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

Percentage Accurate: 99.7% → 99.8%

Time: 7.8s

Alternatives: 4

Speedup: 1.3×

• 25.7% 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, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot \left(2 + \left(\mathsf{neg}\left(x \cdot 3\right)\right)\right)\right), x\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot 3 + \left(\mathsf{neg}\left(x \cdot 3\right)\right) \cdot 3\right), x\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(6 + \left(\mathsf{neg}\left(x \cdot 3\right)\right) \cdot 3\right), x\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot 2 + \left(\mathsf{neg}\left(x \cdot 3\right)\right) \cdot 3\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot 2 + \left(\mathsf{neg}\left(\left(x \cdot 3\right) \cdot 3\right)\right)\right), x\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot 2 - \left(x \cdot 3\right) \cdot 3\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(3 \cdot 2\right), \left(\left(x \cdot 3\right) \cdot 3\right)\right), x\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(6, \left(\left(x \cdot 3\right) \cdot 3\right)\right), x\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(6, \left(x \cdot \left(3 \cdot 3\right)\right)\right), x\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(6, \mathsf{*.f64}\left(x, \left(3 \cdot 3\right)\right)\right), x\right) \]

   11. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(6, \mathsf{*.f64}\left(x, 9\right)\right), x\right) \]

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

   \[\leadsto x \cdot \left(6 - x \cdot 9\right) \]
6. Add Preprocessing

Alternative 2: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;6 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.65)
   (* -9.0 (* x x))
   (if (<= x 0.65) (* 6.0 x) (* x (* x -9.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = -9.0 * (x * x);
	} else if (x <= 0.65) {
		tmp = 6.0 * x;
	} else {
		tmp = x * (x * -9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.65d0)) then
        tmp = (-9.0d0) * (x * x)
    else if (x <= 0.65d0) then
        tmp = 6.0d0 * x
    else
        tmp = x * (x * (-9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = -9.0 * (x * x);
	} else if (x <= 0.65) {
		tmp = 6.0 * x;
	} else {
		tmp = x * (x * -9.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.65:
		tmp = -9.0 * (x * x)
	elif x <= 0.65:
		tmp = 6.0 * x
	else:
		tmp = x * (x * -9.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.65)
		tmp = Float64(-9.0 * Float64(x * x));
	elseif (x <= 0.65)
		tmp = Float64(6.0 * x);
	else
		tmp = Float64(x * Float64(x * -9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.65)
		tmp = -9.0 * (x * x);
	elseif (x <= 0.65)
		tmp = 6.0 * x;
	else
		tmp = x * (x * -9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.65], N[(-9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.65], N[(6.0 * x), $MachinePrecision], N[(x * N[(x * -9.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.650000000000000022

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-9, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-9, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 97.4%

         \[\leadsto \mathsf{*.f64}\left(-9, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 97.4%

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

   if -0.650000000000000022 < x < 0.650000000000000022

   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

      1. metadata-eval N/A

         \[\leadsto \left(1 \cdot 6\right) \cdot x \]

      2. lft-mult-inverse N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. remove-double-neg N/A

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

      9. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(6 \cdot \frac{1}{x}\right)\right)\right)\right) \cdot {x}^{\color{blue}{2}} \]

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

          \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(6 \cdot \frac{1}{x}\right)\right) \cdot {x}^{2}\right) \]

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

          \[\leadsto \left(\mathsf{neg}\left(6 \cdot \frac{1}{x}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)} \]

      12. unpow2 N/A

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

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

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

      14. mul-1-neg N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(6 \cdot \frac{1}{x}\right)\right) \cdot \left(-1 \cdot x\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(6 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right) \cdot \left(\color{blue}{-1} \cdot x\right)\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(6 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(-1 \cdot x\right)\right)}\right)\right) \]

      20. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(6 \cdot \left(\frac{1}{\mathsf{neg}\left(x\right)} \cdot \left(\color{blue}{-1} \cdot x\right)\right)\right)\right) \]

      21. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(6 \cdot \left(\frac{1}{-1 \cdot x} \cdot \left(-1 \cdot x\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(6 \cdot 1\right)\right) \]

      23. metadata-eval 97.5%

          \[\leadsto \mathsf{*.f64}\left(x, 6\right) \]

   5. Simplified 97.5%

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

   if 0.650000000000000022 < 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 inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-9 \cdot x\right)}, x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 97.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-9, x\right), x\right) \]

   5. Simplified 97.4%

      \[\leadsto \color{blue}{\left(-9 \cdot x\right)} \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;6 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -9 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;6 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* -9.0 (* x x))))
   (if (<= x -0.65) t_0 (if (<= x 0.65) (* 6.0 x) t_0))))

