Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 5.4s

Alternatives: 5

Speedup: N/A×

• 33.5% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. count-2-rev N/A

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

   5. associate--r+ N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2 \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\left(3 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* -2.0 x) (* x x))))
   (if (<= x -1.5) t_0 (if (<= x 1.5) (* (* 3.0 x) x) t_0))))

C

double code(double x) {
	double t_0 = (-2.0 * x) * (x * x);
	double tmp;
	if (x <= -1.5) {
		tmp = t_0;
	} else if (x <= 1.5) {
		tmp = (3.0 * x) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-2.0d0) * x) * (x * x)
    if (x <= (-1.5d0)) then
        tmp = t_0
    else if (x <= 1.5d0) then
        tmp = (3.0d0 * x) * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (-2.0 * x) * (x * x);
	double tmp;
	if (x <= -1.5) {
		tmp = t_0;
	} else if (x <= 1.5) {
		tmp = (3.0 * x) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (-2.0 * x) * (x * x)
	tmp = 0
	if x <= -1.5:
		tmp = t_0
	elif x <= 1.5:
		tmp = (3.0 * x) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(-2.0 * x) * Float64(x * x))
	tmp = 0.0
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 1.5)
		tmp = Float64(Float64(3.0 * x) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (-2.0 * x) * (x * x);
	tmp = 0.0;
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 1.5)
		tmp = (3.0 * x) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(-2.0 * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5], t$95$0, If[LessEqual[x, 1.5], N[(N[(3.0 * x), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 1.5 < x

   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 96.8

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

   5. Applied rewrites 96.8%

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

   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 98.7%

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{3} \]
   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 98.7

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

   7. Applied rewrites 98.7%

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

4. Final simplification 97.7%

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

Alternative 3: 76.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = (3.0 * x) * 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) then
        tmp = (3.0d0 * x) * 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) {
		tmp = (3.0 * x) * x;
	} else {
		tmp = (-3.0 * x) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.5:
		tmp = (3.0 * x) * x
	else:
		tmp = (-3.0 * 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(-3.0 * 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 = (-3.0 * x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if 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 84.9%

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{3} \]
   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 85.0

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

   7. Applied rewrites 85.0%

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

   if 1.5 < x

   1. Initial program 100.0%

      \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   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 0.2%

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{3} \]
   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 0.2

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

   7. Applied rewrites 0.2%

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

      1. lift-*.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \left(3 \cdot \color{blue}{{x}^{1}}\right) \cdot x \]

      3. remove-double-neg N/A

         \[\leadsto \left(3 \cdot {\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}^{1}\right) \cdot x \]

      4. lift-neg.f64 N/A

         \[\leadsto \left(3 \cdot {\left(\mathsf{neg}\left(\color{blue}{\left(-x\right)}\right)\right)}^{1}\right) \cdot x \]

      5. neg-mul-1 N/A

         \[\leadsto \left(3 \cdot {\color{blue}{\left(-1 \cdot \left(-x\right)\right)}}^{1}\right) \cdot x \]

      6. unpow-prod-down N/A

         \[\leadsto \left(3 \cdot \color{blue}{\left({-1}^{1} \cdot {\left(-x\right)}^{1}\right)}\right) \cdot x \]

      7. metadata-eval N/A

         \[\leadsto \left(3 \cdot \left(\color{blue}{-1} \cdot {\left(-x\right)}^{1}\right)\right) \cdot x \]

      8. metadata-eval N/A

         \[\leadsto \left(3 \cdot \left(-1 \cdot {\left(-x\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)\right) \cdot x \]

      9. sqrt-pow1 N/A

         \[\leadsto \left(3 \cdot \left(-1 \cdot \color{blue}{\sqrt{{\left(-x\right)}^{2}}}\right)\right) \cdot x \]

      10. pow2 N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}\right)\right) \cdot x \]

      11. lift-neg.f64 N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(-x\right)}\right)\right) \cdot x \]

      12. lift-neg.f64 N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right)\right) \cdot x \]

      13. sqr-neg-rev N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\color{blue}{x \cdot x}}\right)\right) \cdot x \]

      14. pow2 N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\color{blue}{{x}^{2}}}\right)\right) \cdot x \]

      15. sqrt-pow1 N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \color{blue}{{x}^{\left(\frac{2}{2}\right)}}\right)\right) \cdot x \]

      16. metadata-eval N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot {x}^{\color{blue}{1}}\right)\right) \cdot x \]

      17. unpow1 N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \color{blue}{x}\right)\right) \cdot x \]

      18. neg-mul-1 N/A

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

      19. distribute-rgt-neg-out N/A

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

      20. lift-*.f64 N/A

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

      21. lower-neg.f64 66.3

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

   9. Applied rewrites 66.3%

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

   10. Taylor expanded in x around 0

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

       1. lower-*.f64 66.3

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

   12. Applied rewrites 66.3%

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

Alternative 4: 76.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = 3.0 * (x * 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) then
        tmp = 3.0d0 * (x * 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) {
		tmp = 3.0 * (x * x);
	} else {
		tmp = (-3.0 * x) * x;
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5)
		tmp = Float64(3.0 * Float64(x * 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)
		tmp = 3.0 * (x * x);
	else
		tmp = (-3.0 * x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if 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 84.9%

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

   if 1.5 < x

   1. Initial program 100.0%

      \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   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 0.2%

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{3} \]
   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 0.2

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

   7. Applied rewrites 0.2%

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

      1. lift-*.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \left(3 \cdot \color{blue}{{x}^{1}}\right) \cdot x \]

      3. remove-double-neg N/A

         \[\leadsto \left(3 \cdot {\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}^{1}\right) \cdot x \]

