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

Percentage Accurate: 99.7% → 99.8%

Time: 8.5s

Alternatives: 6

Speedup: 1.6×

• 25.9% 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, 1.1× 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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

   5. associate-*l* N/A

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

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

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

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

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

   8. metadata-eval N/A

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

   9. swap-sqr N/A

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

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

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

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

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

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

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

   13. metadata-eval N/A

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

   14. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right) \leq -10:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (* 3.0 (- 2.0 (* x 3.0)))) -10.0) (* x (* x -9.0)) (* x 6.0)))

C

double code(double x) {
	double tmp;
	if ((x * (3.0 * (2.0 - (x * 3.0)))) <= -10.0) {
		tmp = x * (x * -9.0);
	} else {
		tmp = x * 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * (3.0d0 * (2.0d0 - (x * 3.0d0)))) <= (-10.0d0)) then
        tmp = x * (x * (-9.0d0))
    else
        tmp = x * 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * (3.0 * (2.0 - (x * 3.0)))) <= -10.0) {
		tmp = x * (x * -9.0);
	} else {
		tmp = x * 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * (3.0 * (2.0 - (x * 3.0)))) <= -10.0:
		tmp = x * (x * -9.0)
	else:
		tmp = x * 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * Float64(3.0 * Float64(2.0 - Float64(x * 3.0)))) <= -10.0)
		tmp = Float64(x * Float64(x * -9.0));
	else
		tmp = Float64(x * 6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * (3.0 * (2.0 - (x * 3.0)))) <= -10.0)
		tmp = x * (x * -9.0);
	else
		tmp = x * 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -10.0], N[(x * N[(x * -9.0), $MachinePrecision]), $MachinePrecision], N[(x * 6.0), $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) < -10

   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 inf

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

      1. *-lowering-*.f64 96.7

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

   5. Simplified 96.7%

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

   if -10 < (*.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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. lft-mult-inverse N/A

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

      4. associate-*l* N/A

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

      5. remove-double-neg N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

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

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

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

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

      11. associate-*l* N/A

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

      12. distribute-neg-frac2 N/A

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

      13. mul-1-neg N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval 97.0

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

   5. Simplified 97.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right) \leq -10:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right) \leq -10:\\ \;\;\;\;\left(x \cdot x\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (* 3.0 (- 2.0 (* x 3.0)))) -10.0) (* (* x x) -9.0) (* x 6.0)))

C

double code(double x) {
	double tmp;
	if ((x * (3.0 * (2.0 - (x * 3.0)))) <= -10.0) {
		tmp = (x * x) * -9.0;
	} else {
		tmp = x * 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * (3.0d0 * (2.0d0 - (x * 3.0d0)))) <= (-10.0d0)) then
        tmp = (x * x) * (-9.0d0)
    else
        tmp = x * 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * (3.0 * (2.0 - (x * 3.0)))) <= -10.0) {
		tmp = (x * x) * -9.0;
	} else {
		tmp = x * 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * (3.0 * (2.0 - (x * 3.0)))) <= -10.0:
		tmp = (x * x) * -9.0
	else:
		tmp = x * 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * Float64(3.0 * Float64(2.0 - Float64(x * 3.0)))) <= -10.0)
		tmp = Float64(Float64(x * x) * -9.0);
	else
		tmp = Float64(x * 6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * (3.0 * (2.0 - (x * 3.0)))) <= -10.0)
		tmp = (x * x) * -9.0;
	else
		tmp = x * 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -10.0], N[(N[(x * x), $MachinePrecision] * -9.0), $MachinePrecision], N[(x * 6.0), $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) < -10

   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 inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 96.6

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

   5. Simplified 96.6%

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

   if -10 < (*.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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. lft-mult-inverse N/A

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

      4. associate-*l* N/A

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

      5. remove-double-neg N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

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

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

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

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

      11. associate-*l* N/A

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

      12. distribute-neg-frac2 N/A

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

      13. mul-1-neg N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval 97.0

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

   5. Simplified 97.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right) \leq -10:\\ \;\;\;\;\left(x \cdot x\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x * N[(x * -9.0 + 6.0), $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. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

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

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

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

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 5: 49.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 6.0

Julia

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

MATLAB

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

Wolfram

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. lft-mult-inverse N/A

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

   4. associate-*l* N/A

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

   5. remove-double-neg N/A

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

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

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

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

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

   8. mul-1-neg N/A

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

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

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

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

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

   11. associate-*l* N/A

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

   12. distribute-neg-frac2 N/A

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

   13. mul-1-neg N/A

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

   14. lft-mult-inverse N/A

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

   15. metadata-eval 46.8

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

5. Simplified 46.8%

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

Alternative 6: 2.3% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x) :precision binary64 4.0)

C

double code(double x) {
	return 4.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 4.0d0
end function

Java

public static double code(double x) {
	return 4.0;
}

Python

def code(x):
	return 4.0

Julia

function code(x)
	return 4.0
end

MATLAB

function tmp = code(x)
	tmp = 4.0;
end

Wolfram

code[x_] := 4.0

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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

   5. associate-*l* N/A

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

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

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

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

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

   8. metadata-eval N/A

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

   9. swap-sqr N/A

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

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

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

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

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

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

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

   13. metadata-eval N/A

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

   14. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

   1. associate-*l* N/A

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

   2. distribute-lft-in N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. associate-*r/ N/A

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

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

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

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

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

   8. sub-neg N/A

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

   9. swap-sqr N/A

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

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

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

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

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

   12. metadata-eval N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{81}, \mathsf{neg}\left(6 \cdot 6\right)\right)}{x \cdot -9 - 6} \]

   13. metadata-eval N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x \cdot x, 81, \mathsf{neg}\left(\color{blue}{36}\right)\right)}{x \cdot -9 - 6} \]

   14. metadata-eval N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x \cdot x, 81, \color{blue}{-36}\right)}{x \cdot -9 - 6} \]

   15. sub-neg N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x \cdot x, 81, -36\right)}{\color{blue}{x \cdot -9 + \left(\mathsf{neg}\left(6\right)\right)}} \]

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

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x \cdot x, 81, -36\right)}{\color{blue}{\mathsf{fma}\left(x, -9, \mathsf{neg}\left(6\right)\right)}} \]

   17. metadata-eval 91.5

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x \cdot x, 81, -36\right)}{\mathsf{fma}\left(x, -9, \color{blue}{-6}\right)} \]

6. Applied egg-rr 91.5%

   \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(x \cdot x, 81, -36\right)}{\mathsf{fma}\left(x, -9, -6\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{-36 \cdot x}}{\mathsf{fma}\left(x, -9, -6\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot -36}}{\mathsf{fma}\left(x, -9, -6\right)} \]

   2. *-lowering-*.f64 45.4

      \[\leadsto \frac{\color{blue}{x \cdot -36}}{\mathsf{fma}\left(x, -9, -6\right)} \]

9. Simplified 45.4%

   \[\leadsto \frac{\color{blue}{x \cdot -36}}{\mathsf{fma}\left(x, -9, -6\right)} \]

10. Taylor expanded in x around inf

    \[\leadsto \color{blue}{4} \]
11. Step-by-step derivation

12. Simplified 2.2%

    \[\leadsto \color{blue}{4} \]
13. 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 2024204 
(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))