Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 7.1s

Alternatives: 6

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. distribute-rgt-in N/A

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

   3. associate-*r* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. associate-*l* N/A

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

   8. associate-*l* N/A

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

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

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-commutative N/A

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

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

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

   14. associate-*r* N/A

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

   15. *-lowering-*.f64 N/A

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

   16. *-lowering-*.f64 N/A

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

   17. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)\\ t_1 := \left(x \cdot x\right) \cdot \left(x \cdot -2\right)\\ \mathbf{if}\;t\_0 \leq -100000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;x \cdot \left(3 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (- 3.0 (* x 2.0)))) (t_1 (* (* x x) (* x -2.0))))
   (if (<= t_0 -100000.0) t_1 (if (<= t_0 20.0) (* x (* 3.0 x)) t_1))))

C

double code(double x) {
	double t_0 = (x * x) * (3.0 - (x * 2.0));
	double t_1 = (x * x) * (x * -2.0);
	double tmp;
	if (t_0 <= -100000.0) {
		tmp = t_1;
	} else if (t_0 <= 20.0) {
		tmp = x * (3.0 * 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 = (x * x) * (3.0d0 - (x * 2.0d0))
    t_1 = (x * x) * (x * (-2.0d0))
    if (t_0 <= (-100000.0d0)) then
        tmp = t_1
    else if (t_0 <= 20.0d0) then
        tmp = x * (3.0d0 * x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (3.0 - (x * 2.0));
	double t_1 = (x * x) * (x * -2.0);
	double tmp;
	if (t_0 <= -100000.0) {
		tmp = t_1;
	} else if (t_0 <= 20.0) {
		tmp = x * (3.0 * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (3.0 - (x * 2.0))
	t_1 = (x * x) * (x * -2.0)
	tmp = 0
	if t_0 <= -100000.0:
		tmp = t_1
	elif t_0 <= 20.0:
		tmp = x * (3.0 * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(3.0 - Float64(x * 2.0)))
	t_1 = Float64(Float64(x * x) * Float64(x * -2.0))
	tmp = 0.0
	if (t_0 <= -100000.0)
		tmp = t_1;
	elseif (t_0 <= 20.0)
		tmp = Float64(x * Float64(3.0 * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (3.0 - (x * 2.0));
	t_1 = (x * x) * (x * -2.0);
	tmp = 0.0;
	if (t_0 <= -100000.0)
		tmp = t_1;
	elseif (t_0 <= 20.0)
		tmp = x * (3.0 * x);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100000.0], t$95$1, If[LessEqual[t$95$0, 20.0], N[(x * N[(3.0 * x), $MachinePrecision]), $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)))) < -1e5 or 20 < (*.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. *-lowering-*.f64 97.9

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

   5. Simplified 97.9%

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

   if -1e5 < (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64)))) < 20

   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

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

      1. metadata-eval N/A

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

      2. lft-mult-inverse N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. remove-double-neg N/A

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

      11. distribute-lft-neg-out N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

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

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

      19. associate-*r* N/A

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

   5. Simplified 97.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \leq -100000:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -2\right)\\ \mathbf{elif}\;\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \leq 20:\\ \;\;\;\;x \cdot \left(3 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \leq -100000:\\ \;\;\;\;\left(x \cdot x\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(3 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (* x x) (- 3.0 (* x 2.0))) -100000.0)
   (* (* x x) -3.0)
   (* x (* 3.0 x))))

C

double code(double x) {
	double tmp;
	if (((x * x) * (3.0 - (x * 2.0))) <= -100000.0) {
		tmp = (x * x) * -3.0;
	} else {
		tmp = x * (3.0 * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x * x) * (3.0d0 - (x * 2.0d0))) <= (-100000.0d0)) then
        tmp = (x * x) * (-3.0d0)
    else
        tmp = x * (3.0d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x * x) * (3.0 - (x * 2.0))) <= -100000.0) {
		tmp = (x * x) * -3.0;
	} else {
		tmp = x * (3.0 * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x * x) * (3.0 - (x * 2.0))) <= -100000.0:
		tmp = (x * x) * -3.0
	else:
		tmp = x * (3.0 * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x * x) * Float64(3.0 - Float64(x * 2.0))) <= -100000.0)
		tmp = Float64(Float64(x * x) * -3.0);
	else
		tmp = Float64(x * Float64(3.0 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x * x) * (3.0 - (x * 2.0))) <= -100000.0)
		tmp = (x * x) * -3.0;
	else
		tmp = x * (3.0 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -100000.0], N[(N[(x * x), $MachinePrecision] * -3.0), $MachinePrecision], N[(x * N[(3.0 * x), $MachinePrecision]), $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)))) < -1e5

   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. Simplified 0.3%

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

      1. metadata-eval N/A

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

      2. div-inv N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{\frac{1}{3}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{\frac{1}{3}}} \]

      4. *-lowering-*.f64 0.3

         \[\leadsto \frac{\color{blue}{x \cdot x}}{0.3333333333333333} \]

   7. Applied egg-rr 0.3%

      \[\leadsto \color{blue}{\frac{x \cdot x}{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. frac-2neg N/A

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

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

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

      3. associate-/l* N/A

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

   9. Applied egg-rr 48.8%

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

   if -1e5 < (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64))))

   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

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

      1. metadata-eval N/A

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

      2. lft-mult-inverse N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. remove-double-neg N/A

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

      11. distribute-lft-neg-out N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

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

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

      19. associate-*r* N/A

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

   5. Simplified 86.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \leq -100000:\\ \;\;\;\;\left(x \cdot x\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(3 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-lowering-*.f64 99.9

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 5: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 6: 61.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(3.0 * 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

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

   1. metadata-eval N/A

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

   2. lft-mult-inverse N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. associate-*l* N/A

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

   7. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. remove-double-neg N/A

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

   11. distribute-lft-neg-out N/A

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

   12. *-commutative N/A

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

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

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

   14. mul-1-neg N/A

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 N/A

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

   17. *-commutative N/A

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

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

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

   19. associate-*r* N/A

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

5. Simplified 64.2%

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

6. Final simplification 64.2%

   \[\leadsto x \cdot \left(3 \cdot x\right) \]
7. 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 2024199 
(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))))