Main:bigenough3 from C

Percentage Accurate: 52.5% → 99.8%

Time: 7.7s

Alternatives: 11

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: 52.5% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.1%

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

   1. flip-- 46.3%

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

   2. div-inv 46.3%

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

   3. add-sqr-sqrt 46.3%

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

   4. add-sqr-sqrt 47.0%

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

   5. associate--l+ 47.0%

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

4. Applied egg-rr 47.0%

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

   1. associate-*r/ 47.0%

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

   2. *-rgt-identity 47.0%

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

   3. +-commutative 47.0%

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

   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%

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

   8. +-commutative 99.8%

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

6. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
7. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 50000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 4\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 50000000.0) (- (sqrt (+ 1.0 x)) (sqrt x)) (pow (* x 4.0) -0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 50000000.0) {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	} else {
		tmp = pow((x * 4.0), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 50000000.0d0) then
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    else
        tmp = (x * 4.0d0) ** (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 50000000.0) {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	} else {
		tmp = Math.pow((x * 4.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 50000000.0:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	else:
		tmp = math.pow((x * 4.0), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 50000000.0)
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	else
		tmp = Float64(x * 4.0) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 50000000.0)
		tmp = sqrt((1.0 + x)) - sqrt(x);
	else
		tmp = (x * 4.0) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 50000000.0], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Power[N[(x * 4.0), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e7

   1. Initial program 99.6%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   if 5e7 < x

   1. Initial program 4.5%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. inv-pow 99.5%

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

      4. sqrt-pow1 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. *-rgt-identity 99.7%

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

      2. metadata-eval 99.7%

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

      3. pow-flip 99.3%

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

      4. pow1/2 99.3%

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

      5. div-inv 99.3%

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

      6. metadata-eval 99.3%

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

      7. sqrt-div 99.5%

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

      8. pow1/2 99.5%

         \[\leadsto \color{blue}{{\left(\frac{0.25}{x}\right)}^{0.5}} \]

      9. clear-num 99.5%

         \[\leadsto {\color{blue}{\left(\frac{1}{\frac{x}{0.25}}\right)}}^{0.5} \]

      10. inv-pow 99.5%

          \[\leadsto {\color{blue}{\left({\left(\frac{x}{0.25}\right)}^{-1}\right)}}^{0.5} \]

      11. pow-pow 99.7%

          \[\leadsto \color{blue}{{\left(\frac{x}{0.25}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      12. div-inv 99.7%

          \[\leadsto {\color{blue}{\left(x \cdot \frac{1}{0.25}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      13. metadata-eval 99.7%

          \[\leadsto {\left(x \cdot \color{blue}{4}\right)}^{\left(-1 \cdot 0.5\right)} \]

      14. metadata-eval 99.7%

          \[\leadsto {\left(x \cdot 4\right)}^{\color{blue}{-0.5}} \]

   7. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{{\left(x \cdot 4\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 50000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 4\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 4\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.3)
   (- (+ 1.0 (* x (+ 0.5 (* x (- (* x 0.0625) 0.125))))) (sqrt x))
   (pow (* x 4.0) -0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.3) {
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - sqrt(x);
	} else {
		tmp = pow((x * 4.0), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.3d0) then
        tmp = (1.0d0 + (x * (0.5d0 + (x * ((x * 0.0625d0) - 0.125d0))))) - sqrt(x)
    else
        tmp = (x * 4.0d0) ** (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.3) {
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - Math.sqrt(x);
	} else {
		tmp = Math.pow((x * 4.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.3:
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - math.sqrt(x)
	else:
		tmp = math.pow((x * 4.0), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.3)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.0625) - 0.125))))) - sqrt(x));
	else
		tmp = Float64(x * 4.0) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.3)
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - sqrt(x);
	else
		tmp = (x * 4.0) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.3], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Power[N[(x * 4.0), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000004

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) - \sqrt{x}} \]

   if 1.30000000000000004 < x

   1. Initial program 5.5%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.8%

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

      1. *-un-lft-identity 98.8%

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

      2. *-commutative 98.8%

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

      3. inv-pow 98.8%

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

      4. sqrt-pow1 98.9%

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

      5. metadata-eval 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. metadata-eval 98.9%

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

      3. pow-flip 98.6%

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

      4. pow1/2 98.6%

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

      5. div-inv 98.6%

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

      6. metadata-eval 98.6%

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

      7. sqrt-div 98.8%

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

      8. pow1/2 98.8%

         \[\leadsto \color{blue}{{\left(\frac{0.25}{x}\right)}^{0.5}} \]

      9. clear-num 98.8%

         \[\leadsto {\color{blue}{\left(\frac{1}{\frac{x}{0.25}}\right)}}^{0.5} \]

