Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 5.6s

Alternatives: 5

Speedup: 1.1×

• 34.7% 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

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

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

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

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: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq -40000 \lor \neg \left(t\_0 \leq 0.005\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (- 3.0 (* x 2.0)))))
   (if (or (<= t_0 -40000.0) (not (<= t_0 0.005)))
     (* (* x x) (* -2.0 x))
     (* (* 3.0 x) x))))

C

double code(double x) {
	double t_0 = (x * x) * (3.0 - (x * 2.0));
	double tmp;
	if ((t_0 <= -40000.0) || !(t_0 <= 0.005)) {
		tmp = (x * x) * (-2.0 * x);
	} else {
		tmp = (3.0 * x) * x;
	}
	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 <= (-40000.0d0)) .or. (.not. (t_0 <= 0.005d0))) then
        tmp = (x * x) * ((-2.0d0) * x)
    else
        tmp = (3.0d0 * x) * x
    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 <= -40000.0) || !(t_0 <= 0.005)) {
		tmp = (x * x) * (-2.0 * x);
	} else {
		tmp = (3.0 * x) * x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (3.0 - (x * 2.0))
	tmp = 0
	if (t_0 <= -40000.0) or not (t_0 <= 0.005):
		tmp = (x * x) * (-2.0 * x)
	else:
		tmp = (3.0 * x) * x
	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 <= -40000.0) || !(t_0 <= 0.005))
		tmp = Float64(Float64(x * x) * Float64(-2.0 * x));
	else
		tmp = Float64(Float64(3.0 * x) * x);
	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 <= -40000.0) || ~((t_0 <= 0.005)))
		tmp = (x * x) * (-2.0 * x);
	else
		tmp = (3.0 * x) * x;
	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, -40000.0], N[Not[LessEqual[t$95$0, 0.005]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] * N[(-2.0 * x), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * x), $MachinePrecision] * x), $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)))) < -4e4 or 0.0050000000000000001 < (*.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 98.9

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

   5. Applied rewrites 98.9%

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

   if -4e4 < (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64)))) < 0.0050000000000000001

   1. Initial program 99.6%

      \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   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 97.6%

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{3} \]
   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 97.7

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

   7. Applied rewrites 97.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \leq -40000 \lor \neg \left(\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \leq 0.005\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x\right) \cdot x\\ \end{array} \]
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 63.7%

   \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{3} \]
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 63.8

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

7. Applied rewrites 63.8%

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

Alternative 4: 62.6% accurate, 1.7× speedup

Math

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

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 63.7%

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

4. Applied rewrites 69.4%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 48.6

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

7. Applied rewrites 48.6%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 2.9%

    \[\leadsto \left(-9 \cdot x\right) \cdot \color{blue}{-0.5} \]
11. 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 2024320 
(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))))