Main:bigenough3 from C

Percentage Accurate: 53.6% → 99.7%

Time: 8.4s

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: 99.7% 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 54.0%

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

   1. flip-- 54.0%

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

   2. div-inv 54.0%

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

   3. add-sqr-sqrt 54.3%

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

   4. add-sqr-sqrt 54.9%

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

   5. associate--l+ 54.9%

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

4. Applied egg-rr 54.9%

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

   1. associate-*r/ 54.9%

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

   2. *-rgt-identity 54.9%

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

   3. +-commutative 54.9%

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

   4. associate-+l- 99.7%

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

   5. +-inverses 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

   8. +-commutative 99.7%

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

6. Simplified 99.7%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.5%

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

      1. flip-- 4.4%

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

      2. div-inv 4.4%

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

      3. add-sqr-sqrt 4.5%

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

      4. add-sqr-sqrt 5.4%

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

      5. associate--l+ 5.4%

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

   4. Applied egg-rr 5.4%

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

      1. associate-*r/ 5.4%

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

      2. *-rgt-identity 5.4%

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

      3. +-commutative 5.4%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 99.3%

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

      1. *-commutative 99.3%

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

   9. Simplified 99.3%

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

       1. inv-pow 99.3%

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

       2. unpow-prod-down 99.3%

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

       3. inv-pow 99.3%

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

       4. pow1/2 99.3%

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

       5. pow-flip 99.7%

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

       6. metadata-eval 99.7%

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

       7. metadata-eval 99.7%

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

   11. Applied egg-rr 99.7%

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

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

   1. Initial program 99.2%

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

4. Final simplification 99.4%

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

Alternative 3: 98.8% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.3)
   (- (+ 1.0 (* x (+ 0.5 (* x (- (* x 0.0625) 0.125))))) (sqrt x))
   (* (pow x -0.5) 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, -0.5) * 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 ** (-0.5d0)) * 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, -0.5) * 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, -0.5) * 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 ^ -0.5) * 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 ^ -0.5) * 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[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000004

   1. Initial program 99.9%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 1.30000000000000004 < x

   1. Initial program 7.4%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 8.0%

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

      4. add-sqr-sqrt 9.1%

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

      5. associate--l+ 9.1%

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

   4. Applied egg-rr 9.1%

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

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 97.0%

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

      1. *-commutative 97.0%

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

   9. Simplified 97.0%

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

       1. inv-pow 97.0%

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

       2. unpow-prod-down 97.0%

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

       3. inv-pow 97.0%

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

       4. pow1/2 97.0%

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

       5. pow-flip 97.4%

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

       6. metadata-eval 97.4%

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

       7. metadata-eval 97.4%

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

   11. Applied egg-rr 97.4%

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

4. Final simplification 98.5%

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

Alternative 4: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.2)
   (+ 1.0 (- (* x (+ 0.5 (* x -0.125))) (sqrt x)))
   (* (pow x -0.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x));
	} else {
		tmp = pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.2d0) then
        tmp = 1.0d0 + ((x * (0.5d0 + (x * (-0.125d0)))) - sqrt(x))
    else
        tmp = (x ** (-0.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - Math.sqrt(x));
	} else {
		tmp = Math.pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.2:
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - math.sqrt(x))
	else:
		tmp = math.pow(x, -0.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.2)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) - sqrt(x)));
	else
		tmp = Float64((x ^ -0.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.2)
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x));
	else
		tmp = (x ^ -0.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.2], N[(1.0 + N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 99.9%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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+ 99.5%

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

      2. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   if 1.19999999999999996 < x

   1. Initial program 7.4%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 8.0%

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

      4. add-sqr-sqrt 9.1%

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

      5. associate--l+ 9.1%

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

   4. Applied egg-rr 9.1%

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

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 97.0%

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

      1. *-commutative 97.0%

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

   9. Simplified 97.0%

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

       1. inv-pow 97.0%

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

       2. unpow-prod-down 97.0%

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

       3. inv-pow 97.0%

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

       4. pow1/2 97.0%

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

       5. pow-flip 97.4%

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

       6. metadata-eval 97.4%

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

       7. metadata-eval 97.4%

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

   11. Applied egg-rr 97.4%

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

Alternative 5: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. associate--l+ 99.2%

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

      2. *-commutative 99.2%

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

   5. Simplified 99.2%

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

      1. +-commutative 99.2%

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

      2. associate-+l- 99.2%

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

   7. Applied egg-rr 99.2%

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

   if 1 < x

   1. Initial program 7.4%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 8.0%

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

      4. add-sqr-sqrt 9.1%

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

      5. associate--l+ 9.1%

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

   4. Applied egg-rr 9.1%

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

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 97.0%

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

      1. *-commutative 97.0%

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

   9. Simplified 97.0%

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

       1. inv-pow 97.0%

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

       2. unpow-prod-down 97.0%

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

       3. inv-pow 97.0%

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

       4. pow1/2 97.0%

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

       5. pow-flip 97.4%

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

       6. metadata-eval 97.4%

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

       7. metadata-eval 97.4%

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

   11. Applied egg-rr 97.4%

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

4. Final simplification 98.3%

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

Alternative 6: 98.6% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (+ 1.0 (- (* x 0.5) (sqrt x))) (* (pow x -0.5) 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, -0.5) * 0.5;
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. associate--l+ 99.2%