      10. inv-pow 98.8%

          \[\leadsto {\color{blue}{\left({\left(\frac{x}{0.25}\right)}^{-1}\right)}}^{0.5} \]

      11. pow-pow 98.9%

          \[\leadsto \color{blue}{{\left(\frac{x}{0.25}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      12. div-inv 98.9%

          \[\leadsto {\color{blue}{\left(x \cdot \frac{1}{0.25}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      13. metadata-eval 98.9%

          \[\leadsto {\left(x \cdot \color{blue}{4}\right)}^{\left(-1 \cdot 0.5\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto {\left(x \cdot 4\right)}^{\color{blue}{-0.5}} \]

   7. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(x \cdot 4\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 4\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 4\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25)
   (+ 1.0 (- (* x (+ 0.5 (* x -0.125))) (sqrt x)))
   (pow (* x 4.0) -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 * 4.0), -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 * 4.0d0) ** (-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 * 4.0), -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 * 4.0), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) - sqrt(x)));
	else
		tmp = Float64(x * 4.0) ^ -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 * 4.0) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], N[(1.0 + N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(x * 4.0), $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. Add Preprocessing

   3. Taylor expanded in x around 0 99.4%

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

      1. associate--l+ 99.4%

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

      2. *-commutative 99.4%

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

   5. Simplified 99.4%

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

   if 1.25 < x

   1. Initial program 5.5%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.8%

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

      1. *-un-lft-identity 98.8%

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

      2. *-commutative 98.8%

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

      3. inv-pow 98.8%

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

      4. sqrt-pow1 98.9%

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

      5. metadata-eval 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. metadata-eval 98.9%

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

      3. pow-flip 98.6%

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

      4. pow1/2 98.6%

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

      5. div-inv 98.6%

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

      6. metadata-eval 98.6%

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

      7. sqrt-div 98.8%

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

      8. pow1/2 98.8%

         \[\leadsto \color{blue}{{\left(\frac{0.25}{x}\right)}^{0.5}} \]

      9. clear-num 98.8%

         \[\leadsto {\color{blue}{\left(\frac{1}{\frac{x}{0.25}}\right)}}^{0.5} \]

      10. inv-pow 98.8%

          \[\leadsto {\color{blue}{\left({\left(\frac{x}{0.25}\right)}^{-1}\right)}}^{0.5} \]

      11. pow-pow 98.9%

          \[\leadsto \color{blue}{{\left(\frac{x}{0.25}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      12. div-inv 98.9%

          \[\leadsto {\color{blue}{\left(x \cdot \frac{1}{0.25}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      13. metadata-eval 98.9%

          \[\leadsto {\left(x \cdot \color{blue}{4}\right)}^{\left(-1 \cdot 0.5\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto {\left(x \cdot 4\right)}^{\color{blue}{-0.5}} \]

   7. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(x \cdot 4\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + \left(x \cdot 0.5 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 4\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (+ 1.0 (- (* x 0.5) (sqrt x))) (pow (* x 4.0) -0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + ((x * 0.5) - sqrt(x));
	} else {
		tmp = pow((x * 4.0), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 1.0d0 + ((x * 0.5d0) - sqrt(x))
    else
        tmp = (x * 4.0d0) ** (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + ((x * 0.5) - Math.sqrt(x));
	} else {
		tmp = Math.pow((x * 4.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 + ((x * 0.5) - math.sqrt(x))
	else:
		tmp = math.pow((x * 4.0), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 + Float64(Float64(x * 0.5) - sqrt(x)));
	else
		tmp = Float64(x * 4.0) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 + ((x * 0.5) - sqrt(x));
	else
		tmp = (x * 4.0) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(x * 4.0), $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. Add Preprocessing

   3. Taylor expanded in x around 0 98.9%

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

      1. associate--l+ 98.9%

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

      2. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   if 1 < x

   1. Initial program 5.5%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.8%

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

      1. *-un-lft-identity 98.8%

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

      2. *-commutative 98.8%

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

      3. inv-pow 98.8%

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

      4. sqrt-pow1 98.9%

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

      5. metadata-eval 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. metadata-eval 98.9%

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

      3. pow-flip 98.6%

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

      4. pow1/2 98.6%

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

      5. div-inv 98.6%

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

      6. metadata-eval 98.6%

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

      7. sqrt-div 98.8%

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

      8. pow1/2 98.8%

         \[\leadsto \color{blue}{{\left(\frac{0.25}{x}\right)}^{0.5}} \]

      9. clear-num 98.8%

         \[\leadsto {\color{blue}{\left(\frac{1}{\frac{x}{0.25}}\right)}}^{0.5} \]

