Main:bigenough3 from C

Percentage Accurate: 53.6% → 99.8%

Time: 7.5s

Alternatives: 9

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: 53.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.5%

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

   1. flip-- 58.7%

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

   2. add-sqr-sqrt 58.3%

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

   3. add-sqr-sqrt 60.2%

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

   4. div-sub 58.5%

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

4. Applied egg-rr 58.5%

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

   1. div-sub 60.2%

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

   2. +-commutative 60.2%

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

   3. associate--l+ 99.7%

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

   4. +-inverses 99.7%

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

   5. metadata-eval 99.7%

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

   6. +-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. metadata-eval 99.7%

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

   9. rem-square-sqrt 99.7%

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

   10. hypot-undefine 99.7%

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

6. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
7. Step-by-step derivation

   1. +-commutative 99.7%

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

   2. hypot-undefine 99.7%

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

   3. metadata-eval 99.7%

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

   4. add-sqr-sqrt 99.7%

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

8. Applied egg-rr 99.7%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{-0.5}}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 5e-5) (/ (pow x -0.5) 2.0) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = pow(x, -0.5) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 5d-5) then
        tmp = (x ** (-0.5d0)) / 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = Math.pow(x, -0.5) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 5e-5:
		tmp = math.pow(x, -0.5) / 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 5e-5)
		tmp = Float64((x ^ -0.5) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 5e-5)
		tmp = (x ^ -0.5) / 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-5], N[(N[Power[x, -0.5], $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 5.00000000000000024e-5

   1. Initial program 5.4%

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

      1. flip-- 6.1%

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

      2. add-sqr-sqrt 5.1%

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

      3. add-sqr-sqrt 9.0%

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

      4. div-sub 5.4%

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

   4. Applied egg-rr 5.4%

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

      1. div-sub 9.0%

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

      2. +-commutative 9.0%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. +-commutative 99.5%

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

      8. metadata-eval 99.5%

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

      9. rem-square-sqrt 99.5%

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

      10. hypot-undefine 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around inf 98.9%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 98.9%

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

   9. Simplified 98.9%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x} \cdot 2}} \]
   10. Step-by-step derivation

       1. associate-/r* 98.9%

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

       2. pow1/2 98.9%

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

       3. pow-flip 99.3%

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

       4. metadata-eval 99.3%

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

   11. Applied egg-rr 99.3%

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

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{-0.5}}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32)
   (- (+ 1.0 (* x (+ 0.5 (* x (- (* x 0.0625) 0.125))))) (sqrt x))
   (/ (pow x -0.5) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 1.32) {
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - sqrt(x);
	} else {
		tmp = pow(x, -0.5) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.32d0) then
        tmp = (1.0d0 + (x * (0.5d0 + (x * ((x * 0.0625d0) - 0.125d0))))) - sqrt(x)
    else
        tmp = (x ** (-0.5d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.32) {
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.32:
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) / 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.32)
		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 ^ -0.5) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.32)
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - sqrt(x);
	else
		tmp = (x ^ -0.5) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.32], 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[(N[Power[x, -0.5], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.32000000000000006

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.5%

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

   if 1.32000000000000006 < x

   1. Initial program 8.4%

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

      1. flip-- 9.0%

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

      2. add-sqr-sqrt 8.1%

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

      3. add-sqr-sqrt 12.1%

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

      4. div-sub 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. div-sub 12.1%

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

      2. +-commutative 12.1%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. +-commutative 99.5%

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

      8. metadata-eval 99.5%

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

      9. rem-square-sqrt 99.5%

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

      10. hypot-undefine 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around inf 96.5%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 96.5%

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

   9. Simplified 96.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x} \cdot 2}} \]
   10. Step-by-step derivation

       1. associate-/r* 96.5%

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

       2. pow1/2 96.5%

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

       3. pow-flip 97.0%

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

       4. metadata-eval 97.0%

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

   11. Applied egg-rr 97.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.32:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (x * 0.5d0) + (1.0d0 - sqrt(x))
    else
        tmp = (x ** (-0.5d0)) / 2.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] / 2.0), $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 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.5%

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

      3. *-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   if 1 < x

   1. Initial program 8.4%

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

      1. flip-- 9.0%

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

      2. add-sqr-sqrt 8.1%

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

      3. add-sqr-sqrt 12.1%

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

      4. div-sub 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. div-sub 12.1%

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

      2. +-commutative 12.1%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. +-commutative 99.5%

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

      8. metadata-eval 99.5%

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

      9. rem-square-sqrt 99.5%

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

      10. hypot-undefine 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around inf 96.5%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 96.5%

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

   9. Simplified 96.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x} \cdot 2}} \]
   10. Step-by-step derivation

       1. associate-/r* 96.5%

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

       2. pow1/2 96.5%

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

       3. pow-flip 97.0%

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

       4. metadata-eval 97.0%

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

   11. Applied egg-rr 97.0%

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

Alternative 5: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 / (1.0 + sqrt(x));
	} else {
		tmp = pow(x, -0.5) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 1.0d0 / (1.0d0 + sqrt(x))
    else
        tmp = (x ** (-0.5d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 / (1.0 + Math.sqrt(x));
	} else {
		tmp = Math.pow(x, -0.5) / 2.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(1.0 / N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] / 2.0), $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. add-sqr-sqrt 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. div-sub 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. div-sub 99.9%

