sqrt sqr

Percentage Accurate: 50.6% → 100.0%

Time: 5.2s

Alternatives: 3

Speedup: 111.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{\left|x\right|}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (/ (fabs x) x)))

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 - (math.fabs(x) / x)

Julia

function code(x)
	return Float64(1.0 - Float64(abs(x) / x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

   1. cancel-sign-sub-inv 50.0%

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

   2. *-inverses 50.0%

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

   3. distribute-frac-neg2 50.0%

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

   4. sqr-neg 50.0%

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

   5. *-inverses 50.0%

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

   6. cancel-sign-sub 50.0%

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

   7. *-inverses 50.0%

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

   8. *-inverses 50.0%

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

   9. distribute-neg-frac 50.0%

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

   10. *-inverses 50.0%

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

   11. metadata-eval 50.0%

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

   12. associate-*l/ 54.5%

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

   13. neg-mul-1 54.5%

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

   14. distribute-frac-neg2 54.5%

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

   15. distribute-neg-frac 54.5%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 60.0% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -5e-310) 1.0 0.0))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 60.3%

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

      1. cancel-sign-sub-inv 60.3%

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

      2. *-inverses 60.3%

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

      3. distribute-frac-neg2 60.3%

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

      4. sqr-neg 60.3%

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

      5. *-inverses 60.3%

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

      6. cancel-sign-sub 60.3%

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

      7. *-inverses 60.3%

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

      8. *-inverses 60.3%

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

      9. distribute-neg-frac 60.3%

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

      10. *-inverses 60.3%

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

      11. metadata-eval 60.3%

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

      12. associate-*l/ 60.3%

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

      13. neg-mul-1 60.3%

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

      14. distribute-frac-neg2 60.3%

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

      15. distribute-neg-frac 60.3%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 18.8%

      \[\leadsto \color{blue}{1} \]

   if -4.999999999999985e-310 < x

   1. Initial program 40.9%

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

      1. cancel-sign-sub-inv 40.9%

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

      2. *-inverses 40.9%

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

      3. distribute-frac-neg2 40.9%

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

      4. sqr-neg 40.9%

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

      5. *-inverses 40.9%

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

      6. cancel-sign-sub 40.9%

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

      7. *-inverses 40.9%

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

      8. *-inverses 40.9%

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

      9. distribute-neg-frac 40.9%

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

      10. *-inverses 40.9%

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

      11. metadata-eval 40.9%

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

      12. associate-*l/ 49.4%

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

      13. neg-mul-1 49.4%

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

      14. distribute-frac-neg2 49.4%

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

      15. distribute-neg-frac 49.4%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{x - \left|x\right|}{x}} \]
   6. Step-by-step derivation

      1. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{x}{x} - \frac{\left|x\right|}{x}} \]

      2. *-inverses 100.0%

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

      3. rem-square-sqrt 47.0%

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

      4. fabs-sqr 47.0%

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

      5. rem-square-sqrt 100.0%

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

      6. *-inverses 100.0%

         \[\leadsto 1 - \color{blue}{1} \]

      7. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 52.3% accurate, 111.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 50.0%

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

   1. cancel-sign-sub-inv 50.0%

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

   2. *-inverses 50.0%

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

   3. distribute-frac-neg2 50.0%

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

   4. sqr-neg 50.0%

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

   5. *-inverses 50.0%

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

   6. cancel-sign-sub 50.0%

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

   7. *-inverses 50.0%

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

   8. *-inverses 50.0%

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

   9. distribute-neg-frac 50.0%

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

   10. *-inverses 50.0%

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

   11. metadata-eval 50.0%

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

   12. associate-*l/ 54.5%

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

   13. neg-mul-1 54.5%

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

   14. distribute-frac-neg2 54.5%

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

   15. distribute-neg-frac 54.5%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{\frac{x - \left|x\right|}{x}} \]
6. Step-by-step derivation

   1. div-sub 100.0%

      \[\leadsto \color{blue}{\frac{x}{x} - \frac{\left|x\right|}{x}} \]

   2. *-inverses 100.0%

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

   3. rem-square-sqrt 25.2%

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

   4. fabs-sqr 25.2%

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

   5. rem-square-sqrt 55.0%

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

   6. *-inverses 55.0%

      \[\leadsto 1 - \color{blue}{1} \]

   7. metadata-eval 55.0%

      \[\leadsto \color{blue}{0} \]

7. Simplified 55.0%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 100.0% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (< x 0.0) 2.0 0.0))

C

double code(double x) {
	double tmp;
	if (x < 0.0) {
		tmp = 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x < 0.0d0) then
        tmp = 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x < 0.0) {
		tmp = 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x < 0.0:
		tmp = 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x < 0.0)
		tmp = 2.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x < 0.0)
		tmp = 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Less[x, 0.0], 2.0, 0.0]

Reproduce

herbie shell --seed 2024144 
(FPCore (x)
  :name "sqrt sqr"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x 0) 2 0))

  (- (/ x x) (* (/ 1.0 x) (sqrt (* x x)))))