Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 41.8s

Alternatives: 3

Speedup: 1.1×

• 33.7% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x * N[(x * N[(x * -2.0 + 3.0), $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 99.9

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. lower-fma.f64 N/A

       \[\leadsto \left(x \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(2\right), 3\right)}\right) \cdot x \]

   13. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right) \]
6. Add Preprocessing

Alternative 2: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot -2\right)\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x -2.0))))
   (if (<= x -1.5) t_0 (if (<= x 1.5) (* x (* x 3.0)) t_0))))

C

double code(double x) {
	double t_0 = (x * x) * (x * -2.0);
	double tmp;
	if (x <= -1.5) {
		tmp = t_0;
	} else if (x <= 1.5) {
		tmp = x * (x * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * (x * (-2.0d0))
    if (x <= (-1.5d0)) then
        tmp = t_0
    else if (x <= 1.5d0) then
        tmp = x * (x * 3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (x * -2.0);
	double tmp;
	if (x <= -1.5) {
		tmp = t_0;
	} else if (x <= 1.5) {
		tmp = x * (x * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (x * -2.0)
	tmp = 0
	if x <= -1.5:
		tmp = t_0
	elif x <= 1.5:
		tmp = x * (x * 3.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * -2.0))
	tmp = 0.0
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 1.5)
		tmp = Float64(x * Float64(x * 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (x * -2.0);
	tmp = 0.0;
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 1.5)
		tmp = x * (x * 3.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5], t$95$0, If[LessEqual[x, 1.5], N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -1.5 < x < 1.5

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{3 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lft-mult-inverse N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. remove-double-neg N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(3 \cdot \frac{1}{x}\right) \cdot x\right)\right)\right)\right)} \cdot x\right) \]

      11. distribute-lft-neg-out N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{1}{x}\right)\right) \cdot x}\right)\right) \cdot x\right) \]

      12. *-commutative N/A

          \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(3 \cdot \frac{1}{x}\right)\right)}\right)\right) \cdot x\right) \]

      13. distribute-lft-neg-out N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(3 \cdot \frac{1}{x}\right)\right)\right)} \cdot x\right) \]

      14. mul-1-neg N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{\left(-1 \cdot x\right)} \cdot \left(\mathsf{neg}\left(3 \cdot \frac{1}{x}\right)\right)\right) \cdot x\right) \]

      15. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(3 \cdot \frac{1}{x}\right)\right)\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(3 \cdot \frac{1}{x}\right)\right)\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(\left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{x} \cdot 3}\right)\right)\right)\right) \]

      18. distribute-lft-neg-in N/A

          \[\leadsto x \cdot \left(x \cdot \left(\left(-1 \cdot x\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot 3\right)}\right)\right) \]

      19. associate-*r* N/A

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\left(\left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right) \cdot 3\right)}\right) \]

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -2\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.6% accurate, 1.7× 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. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{3 \cdot {x}^{2}} \]
4. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lft-mult-inverse N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. remove-double-neg N/A

       \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(3 \cdot \frac{1}{x}\right) \cdot x\right)\right)\right)\right)} \cdot x\right) \]

   11. distribute-lft-neg-out N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{1}{x}\right)\right) \cdot x}\right)\right) \cdot x\right) \]

   12. *-commutative N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(3 \cdot \frac{1}{x}\right)\right)}\right)\right) \cdot x\right) \]

   13. distribute-lft-neg-out N/A

       \[\leadsto x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(3 \cdot \frac{1}{x}\right)\right)\right)} \cdot x\right) \]

   14. mul-1-neg N/A

       \[\leadsto x \cdot \left(\left(\color{blue}{\left(-1 \cdot x\right)} \cdot \left(\mathsf{neg}\left(3 \cdot \frac{1}{x}\right)\right)\right) \cdot x\right) \]

   15. *-commutative N/A

       \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(3 \cdot \frac{1}{x}\right)\right)\right)\right)} \]

   16. lower-*.f64 N/A

       \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(3 \cdot \frac{1}{x}\right)\right)\right)\right)} \]

   17. *-commutative N/A

       \[\leadsto x \cdot \left(x \cdot \left(\left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{x} \cdot 3}\right)\right)\right)\right) \]

   18. distribute-lft-neg-in N/A

       \[\leadsto x \cdot \left(x \cdot \left(\left(-1 \cdot x\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot 3\right)}\right)\right) \]

   19. associate-*r* N/A

       \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\left(\left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right) \cdot 3\right)}\right) \]

5. Applied rewrites 62.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot 3\right)} \]
6. 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 2024257 
(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))))