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

Percentage Accurate: 99.7% → 99.8%

Time: 9.1s

Alternatives: 5

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. Step-by-step derivation

   1. *-commutative N/A

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

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

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

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

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 97.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (-9.0d0))
    if (x <= (-0.65d0)) then
        tmp = t_0
    else if (x <= 0.65d0) then
        tmp = x * 6.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 97.4%

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

   5. Simplified 97.4%

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

   if -0.650000000000000022 < x < 0.650000000000000022

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. lft-mult-inverse N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

      13. distribute-neg-frac2 N/A

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

      14. mul-1-neg N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval 97.5%

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

   7. Simplified 97.5%

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

4. Final simplification 97.5%

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

Alternative 3: 52.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.65:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.65) (* x 6.0) (/ x -0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if (x <= 0.65) {
		tmp = x * 6.0;
	} else {
		tmp = x / -0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.65d0) then
        tmp = x * 6.0d0
    else
        tmp = x / (-0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.65) {
		tmp = x * 6.0;
	} else {
		tmp = x / -0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.65:
		tmp = x * 6.0
	else:
		tmp = x / -0.16666666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.65)
		tmp = Float64(x * 6.0);
	else
		tmp = Float64(x / -0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.65)
		tmp = x * 6.0;
	else
		tmp = x / -0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.65], N[(x * 6.0), $MachinePrecision], N[(x / -0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.650000000000000022

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. lft-mult-inverse N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

      13. distribute-neg-frac2 N/A

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

      14. mul-1-neg N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval 65.5%

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

   7. Simplified 65.5%

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

   if 0.650000000000000022 < x

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. flip3-+ N/A

         \[\leadsto x \cdot \frac{{6}^{3} + {\left(x \cdot -9\right)}^{3}}{\color{blue}{6 \cdot 6 + \left(\left(x \cdot -9\right) \cdot \left(x \cdot -9\right) - 6 \cdot \left(x \cdot -9\right)\right)}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{6 \cdot 6 + \left(\left(x \cdot -9\right) \cdot \left(x \cdot -9\right) - 6 \cdot \left(x \cdot -9\right)\right)}{{6}^{3} + {\left(x \cdot -9\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{6 \cdot 6 + \left(\left(x \cdot -9\right) \cdot \left(x \cdot -9\right) - 6 \cdot \left(x \cdot -9\right)\right)}{{6}^{3} + {\left(x \cdot -9\right)}^{3}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{6 \cdot 6 + \left(\left(x \cdot -9\right) \cdot \left(x \cdot -9\right) - 6 \cdot \left(x \cdot -9\right)\right)}{{6}^{3} + {\left(x \cdot -9\right)}^{3}}\right)}\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{{6}^{3} + {\left(x \cdot -9\right)}^{3}}{6 \cdot 6 + \left(\left(x \cdot -9\right) \cdot \left(x \cdot -9\right) - 6 \cdot \left(x \cdot -9\right)\right)}}}\right)\right) \]

      6. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{6 + \color{blue}{x \cdot -9}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(6 + x \cdot -9\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \color{blue}{\left(x \cdot -9\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(x, \color{blue}{-9}\right)\right)\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{6 + x \cdot -9}}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\frac{1}{6}}\right) \]
   8. Step-by-step derivation

   9. Simplified 0.5%

      \[\leadsto \frac{x}{\color{blue}{0.16666666666666666}} \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{6}}{x}}} \]

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

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

       3. /-lowering-/.f64 0.5%

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

   11. Applied egg-rr 0.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{0.16666666666666666}{x}}} \]
   12. Step-by-step derivation

       1. div-inv N/A

          \[\leadsto \frac{1}{\frac{1}{6} \cdot \color{blue}{\frac{1}{x}}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{\frac{1}{6}}}{\color{blue}{\frac{1}{x}}} \]

       3. inv-pow N/A

          \[\leadsto \frac{\frac{1}{\frac{1}{6}}}{{x}^{\color{blue}{-1}}} \]

