Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E

Percentage Accurate: 99.7% → 99.8%

Time: 5.4s

Alternatives: 7

Speedup: 1.6×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 6, \left(9 \cdot \left(-x\right)\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 6.0 (* (* 9.0 (- x)) x)))

C

double code(double x) {
	return fma(x, 6.0, ((9.0 * -x) * x));
}

Julia

function code(x)
	return fma(x, 6.0, Float64(Float64(9.0 * Float64(-x)) * x))
end

Wolfram

code[x_] := N[(x * 6.0 + N[(N[(9.0 * (-x)), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. fp-cancel-sub-sign-inv N/A

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

   9. distribute-rgt-in N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval N/A

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

   17. lift-*.f64 N/A

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

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

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. associate-*r* N/A

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

   22. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 6, \left(9 \cdot \left(-x\right)\right) \cdot x\right)} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \leq -2000:\\ \;\;\;\;\left(-9 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (* 3.0 (- 2.0 (* x 3.0))) x) -2000.0) (* (* -9.0 x) x) (* 6.0 x)))

C

double code(double x) {
	double tmp;
	if (((3.0 * (2.0 - (x * 3.0))) * x) <= -2000.0) {
		tmp = (-9.0 * x) * x;
	} else {
		tmp = 6.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((3.0d0 * (2.0d0 - (x * 3.0d0))) * x) <= (-2000.0d0)) then
        tmp = ((-9.0d0) * x) * x
    else
        tmp = 6.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((3.0 * (2.0 - (x * 3.0))) * x) <= -2000.0) {
		tmp = (-9.0 * x) * x;
	} else {
		tmp = 6.0 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((3.0 * (2.0 - (x * 3.0))) * x) <= -2000.0:
		tmp = (-9.0 * x) * x
	else:
		tmp = 6.0 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(3.0 * Float64(2.0 - Float64(x * 3.0))) * x) <= -2000.0)
		tmp = Float64(Float64(-9.0 * x) * x);
	else
		tmp = Float64(6.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((3.0 * (2.0 - (x * 3.0))) * x) <= -2000.0)
		tmp = (-9.0 * x) * x;
	else
		tmp = 6.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], -2000.0], N[(N[(-9.0 * x), $MachinePrecision] * x), $MachinePrecision], N[(6.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (-.f64 #s(literal 2 binary64) (*.f64 x #s(literal 3 binary64)))) x) < -2e3

   1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -2e3 < (*.f64 (*.f64 #s(literal 3 binary64) (-.f64 #s(literal 2 binary64) (*.f64 x #s(literal 3 binary64)))) x)

   1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{6} \cdot x \]
   4. Step-by-step derivation

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{6} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \leq -2000:\\ \;\;\;\;\left(x \cdot x\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;6 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (* 3.0 (- 2.0 (* x 3.0))) x) -2000.0) (* (* x x) -9.0) (* 6.0 x)))

C

double code(double x) {
	double tmp;
	if (((3.0 * (2.0 - (x * 3.0))) * x) <= -2000.0) {
		tmp = (x * x) * -9.0;
	} else {
		tmp = 6.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((3.0d0 * (2.0d0 - (x * 3.0d0))) * x) <= (-2000.0d0)) then
        tmp = (x * x) * (-9.0d0)
    else
        tmp = 6.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((3.0 * (2.0 - (x * 3.0))) * x) <= -2000.0) {
		tmp = (x * x) * -9.0;
	} else {
		tmp = 6.0 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((3.0 * (2.0 - (x * 3.0))) * x) <= -2000.0:
		tmp = (x * x) * -9.0
	else:
		tmp = 6.0 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(3.0 * Float64(2.0 - Float64(x * 3.0))) * x) <= -2000.0)
		tmp = Float64(Float64(x * x) * -9.0);
	else
		tmp = Float64(6.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((3.0 * (2.0 - (x * 3.0))) * x) <= -2000.0)
		tmp = (x * x) * -9.0;
	else
		tmp = 6.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], -2000.0], N[(N[(x * x), $MachinePrecision] * -9.0), $MachinePrecision], N[(6.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (-.f64 #s(literal 2 binary64) (*.f64 x #s(literal 3 binary64)))) x) < -2e3

   1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot -9} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot -9} \]

      3. unpow2 N/A

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

      4. lower-*.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -2e3 < (*.f64 (*.f64 #s(literal 3 binary64) (-.f64 #s(literal 2 binary64) (*.f64 x #s(literal 3 binary64)))) x)

   1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{6} \cdot x \]
   4. Step-by-step derivation

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{6} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 6, \left(x \cdot x\right) \cdot -9\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 6.0 (* (* x x) -9.0)))