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

      2. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 1 < x

   1. Initial program 7.4%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 8.0%

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

      4. add-sqr-sqrt 9.1%

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

      5. associate--l+ 9.1%

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

   4. Applied egg-rr 9.1%

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

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 97.0%

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

      1. *-commutative 97.0%

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

   9. Simplified 97.0%

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

       1. inv-pow 97.0%

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

       2. unpow-prod-down 97.0%

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

       3. inv-pow 97.0%

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

       4. pow1/2 97.0%

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

       5. pow-flip 97.4%

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

       6. metadata-eval 97.4%

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

       7. metadata-eval 97.4%

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

   11. Applied egg-rr 97.4%

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

Alternative 7: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) * 0.5;
	}
	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) * 0.5
	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) * 0.5);
	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) * 0.5;
	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] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.35999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 98.4%

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

   if 0.35999999999999999 < x

   1. Initial program 7.4%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 8.0%

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

      4. add-sqr-sqrt 9.1%

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

      5. associate--l+ 9.1%

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

   4. Applied egg-rr 9.1%

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

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 97.0%

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

      1. *-commutative 97.0%

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

   9. Simplified 97.0%

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

       1. inv-pow 97.0%

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

       2. unpow-prod-down 97.0%

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

       3. inv-pow 97.0%

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

       4. pow1/2 97.0%

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

       5. pow-flip 97.4%

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

       6. metadata-eval 97.4%

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

       7. metadata-eval 97.4%

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

   11. Applied egg-rr 97.4%

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

Alternative 8: 52.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.56:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.56) (- 1.0 (sqrt x)) 1.0))

C

double code(double x) {
	double tmp;
	if (x <= 0.56) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.56d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.56) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.56:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.56)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.56)
		tmp = 1.0 - sqrt(x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.56], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.56000000000000005

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 98.4%

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

   if 0.56000000000000005 < x

   1. Initial program 7.4%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 8.0%

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

      4. add-sqr-sqrt 9.1%

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

      5. associate--l+ 9.1%

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

   4. Applied egg-rr 9.1%

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

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   6. Simplified 99.6%

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

      1. add-exp-log 92.5%

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

      2. log-rec 92.5%

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

   8. Applied egg-rr 92.5%

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

      1. exp-neg 92.5%

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

      2. add-exp-log 99.6%

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

      3. add-sqr-sqrt 99.2%

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

      4. associate-/r* 99.1%

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

   10. Applied egg-rr 6.9%

       \[\leadsto \color{blue}{\frac{\mathsf{hypot}\left({\left(x + 1\right)}^{0.25}, {x}^{0.25}\right)}{\mathsf{hypot}\left({\left(x + 1\right)}^{0.25}, {x}^{0.25}\right)}} \]
   11. Step-by-step derivation

       1. *-inverses 6.9%

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

   12. Simplified 6.9%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.6% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 54.0%

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

   1. flip-- 54.0%

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

   2. div-inv 54.0%

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

   3. add-sqr-sqrt 54.3%

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

   4. add-sqr-sqrt 54.9%

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

   5. associate--l+ 54.9%

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

4. Applied egg-rr 54.9%

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

   1. associate-*r/ 54.9%

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

   2. *-rgt-identity 54.9%

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

   3. +-commutative 54.9%

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

   4. associate-+l- 99.7%

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

   5. +-inverses 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

   8. +-commutative 99.7%

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

6. Simplified 99.7%

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

   1. add-exp-log 96.2%

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

   2. log-rec 96.2%

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

8. Applied egg-rr 96.2%

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

   1. exp-neg 96.2%

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

   2. add-exp-log 99.7%

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

   3. add-sqr-sqrt 99.5%

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

   4. associate-/r* 99.5%

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

10. Applied egg-rr 51.2%

    \[\leadsto \color{blue}{\frac{\mathsf{hypot}\left({\left(x + 1\right)}^{0.25}, {x}^{0.25}\right)}{\mathsf{hypot}\left({\left(x + 1\right)}^{0.25}, {x}^{0.25}\right)}} \]
11. Step-by-step derivation

    1. *-inverses 51.2%

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

12. Simplified 51.2%

    \[\leadsto \color{blue}{1} \]
13. Add Preprocessing

Developer Target 1: 99.7% 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 2024133 
(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)))