Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 5.5s

Alternatives: 5

Speedup: 1.0×

• 32.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x x) (- 3.0 (* x 2.0))))

C

double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (3.0d0 - (x * 2.0d0))
end function

Java

public static double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

Python

def code(x):
	return (x * x) * (3.0 - (x * 2.0))

Julia

function code(x)
	return Float64(Float64(x * x) * Float64(3.0 - Float64(x * 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * (3.0 - (x * 2.0));
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x x) (- 3.0 (* x 2.0))))

C

double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (3.0d0 - (x * 2.0d0))
end function

Java

public static double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

Python

def code(x):
	return (x * x) * (3.0 - (x * 2.0))

Julia

function code(x)
	return Float64(Float64(x * x) * Float64(3.0 - Float64(x * 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * (3.0 - (x * 2.0));
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x x) (- 3.0 (* x 2.0))))

C

double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (3.0d0 - (x * 2.0d0))
end function

Java

public static double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

Python

def code(x):
	return (x * x) * (3.0 - (x * 2.0))

Julia

function code(x)
	return Float64(Float64(x * x) * Float64(3.0 - Float64(x * 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * (3.0 - (x * 2.0));
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 1.5)))
   (* x (* x (* x -2.0)))
   (* x (/ x 0.3333333333333333))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.5)) {
		tmp = x * (x * (x * -2.0));
	} else {
		tmp = x * (x / 0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.5d0)) .or. (.not. (x <= 1.5d0))) then
        tmp = x * (x * (x * (-2.0d0)))
    else
        tmp = x * (x / 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.5)) {
		tmp = x * (x * (x * -2.0));
	} else {
		tmp = x * (x / 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.5) or not (x <= 1.5):
		tmp = x * (x * (x * -2.0))
	else:
		tmp = x * (x / 0.3333333333333333)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.5) || !(x <= 1.5))
		tmp = Float64(x * Float64(x * Float64(x * -2.0)));
	else
		tmp = Float64(x * Float64(x / 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.5) || ~((x <= 1.5)))
		tmp = x * (x * (x * -2.0));
	else
		tmp = x * (x / 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.5], N[Not[LessEqual[x, 1.5]], $MachinePrecision]], N[(x * N[(x * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 1.5 < x

   1. Initial program 99.8%

      \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   2. Step-by-step derivation

      1. associate-*l* 99.8%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.9%

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(-2 \cdot x\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 97.9%

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot -2\right)}\right) \]

   7. Simplified 97.9%

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot -2\right)}\right) \]

   if -1.5 < x < 1.5

   1. Initial program 99.8%

      \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   2. Step-by-step derivation

      1. associate-*l* 99.7%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 88.8%

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(3 \cdot \frac{1}{x} - 2\right)} \]
   6. Step-by-step derivation

      1. sub-neg 88.8%

         \[\leadsto {x}^{3} \cdot \color{blue}{\left(3 \cdot \frac{1}{x} + \left(-2\right)\right)} \]

      2. associate-*r/ 88.9%

         \[\leadsto {x}^{3} \cdot \left(\color{blue}{\frac{3 \cdot 1}{x}} + \left(-2\right)\right) \]

      3. metadata-eval 88.9%

         \[\leadsto {x}^{3} \cdot \left(\frac{\color{blue}{3}}{x} + \left(-2\right)\right) \]

      4. metadata-eval 88.9%

         \[\leadsto {x}^{3} \cdot \left(\frac{3}{x} + \color{blue}{-2}\right) \]

   7. Simplified 88.9%

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{3}{x} + -2\right)} \]

   8. Taylor expanded in x around 0 87.2%

      \[\leadsto {x}^{3} \cdot \color{blue}{\frac{3}{x}} \]
   9. Step-by-step derivation

      1. clear-num 87.2%

         \[\leadsto {x}^{3} \cdot \color{blue}{\frac{1}{\frac{x}{3}}} \]

