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

Percentage Accurate: 99.7% → 99.8%

Time: 10.7s

Alternatives: 6

Speedup: 1.6×

• 24.9% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \leq -20:\\ \;\;\;\;\left(-9 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (* 3.0 (- 2.0 (* x 3.0))) x) -20.0) (* (* -9.0 x) x) (* 6.0 x)))

C

double code(double x) {
	double tmp;
	if (((3.0 * (2.0 - (x * 3.0))) * x) <= -20.0) {
		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) <= (-20.0d0)) 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) <= -20.0) {
		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) <= -20.0:
		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) <= -20.0)
		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) <= -20.0)
		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], -20.0], 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) < -20

   1. Initial program 99.6%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   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 -20 < (*.f64 (*.f64 #s(literal 3 binary64) (-.f64 #s(literal 2 binary64) (*.f64 x #s(literal 3 binary64)))) x)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.9%

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

Alternative 3: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \leq -20:\\ \;\;\;\;\left(x \cdot x\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;6 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (* 3.0 (- 2.0 (* x 3.0))) x) -20.0) (* (* x x) -9.0) (* 6.0 x)))

C

double code(double x) {
	double tmp;
	if (((3.0 * (2.0 - (x * 3.0))) * x) <= -20.0) {
		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) <= (-20.0d0)) 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) <= -20.0) {
		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) <= -20.0:
		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) <= -20.0)
		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) <= -20.0)
		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], -20.0], 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) < -20

   1. Initial program 99.6%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   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 96.9

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

   5. Applied rewrites 96.9%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.9%

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

Alternative 4: 53.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;6 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.68) (* 6.0 x) (* -6.0 x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = 6.0 * x;
	} else {
		tmp = -6.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = 6.0d0 * x
    else
        tmp = (-6.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = 6.0 * x;
	} else {
		tmp = -6.0 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.68:
		tmp = 6.0 * x
	else:
		tmp = -6.0 * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64(6.0 * x);
	else
		tmp = Float64(-6.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = 6.0 * x;
	else
		tmp = -6.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.68], N[(6.0 * x), $MachinePrecision], N[(-6.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   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 62.9%

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

   if 0.680000000000000049 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 0.0%

      \[\leadsto \left(3 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{9 \cdot \left(-x\right)}, \sqrt{-x}, 2\right)}\right) \cdot x \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \left(3 \cdot \color{blue}{\left(\sqrt{9 \cdot \left(-x\right)} \cdot \sqrt{-x} + 2\right)}\right) \cdot x \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{9 \cdot \left(-x\right)} \cdot \sqrt{-x}\right) \cdot 3 + 2 \cdot 3\right)} \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\sqrt{9 \cdot \left(-x\right)} \cdot \left(\sqrt{-x} \cdot 3\right)} + 2 \cdot 3\right) \cdot x \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{9 \cdot \left(-x\right)}} \cdot \left(\sqrt{-x} \cdot 3\right) + 2 \cdot 3\right) \cdot x \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{9 \cdot \left(-x\right)} \cdot \left(\color{blue}{\sqrt{-x}} \cdot 3\right) + 2 \cdot 3\right) \cdot x \]

      7. metadata-eval N/A

         \[\leadsto \left(\sqrt{9 \cdot \left(-x\right)} \cdot \left(\sqrt{-x} \cdot \color{blue}{\sqrt{9}}\right) + 2 \cdot 3\right) \cdot x \]

      8. sqrt-prod N/A

         \[\leadsto \left(\sqrt{9 \cdot \left(-x\right)} \cdot \color{blue}{\sqrt{\left(-x\right) \cdot 9}} + 2 \cdot 3\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{9 \cdot \left(-x\right)} \cdot \sqrt{\color{blue}{9 \cdot \left(-x\right)}} + 2 \cdot 3\right) \cdot x \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{9 \cdot \left(-x\right)} \cdot \sqrt{\color{blue}{9 \cdot \left(-x\right)}} + 2 \cdot 3\right) \cdot x \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\color{blue}{9 \cdot \left(-x\right)} + 2 \cdot 3\right) \cdot x \]

      12. metadata-eval N/A

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

      13. lift-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

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

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

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

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.9

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

   6. Applied rewrites 99.9%

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

   8. Applied rewrites 95.4%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 7.4

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

   11. Applied rewrites 7.4%

       \[\leadsto \color{blue}{-6 \cdot x} \]
3. Recombined 2 regimes into one program.
4. 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 99.8

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

5. Applied rewrites 99.8%

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

Alternative 6: 3.8% 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

4. Applied rewrites 49.8%

   \[\leadsto \left(3 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{9 \cdot \left(-x\right)}, \sqrt{-x}, 2\right)}\right) \cdot x \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

      \[\leadsto \left(3 \cdot \color{blue}{\left(\sqrt{9 \cdot \left(-x\right)} \cdot \sqrt{-x} + 2\right)}\right) \cdot x \]

   3. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{9 \cdot \left(-x\right)} \cdot \sqrt{-x}\right) \cdot 3 + 2 \cdot 3\right)} \cdot x \]

   4. associate-*l* N/A

      \[\leadsto \left(\color{blue}{\sqrt{9 \cdot \left(-x\right)} \cdot \left(\sqrt{-x} \cdot 3\right)} + 2 \cdot 3\right) \cdot x \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{9 \cdot \left(-x\right)}} \cdot \left(\sqrt{-x} \cdot 3\right) + 2 \cdot 3\right) \cdot x \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{9 \cdot \left(-x\right)} \cdot \left(\color{blue}{\sqrt{-x}} \cdot 3\right) + 2 \cdot 3\right) \cdot x \]

   7. metadata-eval N/A

      \[\leadsto \left(\sqrt{9 \cdot \left(-x\right)} \cdot \left(\sqrt{-x} \cdot \color{blue}{\sqrt{9}}\right) + 2 \cdot 3\right) \cdot x \]

   8. sqrt-prod N/A

      \[\leadsto \left(\sqrt{9 \cdot \left(-x\right)} \cdot \color{blue}{\sqrt{\left(-x\right) \cdot 9}} + 2 \cdot 3\right) \cdot x \]

   9. *-commutative N/A

      \[\leadsto \left(\sqrt{9 \cdot \left(-x\right)} \cdot \sqrt{\color{blue}{9 \cdot \left(-x\right)}} + 2 \cdot 3\right) \cdot x \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\sqrt{9 \cdot \left(-x\right)} \cdot \sqrt{\color{blue}{9 \cdot \left(-x\right)}} + 2 \cdot 3\right) \cdot x \]

   11. rem-square-sqrt N/A

       \[\leadsto \left(\color{blue}{9 \cdot \left(-x\right)} + 2 \cdot 3\right) \cdot x \]

   12. metadata-eval N/A

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

   13. lift-*.f64 N/A

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

   14. lift-neg.f64 N/A

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

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

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

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

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

   17. metadata-eval N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f64 99.8

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 56.8%

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

9. Taylor expanded in x around 0

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

    1. lower-*.f64 4.0

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

11. Applied rewrites 4.0%

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

Developer Target 1: 99.7% 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 2024333 
(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))