Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 6

Speedup: 1.1×

• 32.8% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 99.8

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.8

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 1.5))) (* (* x x) (* -2.0 x)) (* (* 3.0 x) x)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.5)) {
		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) :: tmp
    if ((x <= (-1.5d0)) .or. (.not. (x <= 1.5d0))) 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 tmp;
	if ((x <= -1.5) || !(x <= 1.5)) {
		tmp = (x * x) * (-2.0 * x);
	} else {
		tmp = (3.0 * x) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.5) or not (x <= 1.5):
		tmp = (x * x) * (-2.0 * x)
	else:
		tmp = (3.0 * x) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.5) || !(x <= 1.5))
		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)
	tmp = 0.0;
	if ((x <= -1.5) || ~((x <= 1.5)))
		tmp = (x * x) * (-2.0 * x);
	else
		tmp = (3.0 * x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.5], N[Not[LessEqual[x, 1.5]], $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 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 96.7

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

   5. Applied rewrites 96.7%

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

   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

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

   5. Applied rewrites 97.1%

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{3} \]
   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 97.2

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

   7. Applied rewrites 97.2%

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

4. Final simplification 96.9%

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

Alternative 3: 75.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.5], N[(N[(3.0 * x), $MachinePrecision] * x), $MachinePrecision], N[((-x) * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   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 84.4%

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{3} \]
   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 84.5

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

   7. Applied rewrites 84.5%

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

   if 1.5 < x

   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%

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

   6. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\frac{x \cdot \left(9 - 6 \cdot x\right)}{\mathsf{fma}\left(6, x, 3\right)}} \cdot x \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot {x}^{2}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 47.2

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

   9. Applied rewrites 47.2%

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

Alternative 4: 75.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.5], N[(N[(x * x), $MachinePrecision] * 3.0), $MachinePrecision], N[((-x) * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   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 84.4%

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

   if 1.5 < x

   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%

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

   6. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\frac{x \cdot \left(9 - 6 \cdot x\right)}{\mathsf{fma}\left(6, x, 3\right)}} \cdot x \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot {x}^{2}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 47.2

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

   9. Applied rewrites 47.2%

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

Alternative 5: 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 6: 38.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -x * x

Julia

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

MATLAB

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

Wolfram

code[x_] := N[((-x) * 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%

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

6. Applied rewrites 59.2%

   \[\leadsto \color{blue}{\frac{x \cdot \left(9 - 6 \cdot x\right)}{\mathsf{fma}\left(6, x, 3\right)}} \cdot x \]

7. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1 \cdot {x}^{2}} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 38.8

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

9. Applied rewrites 38.8%

   \[\leadsto \color{blue}{\left(-x\right) \cdot x} \]
10. 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 2024338 
(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))))