Main:bigenough3 from C

Percentage Accurate: 54.9% → 99.7%

Time: 10.2s

Alternatives: 7

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× 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]

Derivation

1. Initial program 54.0%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-+.f64 N/A

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

   11. associate--l+ N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. lower-+.f64 N/A

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

   15. metadata-eval N/A

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

   16. *-rgt-identity N/A

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

   17. lower--.f64 N/A

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

   18. lower-+.f64 55.2

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

4. Applied rewrites 55.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.8%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 4e-5) (* 0.5 (sqrt (/ 1.0 x))) t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 4e-5) {
		tmp = 0.5 * sqrt((1.0 / x));
	} 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((x + 1.0d0)) - sqrt(x)
    if (t_0 <= 4d-5) then
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 4e-5) {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 4e-5:
		tmp = 0.5 * math.sqrt((1.0 / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 4e-5)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 4e-5)
		tmp = 0.5 * sqrt((1.0 / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-5], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $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)) < 4.00000000000000033e-5

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 4.00000000000000033e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.4%

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

Developer Target 1: 99.7% accurate, 0.7× 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 2024219 
(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)))