      2. un-div-inv 87.2%

         \[\leadsto \color{blue}{\frac{{x}^{3}}{\frac{x}{3}}} \]

      3. div-inv 87.2%

         \[\leadsto \frac{{x}^{3}}{\color{blue}{x \cdot \frac{1}{3}}} \]

      4. metadata-eval 87.2%

         \[\leadsto \frac{{x}^{3}}{x \cdot \color{blue}{0.3333333333333333}} \]

   10. Applied egg-rr 87.2%

       \[\leadsto \color{blue}{\frac{{x}^{3}}{x \cdot 0.3333333333333333}} \]
   11. Step-by-step derivation

       1. associate-/r* 87.2%

          \[\leadsto \color{blue}{\frac{\frac{{x}^{3}}{x}}{0.3333333333333333}} \]

       2. unpow3 87.2%

          \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot x}}{x}}{0.3333333333333333} \]

       3. unpow2 87.2%

          \[\leadsto \frac{\frac{\color{blue}{{x}^{2}} \cdot x}{x}}{0.3333333333333333} \]

       4. associate-/l* 98.1%

          \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \frac{x}{x}}}{0.3333333333333333} \]

       5. *-inverses 98.1%

          \[\leadsto \frac{{x}^{2} \cdot \color{blue}{1}}{0.3333333333333333} \]

       6. *-rgt-identity 98.1%

          \[\leadsto \frac{\color{blue}{{x}^{2}}}{0.3333333333333333} \]

   12. Simplified 98.1%

       \[\leadsto \color{blue}{\frac{{x}^{2}}{0.3333333333333333}} \]
   13. Step-by-step derivation

       1. pow2 98.1%

          \[\leadsto \frac{\color{blue}{x \cdot x}}{0.3333333333333333} \]

       2. associate-/l* 98.2%

          \[\leadsto \color{blue}{x \cdot \frac{x}{0.3333333333333333}} \]

   14. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{x \cdot \frac{x}{0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x (- 3.0 (* x 2.0)))))

C

double code(double x) {
	return x * (x * (3.0 - (x * 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (3.0d0 - (x * 2.0d0)))
end function

Java

public static double code(double x) {
	return x * (x * (3.0 - (x * 2.0)));
}

Python

def code(x):
	return x * (x * (3.0 - (x * 2.0)))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(3.0 - Float64(x * 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (3.0 - (x * 2.0)));
end

Wolfram

code[x_] := N[(x * N[(x * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 4: 61.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{0.3333333333333333} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (/ x 0.3333333333333333)))

C

double code(double x) {
	return x * (x / 0.3333333333333333);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x / 0.3333333333333333d0)
end function

Java

public static double code(double x) {
	return x * (x / 0.3333333333333333);
}

Python

def code(x):
	return x * (x / 0.3333333333333333)

Julia

function code(x)
	return Float64(x * Float64(x / 0.3333333333333333))
end

MATLAB

function tmp = code(x)
	tmp = x * (x / 0.3333333333333333);
end

Wolfram

code[x_] := N[(x * N[(x / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 94.4%

   \[\leadsto \color{blue}{{x}^{3} \cdot \left(3 \cdot \frac{1}{x} - 2\right)} \]
6. Step-by-step derivation

   1. sub-neg 94.4%

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(3 \cdot \frac{1}{x} + \left(-2\right)\right)} \]

   2. associate-*r/ 94.4%

      \[\leadsto {x}^{3} \cdot \left(\color{blue}{\frac{3 \cdot 1}{x}} + \left(-2\right)\right) \]

   3. metadata-eval 94.4%

      \[\leadsto {x}^{3} \cdot \left(\frac{\color{blue}{3}}{x} + \left(-2\right)\right) \]

   4. metadata-eval 94.4%

      \[\leadsto {x}^{3} \cdot \left(\frac{3}{x} + \color{blue}{-2}\right) \]

