sqrt times

Percentage Accurate: 99.2% → 99.2%

Time: 3.5s

Alternatives: 5

Speedup: 0.5×

Specification

Math

\[\sqrt{x - 1} \cdot \sqrt{x} \]

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)
use fmin_fmax_functions
    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

\[\sqrt{x - 1} \cdot \sqrt{x} \]

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)
use fmin_fmax_functions
    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: 99.2% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* (sqrt (/ 1.0 (pow (- x 1.0) -1.0))) (sqrt x)))

C

double code(double x) {
	return sqrt((1.0 / pow((x - 1.0), -1.0))) * sqrt(x);
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = sqrt((1.0d0 / ((x - 1.0d0) ** (-1.0d0)))) * sqrt(x)
end function

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.pow((x - 1.0), -1.0))) * Math.sqrt(x);
}

Python

def code(x):
	return math.sqrt((1.0 / math.pow((x - 1.0), -1.0))) * math.sqrt(x)

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / (Float64(x - 1.0) ^ -1.0))) * sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / ((x - 1.0) ^ -1.0))) * sqrt(x);
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. unpow1 N/A

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

   2. remove-double-neg N/A

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

   3. pow-neg N/A

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

   4. remove-sound-/ N/A

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

   5. lower-/.f64 N/A

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

   6. remove-sound-pow N/A

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

   7. lower-pow.f64 N/A

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

   8. metadata-eval 99.2%

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

3. Applied rewrites 99.2%

   \[\leadsto \sqrt{\color{blue}{\frac{1}{{\left(x - 1\right)}^{-1}}}} \cdot \sqrt{x} \]
4. Add Preprocessing

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\sqrt{x - 1} \cdot \frac{\sqrt{x \cdot 2}}{\sqrt{2}} \]

FPCore

(FPCore (x)
  :precision binary64
  (* (sqrt (- x 1.0)) (/ (sqrt (* x 2.0)) (sqrt 2.0))))

C

double code(double x) {
	return sqrt((x - 1.0)) * (sqrt((x * 2.0)) / sqrt(2.0));
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = sqrt((x - 1.0d0)) * (sqrt((x * 2.0d0)) / sqrt(2.0d0))
end function

Java

public static double code(double x) {
	return Math.sqrt((x - 1.0)) * (Math.sqrt((x * 2.0)) / Math.sqrt(2.0));
}

Python

def code(x):
	return math.sqrt((x - 1.0)) * (math.sqrt((x * 2.0)) / math.sqrt(2.0))

Julia

function code(x)
	return Float64(sqrt(Float64(x - 1.0)) * Float64(sqrt(Float64(x * 2.0)) / sqrt(2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x - 1.0)) * (sqrt((x * 2.0)) / sqrt(2.0));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. rem-square-sqrt N/A

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

   4. sqrt-unprod N/A

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

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

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

   6. pow1/2 N/A

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

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

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

   8. sqrt-unprod N/A

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

   9. rem-square-sqrt N/A

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

   10. *-rgt-identity N/A

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

   11. *-inverses N/A

       \[\leadsto \sqrt{x - 1} \cdot \sqrt{x \cdot \color{blue}{\frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}} \]

   12. associate-*r/ N/A

       \[\leadsto \sqrt{x - 1} \cdot \sqrt{\color{blue}{\frac{x \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}} \]

   13. sqrt-div N/A

       \[\leadsto \sqrt{x - 1} \cdot \color{blue}{\frac{\sqrt{\left|x \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)\right|}}{\sqrt{\left|e^{0} + e^{\mathsf{neg}\left(0\right)}\right|}}} \]

   14. lower-/.f64 N/A

       \[\leadsto \sqrt{x - 1} \cdot \color{blue}{\frac{\sqrt{\left|x \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)\right|}}{\sqrt{\left|e^{0} + e^{\mathsf{neg}\left(0\right)}\right|}}} \]

3. Applied rewrites 99.1%

   \[\leadsto \sqrt{x - 1} \cdot \color{blue}{\frac{\sqrt{x \cdot 2}}{\sqrt{2}}} \]
4. Add Preprocessing

Alternative 3: 99.0% accurate, 2.9× speedup

Math

\[x - 0.5 \]

FPCore

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

C

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

Fortran

real(8) function code(x)
use fmin_fmax_functions
    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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. distribute-rgt-out-- N/A

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

   8. *-rgt-identity N/A

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

   9. *-rgt-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lower-sqrt.f64 N/A

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

   12. *-rgt-identity N/A

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

   13. *-rgt-identity N/A

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

   14. *-rgt-identity N/A

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

   15. distribute-rgt-out-- N/A

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

   16. lift--.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 52.8%

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

3. Applied rewrites 52.8%

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

4. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 99.0%

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

6. Applied rewrites 99.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. mult-flip-rev N/A

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

   7. sub-to-mult-rev N/A

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

   8. lower--.f64 99.0%

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

8. Applied rewrites 99.0%

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

Alternative 4: 1.1% accurate, 10.6× speedup

Math

\[-0.5 \]

FPCore

(FPCore (x)
  :precision binary64
  -0.5)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -0.5

Julia

function code(x)
	return -0.5
end

MATLAB

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

Wolfram

code[x_] := -0.5

Derivation

1. Initial program 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. distribute-rgt-out-- N/A

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

   8. *-rgt-identity N/A

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

   9. *-rgt-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lower-sqrt.f64 N/A

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

   12. *-rgt-identity N/A

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

   13. *-rgt-identity N/A

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

   14. *-rgt-identity N/A

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

   15. distribute-rgt-out-- N/A

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

   16. lift--.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 52.8%

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

3. Applied rewrites 52.8%

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

4. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 99.0%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{2} \]
8. Step-by-step derivation

9. Applied rewrites 1.1%

   \[\leadsto -0.5 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025326 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))