Data.Random.Dice:roll from dice-0.1

Percentage Accurate: 100.0% → 100.0%

Time: 2.6s

Alternatives: 4

Speedup: 1.0×

• 24.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot x - 1 \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto x \cdot x + -1 \]
4. Add Preprocessing

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 5e-15) -1.0 (* x x)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-15) {
		tmp = -1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 5d-15) then
        tmp = -1.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 5e-15) {
		tmp = -1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 5e-15:
		tmp = -1.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-15)
		tmp = -1.0;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 5e-15)
		tmp = -1.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-15], -1.0, N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999999e-15

   1. Initial program 100.0%

      \[x \cdot x - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 99.9%

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

   if 4.99999999999999999e-15 < (*.f64 x x)

   1. Initial program 100.0%

      \[x \cdot x - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 98.5%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 98.5%

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

Alternative 3: 52.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) -1.0 x))

C

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

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -1.0
	else:
		tmp = x
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], -1.0, x]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

      \[x \cdot x - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 70.8%

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

   if 1 < x

   1. Initial program 100.0%

      \[x \cdot x - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 98.6%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 98.6%

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

      1. pow2 N/A

         \[\leadsto {x}^{\color{blue}{2}} \]

      2. pow-to-exp N/A

         \[\leadsto e^{\log x \cdot 2} \]

      3. *-commutative N/A

         \[\leadsto e^{2 \cdot \log x} \]

      4. count-2 N/A

         \[\leadsto e^{\log x + \log x} \]

      5. flip-+ N/A

         \[\leadsto e^{\frac{\log x \cdot \log x - \log x \cdot \log x}{\log x - \log x}} \]

      6. +-inverses N/A

         \[\leadsto e^{\frac{0}{\log x - \log x}} \]

      7. +-inverses N/A

         \[\leadsto e^{\frac{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}{\log x - \log x}} \]

      8. +-inverses N/A

         \[\leadsto e^{\frac{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}{0}} \]

      9. +-inverses N/A

         \[\leadsto e^{\frac{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}} \]

      10. flip-+ N/A

          \[\leadsto e^{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) + \log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)} \]

      11. log-prod N/A

          \[\leadsto e^{\log \left({x}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)} \]

      12. sqr-pow N/A

          \[\leadsto e^{\log \left({x}^{\left(\frac{2}{2}\right)}\right)} \]

      13. metadata-eval N/A

          \[\leadsto e^{\log \left({x}^{1}\right)} \]

      14. unpow1 N/A

          \[\leadsto e^{\log x} \]

      15. rem-exp-log 7.0%

          \[\leadsto x \]

   7. Applied egg-rr 7.0%

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

Alternative 4: 50.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 100.0%

   \[x \cdot x - 1 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Simplified 51.9%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (x)
  :name "Data.Random.Dice:roll from dice-0.1"
  :precision binary64
  (- (* x x) 1.0))