Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 6.4s

Alternatives: 5

Speedup: 1.1×

• 33.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   7. lower-*.f64 99.8

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

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

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 - 2 \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot -2\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(3 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (- 3.0 (* 2.0 x)) (* x x))) (t_1 (* (* x -2.0) (* x x))))
   (if (<= t_0 -5e+19) t_1 (if (<= t_0 5e-8) (* (* 3.0 x) x) t_1))))

C

double code(double x) {
	double t_0 = (3.0 - (2.0 * x)) * (x * x);
	double t_1 = (x * -2.0) * (x * x);
	double tmp;
	if (t_0 <= -5e+19) {
		tmp = t_1;
	} else if (t_0 <= 5e-8) {
		tmp = (3.0 * x) * x;
	} 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 = (3.0d0 - (2.0d0 * x)) * (x * x)
    t_1 = (x * (-2.0d0)) * (x * x)
    if (t_0 <= (-5d+19)) then
        tmp = t_1
    else if (t_0 <= 5d-8) then
        tmp = (3.0d0 * x) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (3.0 - (2.0 * x)) * (x * x);
	double t_1 = (x * -2.0) * (x * x);
	double tmp;
	if (t_0 <= -5e+19) {
		tmp = t_1;
	} else if (t_0 <= 5e-8) {
		tmp = (3.0 * x) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (3.0 - (2.0 * x)) * (x * x)
	t_1 = (x * -2.0) * (x * x)
	tmp = 0
	if t_0 <= -5e+19:
		tmp = t_1
	elif t_0 <= 5e-8:
		tmp = (3.0 * x) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(3.0 - Float64(2.0 * x)) * Float64(x * x))
	t_1 = Float64(Float64(x * -2.0) * Float64(x * x))
	tmp = 0.0
	if (t_0 <= -5e+19)
		tmp = t_1;
	elseif (t_0 <= 5e-8)
		tmp = Float64(Float64(3.0 * x) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (3.0 - (2.0 * x)) * (x * x);
	t_1 = (x * -2.0) * (x * x);
	tmp = 0.0;
	if (t_0 <= -5e+19)
		tmp = t_1;
	elseif (t_0 <= 5e-8)
		tmp = (3.0 * x) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(3.0 - N[(2.0 * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * -2.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+19], t$95$1, If[LessEqual[t$95$0, 5e-8], N[(N[(3.0 * x), $MachinePrecision] * x), $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)))) < -5e19 or 4.9999999999999998e-8 < (*.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 99.4

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

   5. Applied rewrites 99.4%

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

   if -5e19 < (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64)))) < 4.9999999999999998e-8

   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

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

   5. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 98.9

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

   7. Applied rewrites 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 - 2 \cdot x\right) \cdot \left(x \cdot x\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\left(x \cdot -2\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;\left(3 - 2 \cdot x\right) \cdot \left(x \cdot x\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(3 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot -2\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Applied rewrites 68.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 68.6

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

7. Applied rewrites 68.6%

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

Alternative 4: 62.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Applied rewrites 68.6%

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

6. Final simplification 68.6%

   \[\leadsto 3 \cdot \left(x \cdot x\right) \]
7. Add Preprocessing

Alternative 5: 3.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(9 \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 9.0 x) 0.5))

C

double code(double x) {
	return (9.0 * x) * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (9.0d0 * x) * 0.5d0
end function

Java

public static double code(double x) {
	return (9.0 * x) * 0.5;
}

Python

def code(x):
	return (9.0 * x) * 0.5

Julia

function code(x)
	return Float64(Float64(9.0 * x) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = (9.0 * x) * 0.5;
end

Wolfram

code[x_] := N[(N[(9.0 * x), $MachinePrecision] * 0.5), $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. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 95.9%

   \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(-4, x \cdot x, 9\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(2, x, 3\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\left(\color{blue}{9} \cdot x\right) \cdot x}{\mathsf{fma}\left(2, x, 3\right)} \]
6. Step-by-step derivation

7. Applied rewrites 50.6%

   \[\leadsto \frac{\left(\color{blue}{9} \cdot x\right) \cdot x}{\mathsf{fma}\left(2, x, 3\right)} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(9 \cdot x\right) \cdot x}{\mathsf{fma}\left(2, x, 3\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(9 \cdot x\right) \cdot x}}{\mathsf{fma}\left(2, x, 3\right)} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\left(9 \cdot x\right) \cdot \frac{x}{\mathsf{fma}\left(2, x, 3\right)}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(2, x, 3\right)} \cdot \left(9 \cdot x\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(2, x, 3\right)} \cdot \left(9 \cdot x\right)} \]

   6. lower-/.f64 50.7

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(2, x, 3\right)}} \cdot \left(9 \cdot x\right) \]

9. Applied rewrites 50.7%

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(2, x, 3\right)} \cdot \left(9 \cdot x\right)} \]

10. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(9 \cdot x\right) \]
11. Step-by-step derivation

12. Applied rewrites 3.0%

    \[\leadsto \color{blue}{0.5} \cdot \left(9 \cdot x\right) \]

13. Final simplification 3.0%

    \[\leadsto \left(9 \cdot x\right) \cdot 0.5 \]
14. 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 2024295 
(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))))