C

double code(double x) {
	return fma(x, 6.0, ((x * x) * -9.0));
}

Julia

function code(x)
	return fma(x, 6.0, Float64(Float64(x * x) * -9.0))
end

Wolfram

code[x_] := N[(x * 6.0 + N[(N[(x * x), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. fp-cancel-sub-sign-inv N/A

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

   9. distribute-rgt-in N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval N/A

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

   17. lift-*.f64 N/A

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

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

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. associate-*r* N/A

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

   22. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

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

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 6, \left(x \cdot x\right) \cdot -9\right)} \]
7. Add Preprocessing

Alternative 5: 99.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-9, x, 6\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fma -9.0 x 6.0) x))

C

double code(double x) {
	return fma(-9.0, x, 6.0) * x;
}

Julia

function code(x)
	return Float64(fma(-9.0, x, 6.0) * x)
end

Wolfram

code[x_] := N[(N[(-9.0 * x + 6.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(6 + -9 \cdot x\right)} \cdot x \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(-9 \cdot x + 6\right)} \cdot x \]

   2. lower-fma.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 6: 50.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ 6 \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 6.0 x))

C

double code(double x) {
	return 6.0 * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0 * x
end function

Java

public static double code(double x) {
	return 6.0 * x;
}

Python

def code(x):
	return 6.0 * x

Julia

function code(x)
	return Float64(6.0 * x)
end

MATLAB

function tmp = code(x)
	tmp = 6.0 * x;
end

Wolfram

code[x_] := N[(6.0 * x), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{6} \cdot x \]
4. Step-by-step derivation

5. Applied rewrites 53.9%

   \[\leadsto \color{blue}{6} \cdot x \]
6. Add Preprocessing

Alternative 7: 3.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ -6 \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* -6.0 x))

C

double code(double x) {
	return -6.0 * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-6.0d0) * x
end function

Java

public static double code(double x) {
	return -6.0 * x;
}

Python

def code(x):
	return -6.0 * x

Julia

function code(x)
	return Float64(-6.0 * x)
end

MATLAB

function tmp = code(x)
	tmp = -6.0 * x;
end

Wolfram

code[x_] := N[(-6.0 * x), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. fp-cancel-sub-sign-inv N/A

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

   9. distribute-rgt-in N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval N/A

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

   17. lift-*.f64 N/A

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

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

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. associate-*r* N/A

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

   22. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 47.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-6 \cdot x} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(6\right)\right)} \cdot x \]

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

      \[\leadsto \color{blue}{\mathsf{neg}\left(6 \cdot x\right)} \]

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

      \[\leadsto \color{blue}{6 \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

   4. metadata-eval N/A

      \[\leadsto \color{blue}{\left(6 \cdot 1\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   5. lft-mult-inverse N/A

      \[\leadsto \left(6 \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\left(6 \cdot \frac{1}{x}\right) \cdot x\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

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

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(6 \cdot \frac{1}{x}\right) \cdot x\right) \cdot x\right)} \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(6 \cdot \frac{1}{x}\right) \cdot x\right)\right) \cdot x} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(6 \cdot \frac{1}{x}\right) \cdot x\right)\right) \cdot x} \]

   10. associate-*l* N/A

       \[\leadsto \left(\mathsf{neg}\left(\color{blue}{6 \cdot \left(\frac{1}{x} \cdot x\right)}\right)\right) \cdot x \]

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

       \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(6\right)\right) \cdot \left(\frac{1}{x} \cdot x\right)\right)} \cdot x \]

   12. metadata-eval N/A

       \[\leadsto \left(\color{blue}{-6} \cdot \left(\frac{1}{x} \cdot x\right)\right) \cdot x \]

   13. lft-mult-inverse N/A

       \[\leadsto \left(-6 \cdot \color{blue}{1}\right) \cdot x \]

   14. metadata-eval 3.6

       \[\leadsto \color{blue}{-6} \cdot x \]

9. Applied rewrites 3.6%

   \[\leadsto \color{blue}{-6 \cdot x} \]
10. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 6 \cdot x - 9 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* 6.0 x) (* 9.0 (* x x))))

C

double code(double x) {
	return (6.0 * x) - (9.0 * (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (6.0d0 * x) - (9.0d0 * (x * x))
end function

Java

public static double code(double x) {
	return (6.0 * x) - (9.0 * (x * x));
}

Python

def code(x):
	return (6.0 * x) - (9.0 * (x * x))

Julia

function code(x)
	return Float64(Float64(6.0 * x) - Float64(9.0 * Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * x) - (9.0 * (x * x));
end

Wolfram

code[x_] := N[(N[(6.0 * x), $MachinePrecision] - N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024332 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* 6 x) (* 9 (* x x))))

  (* (* 3.0 (- 2.0 (* x 3.0))) x))