Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 8.9s

Alternatives: 5

Speedup: 1.1×

• 33.2% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
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. *-commutative N/A

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

   3. lower-*.f64 99.9

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)\\ t_1 := \left(x \cdot x\right) \cdot \left(x \cdot -2\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\left(x \cdot x\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (- 3.0 (* x 2.0)))) (t_1 (* (* x x) (* x -2.0))))
   (if (<= t_0 -2e+18) t_1 (if (<= t_0 0.2) (* (* x x) 3.0) t_1))))

C

double code(double x) {
	double t_0 = (x * x) * (3.0 - (x * 2.0));
	double t_1 = (x * x) * (x * -2.0);
	double tmp;
	if (t_0 <= -2e+18) {
		tmp = t_1;
	} else if (t_0 <= 0.2) {
		tmp = (x * x) * 3.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * x) * (3.0d0 - (x * 2.0d0))
    t_1 = (x * x) * (x * (-2.0d0))
    if (t_0 <= (-2d+18)) then
        tmp = t_1
    else if (t_0 <= 0.2d0) then
        tmp = (x * x) * 3.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (3.0 - (x * 2.0));
	double t_1 = (x * x) * (x * -2.0);
	double tmp;
	if (t_0 <= -2e+18) {
		tmp = t_1;
	} else if (t_0 <= 0.2) {
		tmp = (x * x) * 3.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (3.0 - (x * 2.0))
	t_1 = (x * x) * (x * -2.0)
	tmp = 0
	if t_0 <= -2e+18:
		tmp = t_1
	elif t_0 <= 0.2:
		tmp = (x * x) * 3.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(3.0 - Float64(x * 2.0)))
	t_1 = Float64(Float64(x * x) * Float64(x * -2.0))
	tmp = 0.0
	if (t_0 <= -2e+18)
		tmp = t_1;
	elseif (t_0 <= 0.2)
		tmp = Float64(Float64(x * x) * 3.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (3.0 - (x * 2.0));
	t_1 = (x * x) * (x * -2.0);
	tmp = 0.0;
	if (t_0 <= -2e+18)
		tmp = t_1;
	elseif (t_0 <= 0.2)
		tmp = (x * x) * 3.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+18], t$95$1, If[LessEqual[t$95$0, 0.2], N[(N[(x * x), $MachinePrecision] * 3.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64)))) < -2e18 or 0.20000000000000001 < (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64))))

   1. Initial program 99.9%

      \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   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 98.1

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

   5. Applied rewrites 98.1%

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

   if -2e18 < (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64)))) < 0.20000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -2\right)\\ \mathbf{elif}\;\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \leq 0.2:\\ \;\;\;\;\left(x \cdot x\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.9%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
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 99.8

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

   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 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 4: 62.3% accurate, 1.7× speedup

Math

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

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(Float64(x * x) * 3.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 63.9%

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

Alternative 5: 62.3% 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 99.9%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
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 63.8%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot 3\right)} \]
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 2024233 
(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))))