Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 9.6s

Alternatives: 7

Speedup: 1.1×

• 34.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (3.0d0 - (x * 2.0d0))
end function

Java

public static double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

Python

def code(x):
	return (x * x) * (3.0 - (x * 2.0))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (x * x) * (3.0 - (x * 2.0));
end

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (3.0d0 - (x * 2.0d0))
end function

Java

public static double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

Python

def code(x):
	return (x * x) * (3.0 - (x * 2.0))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (x * x) * (3.0 - (x * 2.0));
end

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

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

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

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

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

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

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

   7. metadata-eval N/A

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

   8. *-lowering-*.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 97.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = (x * x) * (x * -2.0);
	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 = (x * x) * (x * (-2.0d0))
    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 = (x * x) * (x * -2.0);
	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 = (x * x) * (x * -2.0)
	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(x * x) * Float64(x * -2.0))
	tmp = 0.0
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 1.5)
		tmp = Float64(3.0 * Float64(x * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (x * -2.0);
	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[(x * x), $MachinePrecision] * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5], t$95$0, If[LessEqual[x, 1.5], N[(3.0 * N[(x * x), $MachinePrecision]), $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. *-lowering-*.f64 97.2

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

   5. Simplified 97.2%

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

   if -1.5 < 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. Simplified 96.6%

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

4. Final simplification 96.9%

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

Alternative 3: 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 99.9%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
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 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 4: 61.3% 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 99.9%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
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 64.5%

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

6. Final simplification 64.5%

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

Alternative 5: 61.4% accurate, 1.7× speedup

Math

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

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(x * Float64(x * 3.0))
end

MATLAB

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

Wolfram

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

Derivation

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 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 64.5%

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

Alternative 6: 5.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, -13.5, 6.75\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x -13.5 6.75))

C

double code(double x) {
	return fma(x, -13.5, 6.75);
}

Julia

function code(x)
	return fma(x, -13.5, 6.75)
end

Wolfram

code[x_] := N[(x * -13.5 + 6.75), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
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. +-commutative N/A

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

   3. neg-sub0 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\left(0 - x \cdot 2\right)} + 3\right) \]

   4. associate-+l- N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(0 - \left(x \cdot 2 - 3\right)\right)} \]

   5. flip3-- N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\frac{{0}^{3} - {\left(x \cdot 2 - 3\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)}} \]

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

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\frac{{0}^{3} - {\left(x \cdot 2 - 3\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{\color{blue}{0} - {\left(x \cdot 2 - 3\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

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

      \[\leadsto \left(x \cdot x\right) \cdot \frac{\color{blue}{0 - {\left(x \cdot 2 - 3\right)}^{3}}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - \color{blue}{{\left(x \cdot 2 - 3\right)}^{3}}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

   10. sub-neg N/A

       \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\color{blue}{\left(x \cdot 2 + \left(\mathsf{neg}\left(3\right)\right)\right)}}^{3}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

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

       \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\color{blue}{\left(\mathsf{fma}\left(x, 2, \mathsf{neg}\left(3\right)\right)\right)}}^{3}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

   12. metadata-eval N/A

       \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, \color{blue}{-3}\right)\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

   13. metadata-eval N/A

       \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{\color{blue}{0} + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{\color{blue}{0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)}} \]

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

       \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{\mathsf{fma}\left(x \cdot 2 - 3, x \cdot 2 - 3, 0 \cdot \left(x \cdot 2 - 3\right)\right)}} \]

4. Applied egg-rr 70.2%

   \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -3\right), \mathsf{fma}\left(x, 2, -3\right), 0 \cdot \mathsf{fma}\left(x, 2, -3\right)\right)}} \]

5. Taylor expanded in x around inf

   \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{4 \cdot {x}^{2}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{{x}^{2} \cdot 4}} \]

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

      \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{{x}^{2} \cdot 4}} \]

   3. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{\left(x \cdot x\right)} \cdot 4} \]

   4. *-lowering-*.f64 24.2

      \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{\left(x \cdot x\right)} \cdot 4} \]

7. Simplified 24.2%

   \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{\left(x \cdot x\right) \cdot 4}} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{27}{4} + \frac{-27}{2} \cdot x} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-27}{2} \cdot x + \frac{27}{4}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{-27}{2}} + \frac{27}{4} \]

   3. accelerator-lowering-fma.f64 5.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -13.5, 6.75\right)} \]

10. Simplified 5.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, -13.5, 6.75\right)} \]
11. Add Preprocessing

Alternative 7: 3.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 6.75 \end{array} \]

FPCore

(FPCore (x) :precision binary64 6.75)

C

double code(double x) {
	return 6.75;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.75d0
end function

Java

public static double code(double x) {
	return 6.75;
}

Python

def code(x):
	return 6.75

Julia

function code(x)
	return 6.75
end

MATLAB

function tmp = code(x)
	tmp = 6.75;
end

Wolfram

code[x_] := 6.75

Derivation

1. Initial program 99.9%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
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. +-commutative N/A

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

   3. neg-sub0 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\left(0 - x \cdot 2\right)} + 3\right) \]

   4. associate-+l- N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(0 - \left(x \cdot 2 - 3\right)\right)} \]

   5. flip3-- N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\frac{{0}^{3} - {\left(x \cdot 2 - 3\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)}} \]

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

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\frac{{0}^{3} - {\left(x \cdot 2 - 3\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{\color{blue}{0} - {\left(x \cdot 2 - 3\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

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

      \[\leadsto \left(x \cdot x\right) \cdot \frac{\color{blue}{0 - {\left(x \cdot 2 - 3\right)}^{3}}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - \color{blue}{{\left(x \cdot 2 - 3\right)}^{3}}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

   10. sub-neg N/A

       \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\color{blue}{\left(x \cdot 2 + \left(\mathsf{neg}\left(3\right)\right)\right)}}^{3}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

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

       \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\color{blue}{\left(\mathsf{fma}\left(x, 2, \mathsf{neg}\left(3\right)\right)\right)}}^{3}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

   12. metadata-eval N/A

       \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, \color{blue}{-3}\right)\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

   13. metadata-eval N/A

       \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{\color{blue}{0} + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)} \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{\color{blue}{0 + \left(\left(x \cdot 2 - 3\right) \cdot \left(x \cdot 2 - 3\right) + 0 \cdot \left(x \cdot 2 - 3\right)\right)}} \]

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

       \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{\mathsf{fma}\left(x \cdot 2 - 3, x \cdot 2 - 3, 0 \cdot \left(x \cdot 2 - 3\right)\right)}} \]

4. Applied egg-rr 70.2%

   \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -3\right), \mathsf{fma}\left(x, 2, -3\right), 0 \cdot \mathsf{fma}\left(x, 2, -3\right)\right)}} \]

5. Taylor expanded in x around inf

   \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{4 \cdot {x}^{2}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{{x}^{2} \cdot 4}} \]

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

      \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{{x}^{2} \cdot 4}} \]

   3. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{\left(x \cdot x\right)} \cdot 4} \]

   4. *-lowering-*.f64 24.2

      \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{\left(x \cdot x\right)} \cdot 4} \]

7. Simplified 24.2%

   \[\leadsto \left(x \cdot x\right) \cdot \frac{0 - {\left(\mathsf{fma}\left(x, 2, -3\right)\right)}^{3}}{0 + \color{blue}{\left(x \cdot x\right) \cdot 4}} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{27}{4}} \]
9. Step-by-step derivation

10. Simplified 3.5%

    \[\leadsto \color{blue}{6.75} \]
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 2024204 
(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))))