      4. lift-neg.f64 N/A

         \[\leadsto \left(3 \cdot {\left(\mathsf{neg}\left(\color{blue}{\left(-x\right)}\right)\right)}^{1}\right) \cdot x \]

      5. neg-mul-1 N/A

         \[\leadsto \left(3 \cdot {\color{blue}{\left(-1 \cdot \left(-x\right)\right)}}^{1}\right) \cdot x \]

      6. unpow-prod-down N/A

         \[\leadsto \left(3 \cdot \color{blue}{\left({-1}^{1} \cdot {\left(-x\right)}^{1}\right)}\right) \cdot x \]

      7. metadata-eval N/A

         \[\leadsto \left(3 \cdot \left(\color{blue}{-1} \cdot {\left(-x\right)}^{1}\right)\right) \cdot x \]

      8. metadata-eval N/A

         \[\leadsto \left(3 \cdot \left(-1 \cdot {\left(-x\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)\right) \cdot x \]

      9. sqrt-pow1 N/A

         \[\leadsto \left(3 \cdot \left(-1 \cdot \color{blue}{\sqrt{{\left(-x\right)}^{2}}}\right)\right) \cdot x \]

      10. pow2 N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}\right)\right) \cdot x \]

      11. lift-neg.f64 N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(-x\right)}\right)\right) \cdot x \]

      12. lift-neg.f64 N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right)\right) \cdot x \]

      13. sqr-neg-rev N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\color{blue}{x \cdot x}}\right)\right) \cdot x \]

      14. pow2 N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\color{blue}{{x}^{2}}}\right)\right) \cdot x \]

      15. sqrt-pow1 N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \color{blue}{{x}^{\left(\frac{2}{2}\right)}}\right)\right) \cdot x \]

      16. metadata-eval N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot {x}^{\color{blue}{1}}\right)\right) \cdot x \]

      17. unpow1 N/A

          \[\leadsto \left(3 \cdot \left(-1 \cdot \color{blue}{x}\right)\right) \cdot x \]

      18. neg-mul-1 N/A

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

      19. distribute-rgt-neg-out N/A

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

      20. lift-*.f64 N/A

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

      21. lower-neg.f64 66.3

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

   9. Applied rewrites 66.3%

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

   10. Taylor expanded in x around 0

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

       1. lower-*.f64 66.3

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

   12. Applied rewrites 66.3%

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

4. Final simplification 80.4%

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

Alternative 5: 38.8% 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 64.1%

   \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{3} \]
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 64.1

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

7. Applied rewrites 64.1%

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

   1. lift-*.f64 N/A

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

   2. unpow1 N/A

      \[\leadsto \left(3 \cdot \color{blue}{{x}^{1}}\right) \cdot x \]

   3. remove-double-neg N/A

      \[\leadsto \left(3 \cdot {\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}^{1}\right) \cdot x \]

   4. lift-neg.f64 N/A

      \[\leadsto \left(3 \cdot {\left(\mathsf{neg}\left(\color{blue}{\left(-x\right)}\right)\right)}^{1}\right) \cdot x \]

   5. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot {\color{blue}{\left(-1 \cdot \left(-x\right)\right)}}^{1}\right) \cdot x \]

   6. unpow-prod-down N/A

      \[\leadsto \left(3 \cdot \color{blue}{\left({-1}^{1} \cdot {\left(-x\right)}^{1}\right)}\right) \cdot x \]

   7. metadata-eval N/A

      \[\leadsto \left(3 \cdot \left(\color{blue}{-1} \cdot {\left(-x\right)}^{1}\right)\right) \cdot x \]

   8. metadata-eval N/A

      \[\leadsto \left(3 \cdot \left(-1 \cdot {\left(-x\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)\right) \cdot x \]

   9. sqrt-pow1 N/A

      \[\leadsto \left(3 \cdot \left(-1 \cdot \color{blue}{\sqrt{{\left(-x\right)}^{2}}}\right)\right) \cdot x \]

   10. pow2 N/A

       \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}\right)\right) \cdot x \]

   11. lift-neg.f64 N/A

       \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(-x\right)}\right)\right) \cdot x \]

   12. lift-neg.f64 N/A

       \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right)\right) \cdot x \]

   13. sqr-neg-rev N/A

       \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\color{blue}{x \cdot x}}\right)\right) \cdot x \]

   14. pow2 N/A

       \[\leadsto \left(3 \cdot \left(-1 \cdot \sqrt{\color{blue}{{x}^{2}}}\right)\right) \cdot x \]

   15. sqrt-pow1 N/A

       \[\leadsto \left(3 \cdot \left(-1 \cdot \color{blue}{{x}^{\left(\frac{2}{2}\right)}}\right)\right) \cdot x \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot \left(-1 \cdot {x}^{\color{blue}{1}}\right)\right) \cdot x \]

   17. unpow1 N/A

       \[\leadsto \left(3 \cdot \left(-1 \cdot \color{blue}{x}\right)\right) \cdot x \]

   18. neg-mul-1 N/A

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

   19. distribute-rgt-neg-out N/A

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

   20. lift-*.f64 N/A

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

   21. lower-neg.f64 42.1

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

9. Applied rewrites 42.1%

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

10. Taylor expanded in x around 0

    \[\leadsto \color{blue}{-3 \cdot {x}^{2}} \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \color{blue}{-3 \cdot {x}^{2}} \]

    2. unpow2 N/A

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

    3. lower-*.f64 42.1

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

12. Applied rewrites 42.1%

    \[\leadsto \color{blue}{-3 \cdot \left(x \cdot x\right)} \]
13. 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 2024298 
(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))))