sqrt times

Percentage Accurate: 99.2% → 100.0%

Time: 3.4s

Alternatives: 4

Speedup: 1.2×

Specification

Math

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

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: 99.2% accurate, 1.0× speedup

Math

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

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: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+15}:\\ \;\;\;\;\sqrt{\left(x - 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1e+15) (sqrt (* (- x 1.0) x)) x))

C

double code(double x) {
	double tmp;
	if (x <= 1e+15) {
		tmp = sqrt(((x - 1.0) * x));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d+15) then
        tmp = sqrt(((x - 1.0d0) * x))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e+15) {
		tmp = Math.sqrt(((x - 1.0) * x));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e+15:
		tmp = math.sqrt(((x - 1.0) * x))
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+15)
		tmp = sqrt(Float64(Float64(x - 1.0) * x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e+15)
		tmp = sqrt(((x - 1.0) * x));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e+15], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < 1e15

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|x\right|} - 1} \cdot \sqrt{x} \]

      7. rem-sqrt-square-rev N/A

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

      8. pow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. rem-exp-log N/A

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

      13. lower-expm1.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\log x\right)}} \cdot \sqrt{x} \]

      14. lower-log.f64 99.2

          \[\leadsto \sqrt{\mathsf{expm1}\left(\color{blue}{\log x}\right)} \cdot \sqrt{x} \]

   4. Applied rewrites 99.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\log x\right)}} \cdot \sqrt{x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\mathsf{expm1}\left(\log x\right)} \cdot \sqrt{x}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\mathsf{expm1}\left(\log x\right)}} \cdot \sqrt{x} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\mathsf{expm1}\left(\log x\right)} \cdot \color{blue}{\sqrt{x}} \]

      4. sqrt-unprod N/A

         \[\leadsto \color{blue}{\sqrt{\mathsf{expm1}\left(\log x\right) \cdot x}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\mathsf{expm1}\left(\log x\right) \cdot x}} \]

      6. lower-*.f64 99.1

         \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\log x\right) \cdot x}} \]

      7. lift-expm1.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lower--.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if 1e15 < x

   1. Initial program 99.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

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]

Derivation

1. Initial program 99.2%

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

Alternative 3: 99.1% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return x - 0.5;
}

Python

def code(x):
	return x - 0.5

Julia

function code(x)
	return Float64(x - 0.5)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{x}, x, x\right)} \]

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{2}}}{x}, x, x\right) \]

   9. lower-/.f64 98.3

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-0.5}{x}}, x, x\right) \]

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5}{x}, x, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 98.3%

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

Alternative 4: 98.0% accurate, 29.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity 97.9

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

5. Applied rewrites 97.9%

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

Reproduce

herbie shell --seed 2024343 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))