C

double code(double x) {
	double t_0 = -9.0 * (x * x);
	double tmp;
	if (x <= -0.65) {
		tmp = t_0;
	} else if (x <= 0.65) {
		tmp = 6.0 * 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 = (-9.0d0) * (x * x)
    if (x <= (-0.65d0)) then
        tmp = t_0
    else if (x <= 0.65d0) then
        tmp = 6.0d0 * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -9.0 * (x * x);
	double tmp;
	if (x <= -0.65) {
		tmp = t_0;
	} else if (x <= 0.65) {
		tmp = 6.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = -9.0 * (x * x)
	tmp = 0
	if x <= -0.65:
		tmp = t_0
	elif x <= 0.65:
		tmp = 6.0 * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(-9.0 * Float64(x * x))
	tmp = 0.0
	if (x <= -0.65)
		tmp = t_0;
	elseif (x <= 0.65)
		tmp = Float64(6.0 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -9.0 * (x * x);
	tmp = 0.0;
	if (x <= -0.65)
		tmp = t_0;
	elseif (x <= 0.65)
		tmp = 6.0 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.65], t$95$0, If[LessEqual[x, 0.65], N[(6.0 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.650000000000000022 or 0.650000000000000022 < 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 inf

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

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

         \[\leadsto \mathsf{*.f64}\left(-9, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-9, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 97.3%

         \[\leadsto \mathsf{*.f64}\left(-9, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 97.3%

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

   if -0.650000000000000022 < x < 0.650000000000000022

   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

      1. metadata-eval N/A

         \[\leadsto \left(1 \cdot 6\right) \cdot x \]

      2. lft-mult-inverse N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. remove-double-neg N/A

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

      9. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(6 \cdot \frac{1}{x}\right)\right)\right)\right) \cdot {x}^{\color{blue}{2}} \]

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

          \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(6 \cdot \frac{1}{x}\right)\right) \cdot {x}^{2}\right) \]

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

          \[\leadsto \left(\mathsf{neg}\left(6 \cdot \frac{1}{x}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)} \]

      12. unpow2 N/A

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

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

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

      14. mul-1-neg N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(6 \cdot \frac{1}{x}\right)\right) \cdot \left(-1 \cdot x\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(6 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right) \cdot \left(\color{blue}{-1} \cdot x\right)\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(6 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(-1 \cdot x\right)\right)}\right)\right) \]

      20. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(6 \cdot \left(\frac{1}{\mathsf{neg}\left(x\right)} \cdot \left(\color{blue}{-1} \cdot x\right)\right)\right)\right) \]

      21. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(6 \cdot \left(\frac{1}{-1 \cdot x} \cdot \left(-1 \cdot x\right)\right)\right)\right) \]

      22. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(6 \cdot 1\right)\right) \]

      23. metadata-eval 97.5%

          \[\leadsto \mathsf{*.f64}\left(x, 6\right) \]

   5. Simplified 97.5%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;6 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.6% accurate, 3.0× 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 99.8%

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

3. Taylor expanded in x around 0

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

   1. metadata-eval N/A

      \[\leadsto \left(1 \cdot 6\right) \cdot x \]

   2. lft-mult-inverse N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. remove-double-neg N/A

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

   9. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(6 \cdot \frac{1}{x}\right)\right)\right)\right) \cdot {x}^{\color{blue}{2}} \]

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

       \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(6 \cdot \frac{1}{x}\right)\right) \cdot {x}^{2}\right) \]

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

       \[\leadsto \left(\mathsf{neg}\left(6 \cdot \frac{1}{x}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)} \]

   12. unpow2 N/A

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

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

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

   14. mul-1-neg N/A

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

   15. associate-*r* N/A

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

   16. *-commutative N/A

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

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

       \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(6 \cdot \frac{1}{x}\right)\right) \cdot \left(-1 \cdot x\right)\right)}\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \left(\left(6 \cdot \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right) \cdot \left(\color{blue}{-1} \cdot x\right)\right)\right) \]

   19. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(6 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left(-1 \cdot x\right)\right)}\right)\right) \]

   20. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(6 \cdot \left(\frac{1}{\mathsf{neg}\left(x\right)} \cdot \left(\color{blue}{-1} \cdot x\right)\right)\right)\right) \]

   21. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(6 \cdot \left(\frac{1}{-1 \cdot x} \cdot \left(-1 \cdot x\right)\right)\right)\right) \]

   22. lft-mult-inverse N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(6 \cdot 1\right)\right) \]

   23. metadata-eval 48.5%

       \[\leadsto \mathsf{*.f64}\left(x, 6\right) \]

5. Simplified 48.5%

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

6. Final simplification 48.5%

   \[\leadsto 6 \cdot x \]
7. 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 2024191 
(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))