sqrt times

Percentage Accurate: 99.2% → 99.8%

Time: 5.9s

Alternatives: 8

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: 99.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \frac{-0.5 + \frac{-0.125 + \frac{-0.0625 - \frac{0.0390625 + \frac{0.02734375}{x}}{x}}{x}}{x}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (/
    (+
     -0.5
     (/ (+ -0.125 (/ (- -0.0625 (/ (+ 0.0390625 (/ 0.02734375 x)) x)) x)) x))
    x))))

C

double code(double x) {
	return x * (1.0 + ((-0.5 + ((-0.125 + ((-0.0625 - ((0.0390625 + (0.02734375 / x)) / x)) / x)) / x)) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (((-0.5d0) + (((-0.125d0) + (((-0.0625d0) - ((0.0390625d0 + (0.02734375d0 / x)) / x)) / x)) / x)) / x))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((-0.5 + ((-0.125 + ((-0.0625 - ((0.0390625 + (0.02734375 / x)) / x)) / x)) / x)) / x));
}

Python

def code(x):
	return x * (1.0 + ((-0.5 + ((-0.125 + ((-0.0625 - ((0.0390625 + (0.02734375 / x)) / x)) / x)) / x)) / x))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(-0.5 + Float64(Float64(-0.125 + Float64(Float64(-0.0625 - Float64(Float64(0.0390625 + Float64(0.02734375 / x)) / x)) / x)) / x)) / x)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((-0.5 + ((-0.125 + ((-0.0625 - ((0.0390625 + (0.02734375 / x)) / x)) / x)) / x)) / x));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(-0.5 + N[(N[(-0.125 + N[(N[(-0.0625 - N[(N[(0.0390625 + N[(0.02734375 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.0390625 \cdot {\left(\sqrt{-1}\right)}^{2} + -0.02734375 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x}}{x} + 0.0625 \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + -0.125 \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 0.5 \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-0.5 + \frac{-0.125 + \frac{-0.0625 - \frac{0.0390625 + \frac{0.02734375}{x}}{x}}{x}}{x}}{x}\right)} \]

6. Final simplification 99.6%

   \[\leadsto x \cdot \left(1 + \frac{-0.5 + \frac{-0.125 + \frac{-0.0625 - \frac{0.0390625 + \frac{0.02734375}{x}}{x}}{x}}{x}}{x}\right) \]
7. Add Preprocessing

Alternative 2: 99.7% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(-0.5 + \frac{-0.125 + \frac{-0.0625 + \frac{-0.0390625}{x}}{x}}{x}\right) \cdot \frac{1}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (*
    (+ -0.5 (/ (+ -0.125 (/ (+ -0.0625 (/ -0.0390625 x)) x)) x))
    (/ 1.0 x)))))

C

double code(double x) {
	return x * (1.0 + ((-0.5 + ((-0.125 + ((-0.0625 + (-0.0390625 / x)) / x)) / x)) * (1.0 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (((-0.5d0) + (((-0.125d0) + (((-0.0625d0) + ((-0.0390625d0) / x)) / x)) / x)) * (1.0d0 / x)))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((-0.5 + ((-0.125 + ((-0.0625 + (-0.0390625 / x)) / x)) / x)) * (1.0 / x)));
}

Python

def code(x):
	return x * (1.0 + ((-0.5 + ((-0.125 + ((-0.0625 + (-0.0390625 / x)) / x)) / x)) * (1.0 / x)))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(-0.5 + Float64(Float64(-0.125 + Float64(Float64(-0.0625 + Float64(-0.0390625 / x)) / x)) / x)) * Float64(1.0 / x))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((-0.5 + ((-0.125 + ((-0.0625 + (-0.0390625 / x)) / x)) / x)) * (1.0 / x)));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(-0.5 + N[(N[(-0.125 + N[(N[(-0.0625 + N[(-0.0390625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.0390625 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x} + 0.0625 \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + -0.125 \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 0.5 \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-0.5 + \frac{-0.125 + \frac{-0.0625 + \frac{-0.0390625}{x}}{x}}{x}}{x}\right)} \]
6. Step-by-step derivation

   1. div-inv 99.6%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left(-0.5 + \frac{-0.125 + \frac{-0.0625 + \frac{-0.0390625}{x}}{x}}{x}\right) \cdot \frac{1}{x}}\right) \]

7. Applied egg-rr 99.6%

   \[\leadsto x \cdot \left(1 + \color{blue}{\left(-0.5 + \frac{-0.125 + \frac{-0.0625 + \frac{-0.0390625}{x}}{x}}{x}\right) \cdot \frac{1}{x}}\right) \]

8. Final simplification 99.6%

   \[\leadsto x \cdot \left(1 + \left(-0.5 + \frac{-0.125 + \frac{-0.0625 + \frac{-0.0390625}{x}}{x}}{x}\right) \cdot \frac{1}{x}\right) \]
9. Add Preprocessing

Alternative 3: 99.7% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \frac{-0.5 + \frac{-0.125 + \frac{-0.0625 + \frac{-0.0390625}{x}}{x}}{x}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+ 1.0 (/ (+ -0.5 (/ (+ -0.125 (/ (+ -0.0625 (/ -0.0390625 x)) x)) x)) x))))

