sqrt times

Percentage Accurate: 99.2% → 99.6%

Time: 6.4s

Alternatives: 5

Speedup: 29.3×

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: 99.6% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ x - \left(\frac{0.125 + \frac{0.0625}{x}}{x} + 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- x (+ (/ (+ 0.125 (/ 0.0625 x)) x) 0.5)))

C

double code(double x) {
	return x - (((0.125 + (0.0625 / x)) / x) + 0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - (((0.125d0 + (0.0625d0 / x)) / x) + 0.5d0)
end function

Java

public static double code(double x) {
	return x - (((0.125 + (0.0625 / x)) / x) + 0.5);
}

Python

def code(x):
	return x - (((0.125 + (0.0625 / x)) / x) + 0.5)

Julia

function code(x)
	return Float64(x - Float64(Float64(Float64(0.125 + Float64(0.0625 / x)) / x) + 0.5))
end

MATLAB

function tmp = code(x)
	tmp = x - (((0.125 + (0.0625 / x)) / x) + 0.5);
end

Wolfram

code[x_] := N[(x - N[(N[(N[(0.125 + N[(0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-0.125 \cdot {\left(\sqrt{-1}\right)}^{2} + -0.0625 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x}}{x} + 0.5 \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{x - \left(0.5 + \frac{0.125 + \frac{0.0625}{x}}{x}\right) \cdot 1} \]
6. Step-by-step derivation

   1. *-rgt-identity 99.9%

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

   2. +-commutative 99.9%

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

7. Applied egg-rr 99.9%

   \[\leadsto x - \color{blue}{\left(\frac{0.125 + \frac{0.0625}{x}}{x} + 0.5\right)} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ x - \left(0.5 + \frac{0.125}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- x (+ 0.5 (/ 0.125 x))))

C

double code(double x) {
	return x - (0.5 + (0.125 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - (0.5d0 + (0.125d0 / x))
end function

Java

public static double code(double x) {
	return x - (0.5 + (0.125 / x));
}

Python

def code(x):
	return x - (0.5 + (0.125 / x))

Julia

function code(x)
	return Float64(x - Float64(0.5 + Float64(0.125 / x)))
end

MATLAB

function tmp = code(x)
	tmp = x - (0.5 + (0.125 / x));
end

Wolfram

code[x_] := N[(x - N[(0.5 + N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf 99.9%

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

5. Simplified 99.9%

   \[\leadsto \color{blue}{x - \left(0.5 + \frac{0.125}{x}\right) \cdot 1} \]
6. Step-by-step derivation

   1. *-rgt-identity 99.9%

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

   2. +-commutative 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

   \[\leadsto x - \left(0.5 + \frac{0.125}{x}\right) \]
9. Add Preprocessing

Alternative 3: 99.1% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ -1 + \left(x + 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ -1.0 (+ x 0.5)))

C

double code(double x) {
	return -1.0 + (x + 0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) + (x + 0.5d0)
end function

Java

public static double code(double x) {
	return -1.0 + (x + 0.5);
}

Python

def code(x):
	return -1.0 + (x + 0.5)

Julia

function code(x)
	return Float64(-1.0 + Float64(x + 0.5))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 + (x + 0.5);
end

Wolfram

code[x_] := N[(-1.0 + N[(x + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. *-lft-identity 99.8%

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

   4. associate-*r/ 99.8%

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

   5. metadata-eval 99.8%

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

   6. distribute-neg-frac2 99.8%

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

   7. neg-mul-1 99.8%

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

   8. associate-*l/ 99.8%

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

   9. neg-mul-1 99.8%

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

   10. distribute-neg-frac2 99.8%

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

   11. associate-/l* 99.8%

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

   12. *-rgt-identity 99.8%

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

   13. associate-*r/ 99.8%

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

   14. rgt-mult-inverse 99.8%

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

   15. metadata-eval 99.8%

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

   16. metadata-eval 99.8%

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

5. Simplified 99.8%

   \[\leadsto \color{blue}{x + -0.5} \]
6. Step-by-step derivation

   1. expm1-log1p-u 90.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + -0.5\right)\right)} \]

   2. expm1-undefine 90.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + -0.5\right)} - 1} \]

7. Applied egg-rr 90.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + -0.5\right)} - 1} \]
8. Step-by-step derivation

   1. sub-neg 90.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + -0.5\right)} + \left(-1\right)} \]

   2. log1p-undefine 90.8%

      \[\leadsto e^{\color{blue}{\log \left(1 + \left(x + -0.5\right)\right)}} + \left(-1\right) \]

   3. rem-exp-log 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+r+ 99.8%

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

   6. metadata-eval 99.8%

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

   7. +-commutative 99.8%

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

   8. metadata-eval 99.8%

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

   9. +-commutative 99.8%

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

9. Simplified 99.8%

   \[\leadsto \color{blue}{-1 + \left(x + 0.5\right)} \]
10. Add Preprocessing

Alternative 4: 99.1% accurate, 68.3× 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 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. *-lft-identity 99.8%

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

   4. associate-*r/ 99.8%

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

   5. metadata-eval 99.8%

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

   6. distribute-neg-frac2 99.8%

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

   7. neg-mul-1 99.8%

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

   8. associate-*l/ 99.8%

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

   9. neg-mul-1 99.8%

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

   10. distribute-neg-frac2 99.8%

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

   11. associate-/l* 99.8%

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

   12. *-rgt-identity 99.8%

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

   13. associate-*r/ 99.8%

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

   14. rgt-mult-inverse 99.8%

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

   15. metadata-eval 99.8%

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

   16. metadata-eval 99.8%

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

5. Simplified 99.8%

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

Alternative 5: 98.0% accurate, 205.0× 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 98.9%

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

Reproduce

herbie shell --seed 2024157 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))