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

      2. +-commutative 99.9%

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

      3. associate--l+ 99.9%

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

      4. +-inverses 99.9%

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

      5. metadata-eval 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. metadata-eval 99.9%

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

      9. rem-square-sqrt 99.9%

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

      10. hypot-undefine 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 98.2%

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

   if 1 < x

   1. Initial program 8.4%

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

      1. flip-- 9.0%

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

      2. add-sqr-sqrt 8.1%

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

      3. add-sqr-sqrt 12.1%

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

      4. div-sub 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. div-sub 12.1%

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

      2. +-commutative 12.1%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. +-commutative 99.5%

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

      8. metadata-eval 99.5%

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

      9. rem-square-sqrt 99.5%

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

      10. hypot-undefine 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around inf 96.5%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 96.5%

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

   9. Simplified 96.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x} \cdot 2}} \]
   10. Step-by-step derivation

       1. associate-/r* 96.5%

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

       2. pow1/2 96.5%

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

       3. pow-flip 97.0%

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

       4. metadata-eval 97.0%

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

   11. Applied egg-rr 97.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1}{1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.36:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.36) (- 1.0 (sqrt x)) (/ (pow x -0.5) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = pow(x, -0.5) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.36d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = (x ** (-0.5d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.36:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) / 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.36)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = Float64((x ^ -0.5) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.36)
		tmp = 1.0 - sqrt(x);
	else
		tmp = (x ^ -0.5) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.36], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.35999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.2%

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

   if 0.35999999999999999 < x

   1. Initial program 8.4%

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

      1. flip-- 9.0%

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

      2. add-sqr-sqrt 8.1%

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

      3. add-sqr-sqrt 12.1%

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

      4. div-sub 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. div-sub 12.1%

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

      2. +-commutative 12.1%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. +-commutative 99.5%

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

      8. metadata-eval 99.5%

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

      9. rem-square-sqrt 99.5%

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

      10. hypot-undefine 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around inf 96.5%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 96.5%

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

   9. Simplified 96.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x} \cdot 2}} \]
   10. Step-by-step derivation

       1. associate-/r* 96.5%

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

       2. pow1/2 96.5%

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

       3. pow-flip 97.0%

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

       4. metadata-eval 97.0%

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

   11. Applied egg-rr 97.0%

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

Alternative 7: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.36:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.36) (- 1.0 (sqrt x)) (sqrt (/ 0.25 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.36) {
		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 <= 0.36d0) 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 <= 0.36) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = Math.sqrt((0.25 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.36:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = math.sqrt((0.25 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.36)
		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 <= 0.36)
		tmp = 1.0 - sqrt(x);
	else
		tmp = sqrt((0.25 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.36], 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 < 0.35999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.2%

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

   if 0.35999999999999999 < x

   1. Initial program 8.4%

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

      1. flip-- 9.0%

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

      2. add-sqr-sqrt 8.1%

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

      3. add-sqr-sqrt 12.1%

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

      4. div-sub 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. div-sub 12.1%

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

      2. +-commutative 12.1%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. +-commutative 99.5%

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

      8. metadata-eval 99.5%

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

      9. rem-square-sqrt 99.5%

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

      10. hypot-undefine 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around inf 96.5%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 96.5%

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

   9. Simplified 96.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x} \cdot 2}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 96.2%

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

       2. sqrt-unprod 96.5%

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

       3. frac-times 95.6%

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

       4. metadata-eval 95.6%

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

       5. swap-sqr 95.6%

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

       6. add-sqr-sqrt 96.0%

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

       7. metadata-eval 96.0%

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

   11. Applied egg-rr 96.0%

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

       1. *-commutative 96.0%

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

       2. associate-/r* 96.8%

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

       3. metadata-eval 96.8%

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

   13. Simplified 96.8%

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

Alternative 8: 52.1% 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 58.5%

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

   1. flip-- 58.7%

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

   2. add-sqr-sqrt 58.3%

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

   3. add-sqr-sqrt 60.2%

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

   4. div-sub 58.5%

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

4. Applied egg-rr 58.5%

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

   1. div-sub 60.2%

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

   2. +-commutative 60.2%

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

   3. associate--l+ 99.7%

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

   4. +-inverses 99.7%

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

   5. metadata-eval 99.7%

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

   6. +-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. metadata-eval 99.7%

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

   9. rem-square-sqrt 99.7%

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

   10. hypot-undefine 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around inf 47.5%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{x}}} \]
8. Step-by-step derivation

   1. *-commutative 47.5%

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

9. Simplified 47.5%

   \[\leadsto \frac{1}{\color{blue}{\sqrt{x} \cdot 2}} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 47.4%

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

    2. sqrt-unprod 47.5%

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

    3. frac-times 47.1%

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

    4. metadata-eval 47.1%

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

    5. swap-sqr 47.1%

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

    6. add-sqr-sqrt 47.3%

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

    7. metadata-eval 47.3%

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

11. Applied egg-rr 47.3%

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

    1. *-commutative 47.3%

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

    2. associate-/r* 47.7%

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

    3. metadata-eval 47.7%

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

13. Simplified 47.7%

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

Alternative 9: 1.7% 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 Float64(-sqrt(x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in x around 0 54.4%

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

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. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :alt
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

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