sqrt times

Percentage Accurate: 99.2% → 99.6%

Time: 6.8s

Alternatives: 6

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, 15.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + 0.5\right) + \frac{-0.125 + \frac{-0.0625}{x}}{x}\right) + -1 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (+ (+ x 0.5) (/ (+ -0.125 (/ -0.0625 x)) x)) -1.0))

C

double code(double x) {
	return ((x + 0.5) + ((-0.125 + (-0.0625 / x)) / x)) + -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + 0.5d0) + (((-0.125d0) + ((-0.0625d0) / x)) / x)) + (-1.0d0)
end function

Java

public static double code(double x) {
	return ((x + 0.5) + ((-0.125 + (-0.0625 / x)) / x)) + -1.0;
}

Python

def code(x):
	return ((x + 0.5) + ((-0.125 + (-0.0625 / x)) / x)) + -1.0

Julia

function code(x)
	return Float64(Float64(Float64(x + 0.5) + Float64(Float64(-0.125 + Float64(-0.0625 / x)) / x)) + -1.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x + 0.5) + ((-0.125 + (-0.0625 / x)) / x)) + -1.0;
end

Wolfram

code[x_] := N[(N[(N[(x + 0.5), $MachinePrecision] + N[(N[(-0.125 + N[(-0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + -1.0), $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 99.4%

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

   1. expm1-log1p-u 90.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x - \left(0.5 + \frac{0.125 + \frac{0.0625}{x}}{x}\right) \cdot 1\right)\right)} \]

   2. *-rgt-identity 90.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(x - \color{blue}{\left(0.5 + \frac{0.125 + \frac{0.0625}{x}}{x}\right)}\right)\right) \]

   3. associate--r+ 90.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(x - 0.5\right) - \frac{0.125 + \frac{0.0625}{x}}{x}}\right)\right) \]

7. Applied egg-rr 90.4%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x - 0.5\right) - \frac{0.125 + \frac{0.0625}{x}}{x}\right)\right)} \]
8. Step-by-step derivation

   1. expm1-undefine 90.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x - 0.5\right) - \frac{0.125 + \frac{0.0625}{x}}{x}\right)} - 1} \]

   2. sub-neg 90.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x - 0.5\right) - \frac{0.125 + \frac{0.0625}{x}}{x}\right)} + \left(-1\right)} \]

9. Simplified 99.4%

   \[\leadsto \color{blue}{\left(\left(x + 0.5\right) + \frac{-0.125 + \frac{-0.0625}{x}}{x}\right) + -1} \]
10. Add Preprocessing

Alternative 2: 99.6% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x + -0.5), $MachinePrecision] - N[(N[(0.125 + N[(0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $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 99.4%

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

   1. sub-neg 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. distribute-neg-in 99.4%

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

   4. metadata-eval 99.4%

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

7. Applied egg-rr 99.4%

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

   1. associate-+r+ 99.4%

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

   2. sub-neg 99.4%

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

9. Simplified 99.4%

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

Alternative 3: 99.4% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ -1 + \left(\left(x + 0.5\right) - \frac{0.125}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ -1.0 (- (+ x 0.5) (/ 0.125 x))))

C

double code(double x) {
	return -1.0 + ((x + 0.5) - (0.125 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) + ((x + 0.5d0) - (0.125d0 / x))
end function

Java

public static double code(double x) {
	return -1.0 + ((x + 0.5) - (0.125 / x));
}

Python

def code(x):
	return -1.0 + ((x + 0.5) - (0.125 / x))

Julia

function code(x)
	return Float64(-1.0 + Float64(Float64(x + 0.5) - Float64(0.125 / x)))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 + ((x + 0.5) - (0.125 / x));
end

Wolfram

code[x_] := N[(-1.0 + N[(N[(x + 0.5), $MachinePrecision] - 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 99.2%

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

5. Simplified 99.2%

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

   1. expm1-log1p-u 90.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x - \left(0.5 + \frac{0.125}{x}\right) \cdot 1\right)\right)} \]

   2. *-rgt-identity 90.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(x - \color{blue}{\left(0.5 + \frac{0.125}{x}\right)}\right)\right) \]

7. Applied egg-rr 90.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x - \left(0.5 + \frac{0.125}{x}\right)\right)\right)} \]
8. Step-by-step derivation

   1. expm1-undefine 90.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x - \left(0.5 + \frac{0.125}{x}\right)\right)} - 1} \]

   2. sub-neg 90.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x - \left(0.5 + \frac{0.125}{x}\right)\right)} + \left(-1\right)} \]

   3. log1p-undefine 90.2%

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

   4. rem-exp-log 99.2%

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

   5. associate--r+ 99.2%

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

   6. sub-neg 99.2%

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

   7. metadata-eval 99.2%

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

   8. associate-+r- 99.2%

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

   9. +-commutative 99.2%

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

   10. associate-+r+ 99.2%

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

   11. metadata-eval 99.2%

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

   12. +-commutative 99.2%

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

   13. metadata-eval 99.2%

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

9. Simplified 99.2%

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

10. Final simplification 99.2%

    \[\leadsto -1 + \left(\left(x + 0.5\right) - \frac{0.125}{x}\right) \]
11. Add Preprocessing

Alternative 4: 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 99.2%

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

5. Simplified 99.2%

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

   1. *-rgt-identity 99.2%

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

   2. +-commutative 99.2%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 5: 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 98.8%

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

   1. sub-neg 98.8%

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

   2. distribute-rgt-in 98.8%

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

   3. *-lft-identity 98.8%

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

   4. associate-*r/ 98.8%

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

   5. metadata-eval 98.8%

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

   6. distribute-neg-frac2 98.8%

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

   7. neg-mul-1 98.8%

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

   8. associate-*l/ 98.8%

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

   9. neg-mul-1 98.8%

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

   10. distribute-neg-frac2 98.8%

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

   11. associate-/l* 98.8%

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

   12. *-rgt-identity 98.8%

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

   13. associate-*r/ 98.8%

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

   14. rgt-mult-inverse 98.8%

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

   15. metadata-eval 98.8%

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

   16. metadata-eval 98.8%

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

5. Simplified 98.8%

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

Alternative 6: 98.1% 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 97.5%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))