sqrt times

Percentage Accurate: 99.2% → 99.5%

Time: 4.3s

Alternatives: 4

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: 99.5% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

2. Taylor expanded in x around inf 99.9%

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

4. Simplified 99.9%

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

   1. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. frac-2neg 99.9%

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

   4. metadata-eval 99.9%

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

   5. associate-*l/ 99.9%

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

   6. *-un-lft-identity 99.9%

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

6. Applied egg-rr 99.9%

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

   1. +-commutative 99.9%

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

8. Simplified 99.9%

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

9. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{\left(-\left(0.125 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)} + \left(x + -0.5\right) \]
10. Step-by-step derivation

    1. distribute-neg-in 99.9%

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

    2. unsub-neg 99.9%

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

    3. associate-*r/ 99.9%

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

    4. metadata-eval 99.9%

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

    5. distribute-neg-frac 99.9%

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

    6. metadata-eval 99.9%

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

    7. associate-*r/ 99.9%

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

    8. metadata-eval 99.9%

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

    9. unpow2 99.9%

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

    10. associate-/r* 99.9%

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

    11. div-sub 99.9%

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

11. Simplified 99.9%

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

12. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

2. Taylor expanded in x around inf 99.8%

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

   1. sub-neg 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. distribute-neg-in 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. distribute-neg-frac 99.8%

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

   7. metadata-eval 99.8%

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

   8. rem-square-sqrt 0.0%

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

   9. unpow2 0.0%

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

   10. sub-neg 0.0%

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

   11. associate-+l- 0.0%

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

   12. associate-*r/ 0.0%

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

   13. unpow2 0.0%

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

   14. rem-square-sqrt 99.8%

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

   15. metadata-eval 99.8%

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

4. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 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 99.1%

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

2. Taylor expanded in x around inf 99.4%

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

3. Final simplification 99.4%

   \[\leadsto x - 0.5 \]

Alternative 4: 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 99.1%

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

2. Taylor expanded in x around inf 98.3%

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

3. Final simplification 98.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023332 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))