Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 5.8s

Alternatives: 4

Speedup: 1.1×

• 34.2% 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

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

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(fma(-2.0, x, 3.0) * Float64(x * x))
end

Wolfram

code[x_] := N[(N[(-2.0 * x + 3.0), $MachinePrecision] * 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. 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 99.8

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

   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. *-commutative N/A

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

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

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq -20000 \lor \neg \left(t\_0 \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (- 3.0 (* x 2.0)))))
   (if (or (<= t_0 -20000.0) (not (<= t_0 2e-8)))
     (* (* x x) (* -2.0 x))
     (* (* x x) 3.0))))

C

double code(double x) {
	double t_0 = (x * x) * (3.0 - (x * 2.0));
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 2e-8)) {
		tmp = (x * x) * (-2.0 * x);
	} else {
		tmp = (x * x) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * (3.0d0 - (x * 2.0d0))
    if ((t_0 <= (-20000.0d0)) .or. (.not. (t_0 <= 2d-8))) then
        tmp = (x * x) * ((-2.0d0) * x)
    else
        tmp = (x * x) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (3.0 - (x * 2.0));
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 2e-8)) {
		tmp = (x * x) * (-2.0 * x);
	} else {
		tmp = (x * x) * 3.0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (3.0 - (x * 2.0))
	tmp = 0
	if (t_0 <= -20000.0) or not (t_0 <= 2e-8):
		tmp = (x * x) * (-2.0 * x)
	else:
		tmp = (x * x) * 3.0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(3.0 - Float64(x * 2.0)))
	tmp = 0.0
	if ((t_0 <= -20000.0) || !(t_0 <= 2e-8))
		tmp = Float64(Float64(x * x) * Float64(-2.0 * x));
	else
		tmp = Float64(Float64(x * x) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (3.0 - (x * 2.0));
	tmp = 0.0;
	if ((t_0 <= -20000.0) || ~((t_0 <= 2e-8)))
		tmp = (x * x) * (-2.0 * x);
	else
		tmp = (x * x) * 3.0;
	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]}, If[Or[LessEqual[t$95$0, -20000.0], N[Not[LessEqual[t$95$0, 2e-8]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] * N[(-2.0 * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64)))) < -2e4 or 2e-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 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 -2e4 < (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64)))) < 2e-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 98.3%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \leq -20000 \lor \neg \left(\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 61.7% 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 99.8%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
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 59.8%

   \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{3} \]
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 2024317 
(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))))