      10. inv-pow 98.8%

          \[\leadsto {\color{blue}{\left({\left(\frac{x}{0.25}\right)}^{-1}\right)}}^{0.5} \]

      11. pow-pow 98.9%

          \[\leadsto \color{blue}{{\left(\frac{x}{0.25}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      12. div-inv 98.9%

          \[\leadsto {\color{blue}{\left(x \cdot \frac{1}{0.25}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      13. metadata-eval 98.9%

          \[\leadsto {\left(x \cdot \color{blue}{4}\right)}^{\left(-1 \cdot 0.5\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto {\left(x \cdot 4\right)}^{\color{blue}{-0.5}} \]

   7. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(x \cdot 4\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1}{1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 4\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (/ 1.0 (+ 1.0 (sqrt x))) (pow (* x 4.0) -0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 / (1.0 + sqrt(x));
	} else {
		tmp = pow((x * 4.0), -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 * 4.0d0) ** (-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 * 4.0), -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 * 4.0), -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 * 4.0) ^ -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 * 4.0) ^ -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[Power[N[(x * 4.0), $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. Add Preprocessing
   3. 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 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. associate--l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l- 99.9%

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

      5. +-inverses 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 97.5%

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

   if 1 < x

   1. Initial program 5.5%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.8%

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

      1. *-un-lft-identity 98.8%

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

      2. *-commutative 98.8%

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

      3. inv-pow 98.8%

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

      4. sqrt-pow1 98.9%

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

      5. metadata-eval 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. metadata-eval 98.9%

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

      3. pow-flip 98.6%

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

      4. pow1/2 98.6%

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

      5. div-inv 98.6%

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

      6. metadata-eval 98.6%

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

      7. sqrt-div 98.8%

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

      8. pow1/2 98.8%

         \[\leadsto \color{blue}{{\left(\frac{0.25}{x}\right)}^{0.5}} \]

      9. clear-num 98.8%

         \[\leadsto {\color{blue}{\left(\frac{1}{\frac{x}{0.25}}\right)}}^{0.5} \]

      10. inv-pow 98.8%

          \[\leadsto {\color{blue}{\left({\left(\frac{x}{0.25}\right)}^{-1}\right)}}^{0.5} \]

      11. pow-pow 98.9%

          \[\leadsto \color{blue}{{\left(\frac{x}{0.25}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      12. div-inv 98.9%

          \[\leadsto {\color{blue}{\left(x \cdot \frac{1}{0.25}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      13. metadata-eval 98.9%

          \[\leadsto {\left(x \cdot \color{blue}{4}\right)}^{\left(-1 \cdot 0.5\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto {\left(x \cdot 4\right)}^{\color{blue}{-0.5}} \]

   7. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(x \cdot 4\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1}{1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 4\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.35:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 4\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.35) (- 1.0 (sqrt x)) (pow (* x 4.0) -0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 0.35) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = pow((x * 4.0), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.35d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = (x * 4.0d0) ** (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.35) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = Math.pow((x * 4.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.35:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = math.pow((x * 4.0), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.35)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = Float64(x * 4.0) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.35)
		tmp = 1.0 - sqrt(x);
	else
		tmp = (x * 4.0) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.35], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Power[N[(x * 4.0), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.34999999999999998

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 97.5%

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

   if 0.34999999999999998 < x

   1. Initial program 5.5%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.8%

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

      1. *-un-lft-identity 98.8%

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

      2. *-commutative 98.8%

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

      3. inv-pow 98.8%

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

      4. sqrt-pow1 98.9%

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

      5. metadata-eval 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. metadata-eval 98.9%

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

      3. pow-flip 98.6%

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

      4. pow1/2 98.6%

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

      5. div-inv 98.6%

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

      6. metadata-eval 98.6%

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

      7. sqrt-div 98.8%

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

      8. pow1/2 98.8%

         \[\leadsto \color{blue}{{\left(\frac{0.25}{x}\right)}^{0.5}} \]

      9. clear-num 98.8%

         \[\leadsto {\color{blue}{\left(\frac{1}{\frac{x}{0.25}}\right)}}^{0.5} \]

      10. inv-pow 98.8%

          \[\leadsto {\color{blue}{\left({\left(\frac{x}{0.25}\right)}^{-1}\right)}}^{0.5} \]

      11. pow-pow 98.9%

          \[\leadsto \color{blue}{{\left(\frac{x}{0.25}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      12. div-inv 98.9%

          \[\leadsto {\color{blue}{\left(x \cdot \frac{1}{0.25}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      13. metadata-eval 98.9%

          \[\leadsto {\left(x \cdot \color{blue}{4}\right)}^{\left(-1 \cdot 0.5\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto {\left(x \cdot 4\right)}^{\color{blue}{-0.5}} \]