       4. sqr-pow N/A

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

       5. pow-prod-down N/A

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

       6. sqr-neg N/A

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

       7. unpow-prod-down N/A

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

       8. sqr-pow N/A

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

       9. inv-pow N/A

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

       10. distribute-neg-frac2 N/A

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

       11. associate-/r* N/A

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

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

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

       13. div-inv N/A

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

       14. distribute-neg-frac N/A

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

       15. clear-num N/A

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

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

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

       17. metadata-eval 7.1%

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

   13. Applied egg-rr 7.1%

       \[\leadsto \color{blue}{\frac{x}{-0.16666666666666666}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 50.6% accurate, 3.0× 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 99.8%

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

   1. *-commutative N/A

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

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

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

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

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. lft-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. associate-*l* N/A

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

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

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

   8. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. associate-*l* N/A

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

   13. distribute-neg-frac2 N/A

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

   14. mul-1-neg N/A

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

   15. lft-mult-inverse N/A

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

   16. metadata-eval 48.5%

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

7. Simplified 48.5%

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

Alternative 5: 2.3% accurate, 9.0× 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 99.8%

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

   1. *-commutative N/A

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

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

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

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

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative N/A

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

   2. +-commutative N/A

      \[\leadsto \left(x \cdot -9 + 6\right) \cdot x \]

   3. flip-+ N/A

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

   4. associate-*l/ N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(x \cdot -9\right) \cdot \left(x \cdot -9\right) - 6 \cdot 6\right) \cdot x\right), \color{blue}{\left(x \cdot -9 - 6\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot -9\right) \cdot \left(x \cdot -9\right) - 6 \cdot 6\right), x\right), \left(\color{blue}{x \cdot -9} - 6\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot -9\right) \cdot \left(x \cdot -9\right) + \left(\mathsf{neg}\left(6 \cdot 6\right)\right)\right), x\right), \left(\color{blue}{x} \cdot -9 - 6\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right), \left(\mathsf{neg}\left(6 \cdot 6\right)\right)\right), x\right), \left(\color{blue}{x} \cdot -9 - 6\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(-9 \cdot \left(x \cdot -9\right)\right)\right), \left(\mathsf{neg}\left(6 \cdot 6\right)\right)\right), x\right), \left(x \cdot -9 - 6\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(\left(x \cdot -9\right) \cdot -9\right)\right), \left(\mathsf{neg}\left(6 \cdot 6\right)\right)\right), x\right), \left(x \cdot -9 - 6\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(x \cdot -9\right) \cdot -9\right)\right), \left(\mathsf{neg}\left(6 \cdot 6\right)\right)\right), x\right), \left(x \cdot -9 - 6\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(-9 \cdot -9\right)\right)\right), \left(\mathsf{neg}\left(6 \cdot 6\right)\right)\right), x\right), \left(x \cdot -9 - 6\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(-9 \cdot -9\right)\right)\right), \left(\mathsf{neg}\left(6 \cdot 6\right)\right)\right), x\right), \left(x \cdot -9 - 6\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 81\right)\right), \left(\mathsf{neg}\left(6 \cdot 6\right)\right)\right), x\right), \left(x \cdot -9 - 6\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 81\right)\right), \left(\mathsf{neg}\left(36\right)\right)\right), x\right), \left(x \cdot -9 - 6\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 81\right)\right), -36\right), x\right), \left(x \cdot -9 - 6\right)\right) \]

   17. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 81\right)\right), -36\right), x\right), \left(x \cdot -9 + \color{blue}{\left(\mathsf{neg}\left(6\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 81\right)\right), -36\right), x\right), \mathsf{+.f64}\left(\left(x \cdot -9\right), \color{blue}{\left(\mathsf{neg}\left(6\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 81\right)\right), -36\right), x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -9\right), \left(\mathsf{neg}\left(\color{blue}{6}\right)\right)\right)\right) \]

   20. metadata-eval 93.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 81\right)\right), -36\right), x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -9\right), -6\right)\right) \]

6. Applied egg-rr 93.3%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-36 \cdot x\right)}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -9\right), -6\right)\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot -36\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, -9\right)}, -6\right)\right) \]

   2. *-lowering-*.f64 47.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, -36\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, -9\right)}, -6\right)\right) \]

9. Simplified 47.3%

   \[\leadsto \frac{\color{blue}{x \cdot -36}}{x \cdot -9 + -6} \]

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: 99.7% 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))