7. Simplified 94.4%

   \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{3}{x} + -2\right)} \]

8. Taylor expanded in x around 0 56.6%

   \[\leadsto {x}^{3} \cdot \color{blue}{\frac{3}{x}} \]
9. Step-by-step derivation

   1. clear-num 56.5%

      \[\leadsto {x}^{3} \cdot \color{blue}{\frac{1}{\frac{x}{3}}} \]

   2. un-div-inv 56.6%

      \[\leadsto \color{blue}{\frac{{x}^{3}}{\frac{x}{3}}} \]

   3. div-inv 56.5%

      \[\leadsto \frac{{x}^{3}}{\color{blue}{x \cdot \frac{1}{3}}} \]

   4. metadata-eval 56.5%

      \[\leadsto \frac{{x}^{3}}{x \cdot \color{blue}{0.3333333333333333}} \]

10. Applied egg-rr 56.5%

    \[\leadsto \color{blue}{\frac{{x}^{3}}{x \cdot 0.3333333333333333}} \]
11. Step-by-step derivation

    1. associate-/r* 56.5%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{3}}{x}}{0.3333333333333333}} \]

    2. unpow3 56.5%

       \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot x}}{x}}{0.3333333333333333} \]

    3. unpow2 56.5%

       \[\leadsto \frac{\frac{\color{blue}{{x}^{2}} \cdot x}{x}}{0.3333333333333333} \]

    4. associate-/l* 60.2%

       \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \frac{x}{x}}}{0.3333333333333333} \]

    5. *-inverses 60.2%

       \[\leadsto \frac{{x}^{2} \cdot \color{blue}{1}}{0.3333333333333333} \]

    6. *-rgt-identity 60.2%

       \[\leadsto \frac{\color{blue}{{x}^{2}}}{0.3333333333333333} \]

12. Simplified 60.2%

    \[\leadsto \color{blue}{\frac{{x}^{2}}{0.3333333333333333}} \]
13. Step-by-step derivation

    1. pow2 60.2%

       \[\leadsto \frac{\color{blue}{x \cdot x}}{0.3333333333333333} \]

    2. associate-/l* 60.3%

       \[\leadsto \color{blue}{x \cdot \frac{x}{0.3333333333333333}} \]

14. Applied egg-rr 60.3%

    \[\leadsto \color{blue}{x \cdot \frac{x}{0.3333333333333333}} \]
15. Add Preprocessing

Alternative 5: 61.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot 3\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x 3.0)))

C

double code(double x) {
	return x * (x * 3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * 3.0d0)
end function

Java

public static double code(double x) {
	return x * (x * 3.0);
}

Python

def code(x):
	return x * (x * 3.0)

Julia

function code(x)
	return Float64(x * Float64(x * 3.0))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * 3.0);
end

Wolfram

code[x_] := N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 60.3%

   \[\leadsto x \cdot \color{blue}{\left(3 \cdot x\right)} \]
6. Step-by-step derivation

   1. *-commutative 60.3%

      \[\leadsto x \cdot \color{blue}{\left(x \cdot 3\right)} \]

7. Simplified 60.3%

   \[\leadsto x \cdot \color{blue}{\left(x \cdot 3\right)} \]
8. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x (- 3.0 (* x 2.0)))))

C

double code(double x) {
	return x * (x * (3.0 - (x * 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (3.0d0 - (x * 2.0d0)))
end function

Java

public static double code(double x) {
	return x * (x * (3.0 - (x * 2.0)));
}

Python

def code(x):
	return x * (x * (3.0 - (x * 2.0)))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(3.0 - Float64(x * 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (3.0 - (x * 2.0)));
end

Wolfram

code[x_] := N[(x * N[(x * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024170 
(FPCore (x)
  :name "Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (* x (- 3 (* x 2)))))

  (* (* x x) (- 3.0 (* x 2.0))))