.5%

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

      5. associate--l+ 7.5%

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

   3. Applied egg-rr 7.5%

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

      1. associate-*r/ 7.5%

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

      2. *-rgt-identity 7.5%

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

      3. +-commutative 7.5%

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

      4. associate-+l- 99.5%

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

      5. +-inverses 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   5. Simplified 99.5%

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

      1. flip3-+ 70.1%

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

      2. associate-/r/ 70.0%

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

      3. sqrt-pow2 70.0%

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

      4. metadata-eval 70.0%

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

      5. +-commutative 70.0%

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

      6. sqrt-pow2 69.9%

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

      7. metadata-eval 69.9%

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

      8. add-sqr-sqrt 70.2%

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

      9. add-sqr-sqrt 69.9%

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

      10. associate-+r- 69.9%

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

      11. +-commutative 69.9%

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

      12. sqrt-unprod 52.5%

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

   7. Applied egg-rr 52.5%

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

   8. Taylor expanded in x around inf 68.5%

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

   9. Taylor expanded in x around inf 98.1%

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

       1. rem-exp-log 90.9%

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

       2. exp-neg 90.9%

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

       3. unpow1/2 90.9%

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

       4. exp-prod 90.9%

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

       5. distribute-lft-neg-out 90.9%

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

       6. *-commutative 90.9%

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

       7. distribute-lft-neg-in 90.9%

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

       8. metadata-eval 90.9%

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

       9. log-pow 90.9%

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

       10. rem-exp-log 98.4%

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

   11. Simplified 98.4%

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

4. Final simplification 97.0%

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

Alternative 6: 3.5% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 51.3%

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

   1. add-cube-cbrt 51.3%

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

   2. sqrt-prod 51.3%

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}} - \sqrt{x} \]

   3. add-cube-cbrt 51.2%

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

   4. sqrt-prod 51.1%

      \[\leadsto \sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}} - \color{blue}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}} \]

   5. prod-diff 51.1%

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

   6. pow2 51.1%

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

   7. pow2 51.1%

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

3. Applied egg-rr 51.1%

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

5. Simplified 51.2%

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

6. Taylor expanded in x around inf 3.5%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 3.5%

   \[\leadsto 0 \]

Alternative 7: 51.1% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 51.3%

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

2. Taylor expanded in x around 0 49.4%

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

3. Final simplification 49.4%

   \[\leadsto 1 \]

Developer target: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 66000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 66000000.0)
   (- (sqrt (+ 1.0 x)) (sqrt x))
   (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 66000000.0) {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	} else {
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 66000000.0d0) then
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    else
        tmp = 1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 66000000.0) {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	} else {
		tmp = 1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 66000000.0:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	else:
		tmp = 1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 66000000.0)
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 66000000.0)
		tmp = sqrt((1.0 + x)) - sqrt(x);
	else
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 66000000.0], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023318 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (if (<= x 66000000.0) (- (sqrt (+ 1.0 x)) (sqrt x)) (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))))

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