C

double code(double x) {
	return x * (1.0 + ((-0.5 + ((-0.125 + ((-0.0625 + (-0.0390625 / x)) / x)) / x)) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (((-0.5d0) + (((-0.125d0) + (((-0.0625d0) + ((-0.0390625d0) / x)) / x)) / x)) / x))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((-0.5 + ((-0.125 + ((-0.0625 + (-0.0390625 / x)) / x)) / x)) / x));
}

Python

def code(x):
	return x * (1.0 + ((-0.5 + ((-0.125 + ((-0.0625 + (-0.0390625 / x)) / x)) / x)) / x))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(-0.5 + Float64(Float64(-0.125 + Float64(Float64(-0.0625 + Float64(-0.0390625 / x)) / x)) / x)) / x)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((-0.5 + ((-0.125 + ((-0.0625 + (-0.0390625 / x)) / x)) / x)) / x));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(-0.5 + N[(N[(-0.125 + N[(N[(-0.0625 + N[(-0.0390625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.0390625 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x} + 0.0625 \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + -0.125 \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 0.5 \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-0.5 + \frac{-0.125 + \frac{-0.0625 + \frac{-0.0390625}{x}}{x}}{x}}{x}\right)} \]

6. Final simplification 99.6%

   \[\leadsto x \cdot \left(1 + \frac{-0.5 + \frac{-0.125 + \frac{-0.0625 + \frac{-0.0390625}{x}}{x}}{x}}{x}\right) \]
7. Add Preprocessing

Alternative 4: 99.6% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.0390625 \cdot {\left(\sqrt{-1}\right)}^{2} + -0.02734375 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x}}{x} + 0.0625 \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + -0.125 \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 0.5 \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-0.5 + \frac{-0.125 + \frac{-0.0625 - \frac{0.0390625 + \frac{0.02734375}{x}}{x}}{x}}{x}}{x}\right)} \]

6. Taylor expanded in x around inf 99.5%

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

   1. sub-neg 99.5%

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

   2. associate-*r/ 99.5%

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

   3. neg-mul-1 99.5%

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

   4. distribute-neg-in 99.5%

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

   5. metadata-eval 99.5%

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

   6. unsub-neg 99.5%

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

   7. associate-*r/ 99.5%

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

   8. metadata-eval 99.5%

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

   9. metadata-eval 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 5: 99.6% accurate, 18.6× speedup

Math

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

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(x + Float64(-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[(x + N[(-0.5 + N[(N[(-0.125 + N[(-0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. add-log-exp 6.4%

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

   2. *-un-lft-identity 6.4%

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

   3. log-prod 6.4%

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

   4. metadata-eval 6.4%

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

   5. add-log-exp 99.3%

      \[\leadsto 0 + \color{blue}{\sqrt{x - 1} \cdot \sqrt{x}} \]

   6. *-commutative 99.3%

      \[\leadsto 0 + \color{blue}{\sqrt{x} \cdot \sqrt{x - 1}} \]

   7. sqrt-unprod 51.9%

      \[\leadsto 0 + \color{blue}{\sqrt{x \cdot \left(x - 1\right)}} \]

   8. sub-neg 51.9%

      \[\leadsto 0 + \sqrt{x \cdot \color{blue}{\left(x + \left(-1\right)\right)}} \]

   9. metadata-eval 51.9%

      \[\leadsto 0 + \sqrt{x \cdot \left(x + \color{blue}{-1}\right)} \]

4. Applied egg-rr 51.9%

   \[\leadsto \color{blue}{0 + \sqrt{x \cdot \left(x + -1\right)}} \]
5. Step-by-step derivation

   1. +-lft-identity 51.9%

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

6. Simplified 51.9%

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

7. Taylor expanded in x around inf 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 6: 99.1% accurate, 29.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x (- 1.0 (/ 0.5 x))))

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * (1.0 - (0.5 / x))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf 99.3%

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

   1. associate-*r/ 99.3%

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

   2. metadata-eval 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

   \[\leadsto x \cdot \left(1 - \frac{0.5}{x}\right) \]
7. Add Preprocessing

Alternative 7: 99.5% 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 99.3%

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

   1. add-log-exp 6.4%

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

   2. *-un-lft-identity 6.4%

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

   3. log-prod 6.4%

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

   4. metadata-eval 6.4%

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

   5. add-log-exp 99.3%

      \[\leadsto 0 + \color{blue}{\sqrt{x - 1} \cdot \sqrt{x}} \]

   6. *-commutative 99.3%

      \[\leadsto 0 + \color{blue}{\sqrt{x} \cdot \sqrt{x - 1}} \]

   7. sqrt-unprod 51.9%

      \[\leadsto 0 + \color{blue}{\sqrt{x \cdot \left(x - 1\right)}} \]

   8. sub-neg 51.9%

      \[\leadsto 0 + \sqrt{x \cdot \color{blue}{\left(x + \left(-1\right)\right)}} \]

   9. metadata-eval 51.9%

      \[\leadsto 0 + \sqrt{x \cdot \left(x + \color{blue}{-1}\right)} \]

4. Applied egg-rr 51.9%

   \[\leadsto \color{blue}{0 + \sqrt{x \cdot \left(x + -1\right)}} \]
5. Step-by-step derivation

   1. +-lft-identity 51.9%

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

6. Simplified 51.9%

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

7. Taylor expanded in x around inf 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in x around inf 99.5%

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

11. Final simplification 99.5%

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

Alternative 8: 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 99.3%

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

3. Taylor expanded in x around inf 98.6%

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

4. Final simplification 98.6%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))