   7. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(x \cdot 4\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.35:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.35) (- 1.0 (sqrt x)) (sqrt (/ 0.25 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.35) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = sqrt((0.25 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.35d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = sqrt((0.25d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.35) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = Math.sqrt((0.25 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.35:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = math.sqrt((0.25 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.35)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = sqrt(Float64(0.25 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.35)
		tmp = 1.0 - sqrt(x);
	else
		tmp = sqrt((0.25 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.35], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.25 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.34999999999999998

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 97.5%

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

   if 0.34999999999999998 < x

   1. Initial program 5.5%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.8%

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

      1. add-sqr-sqrt 98.1%

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

      2. sqrt-unprod 98.8%

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

      3. *-commutative 98.8%

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

      4. *-commutative 98.8%

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

      5. swap-sqr 98.8%

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

      6. add-sqr-sqrt 98.8%

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

      7. metadata-eval 98.8%

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

   5. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x} \cdot 0.25}} \]
   6. Step-by-step derivation

      1. associate-*l/ 98.8%

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

      2. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

Alternative 9: 53.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.25}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (/ 0.25 x)))

C

double code(double x) {
	return sqrt((0.25 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((0.25d0 / x))
end function

Java

public static double code(double x) {
	return Math.sqrt((0.25 / x));
}

Python

def code(x):
	return math.sqrt((0.25 / x))

Julia

function code(x)
	return sqrt(Float64(0.25 / x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((0.25 / x));
end

Wolfram

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

Derivation

1. Initial program 46.1%

   \[\sqrt{x + 1} - \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 59.2%

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

   1. add-sqr-sqrt 58.9%

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

   2. sqrt-unprod 59.2%

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

   3. *-commutative 59.2%

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

   4. *-commutative 59.2%

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

   5. swap-sqr 59.2%

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

   6. add-sqr-sqrt 59.2%

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

   7. metadata-eval 59.2%

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

5. Applied egg-rr 59.2%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x} \cdot 0.25}} \]
6. Step-by-step derivation

   1. associate-*l/ 59.2%

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

   2. metadata-eval 59.2%

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

7. Simplified 59.2%

   \[\leadsto \color{blue}{\sqrt{\frac{0.25}{x}}} \]
8. Add Preprocessing

Alternative 10: 13.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow x -0.5))

C

double code(double x) {
	return pow(x, -0.5);
}

Fortran

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

Java

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

Python

def code(x):
	return math.pow(x, -0.5)

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

function tmp = code(x)
	tmp = x ^ -0.5;
end

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 46.1%

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

   1. flip-- 46.3%

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

   2. div-inv 46.3%

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

   3. add-sqr-sqrt 46.3%

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

   4. add-sqr-sqrt 47.0%

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

   5. associate--l+ 47.0%

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

4. Applied egg-rr 47.0%

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

   1. associate-*r/ 47.0%

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

   2. *-rgt-identity 47.0%

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

   3. +-commutative 47.0%

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

   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%

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

   8. +-commutative 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in x around 0 52.6%

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

8. Taylor expanded in x around inf 13.6%

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

   1. unpow-1 13.6%

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

   2. metadata-eval 13.6%

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

   3. pow-sqr 13.6%

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

   4. rem-sqrt-square 13.6%

      \[\leadsto \color{blue}{\left|{x}^{-0.5}\right|} \]

   5. rem-square-sqrt 13.6%

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

   6. fabs-sqr 13.6%

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

   7. rem-square-sqrt 13.6%

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

10. Simplified 13.6%

    \[\leadsto \color{blue}{{x}^{-0.5}} \]
11. Add Preprocessing

Alternative 11: 6.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt x))

C

double code(double x) {
	return sqrt(x);
}

Fortran

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

Java

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

Python

def code(x):
	return math.sqrt(x)

Julia

function code(x)
	return sqrt(x)
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[x], $MachinePrecision]

Derivation

1. Initial program 46.1%

   \[\sqrt{x + 1} - \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 42.8%

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

4. Taylor expanded in x around inf 1.7%

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

   1. neg-mul-1 1.7%

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

6. Simplified 1.7%

   \[\leadsto \color{blue}{-\sqrt{x}} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 6.1%

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

   3. sqr-neg 6.1%

      \[\leadsto \sqrt{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

   4. add-sqr-sqrt 6.1%

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

   5. pow1/2 6.1%

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

8. Applied egg-rr 6.1%

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

9. Taylor expanded in x around 0 6.1%

   \[\leadsto \color{blue}{\sqrt{x}} \]
10. Add Preprocessing

Developer Target 1: 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 2024181 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ (sqrt (+ x 1)) (sqrt x))))

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