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

Percentage Accurate: 99.7% → 99.7%

Time: 8.1s

Alternatives: 5

Speedup: 1.3×

• 24.6% 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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. cancel-sign-sub-inv 99.7%

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

   3. distribute-rgt-in 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-lft-neg-out 99.7%

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

   6. distribute-rgt-neg-in 99.7%

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

   7. associate-*l* 99.8%

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

   8. metadata-eval 99.8%

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

   9. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. distribute-lft-in 99.8%

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

   2. flip-+ 42.2%

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

   3. *-commutative 42.2%

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

   4. *-commutative 42.2%

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

   5. *-commutative 42.2%

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

   6. associate-*l* 42.2%

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

   7. pow2 42.2%

      \[\leadsto \frac{\left(x \cdot 6\right) \cdot \left(x \cdot 6\right) - \left(-9 \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\left(x \cdot -9\right) \cdot x\right)}{x \cdot 6 - x \cdot \left(x \cdot -9\right)} \]

   8. *-commutative 42.2%

      \[\leadsto \frac{\left(x \cdot 6\right) \cdot \left(x \cdot 6\right) - \left(-9 \cdot {x}^{2}\right) \cdot \left(\color{blue}{\left(-9 \cdot x\right)} \cdot x\right)}{x \cdot 6 - x \cdot \left(x \cdot -9\right)} \]

   9. associate-*l* 42.2%

      \[\leadsto \frac{\left(x \cdot 6\right) \cdot \left(x \cdot 6\right) - \left(-9 \cdot {x}^{2}\right) \cdot \color{blue}{\left(-9 \cdot \left(x \cdot x\right)\right)}}{x \cdot 6 - x \cdot \left(x \cdot -9\right)} \]

   10. pow2 42.2%

       \[\leadsto \frac{\left(x \cdot 6\right) \cdot \left(x \cdot 6\right) - \left(-9 \cdot {x}^{2}\right) \cdot \left(-9 \cdot \color{blue}{{x}^{2}}\right)}{x \cdot 6 - x \cdot \left(x \cdot -9\right)} \]

   11. *-commutative 42.2%

       \[\leadsto \frac{\left(x \cdot 6\right) \cdot \left(x \cdot 6\right) - \left(-9 \cdot {x}^{2}\right) \cdot \left(-9 \cdot {x}^{2}\right)}{x \cdot 6 - \color{blue}{\left(x \cdot -9\right) \cdot x}} \]

   12. *-commutative 42.2%

       \[\leadsto \frac{\left(x \cdot 6\right) \cdot \left(x \cdot 6\right) - \left(-9 \cdot {x}^{2}\right) \cdot \left(-9 \cdot {x}^{2}\right)}{x \cdot 6 - \color{blue}{\left(-9 \cdot x\right)} \cdot x} \]

   13. associate-*l* 42.2%

       \[\leadsto \frac{\left(x \cdot 6\right) \cdot \left(x \cdot 6\right) - \left(-9 \cdot {x}^{2}\right) \cdot \left(-9 \cdot {x}^{2}\right)}{x \cdot 6 - \color{blue}{-9 \cdot \left(x \cdot x\right)}} \]

   14. pow2 42.2%

       \[\leadsto \frac{\left(x \cdot 6\right) \cdot \left(x \cdot 6\right) - \left(-9 \cdot {x}^{2}\right) \cdot \left(-9 \cdot {x}^{2}\right)}{x \cdot 6 - -9 \cdot \color{blue}{{x}^{2}}} \]

6. Applied egg-rr 42.2%

   \[\leadsto \color{blue}{\frac{\left(x \cdot 6\right) \cdot \left(x \cdot 6\right) - \left(-9 \cdot {x}^{2}\right) \cdot \left(-9 \cdot {x}^{2}\right)}{x \cdot 6 - -9 \cdot {x}^{2}}} \]
7. Step-by-step derivation

   1. difference-of-squares 42.2%

      \[\leadsto \frac{\color{blue}{\left(x \cdot 6 + -9 \cdot {x}^{2}\right) \cdot \left(x \cdot 6 - -9 \cdot {x}^{2}\right)}}{x \cdot 6 - -9 \cdot {x}^{2}} \]

   2. +-commutative 42.2%

      \[\leadsto \frac{\color{blue}{\left(-9 \cdot {x}^{2} + x \cdot 6\right)} \cdot \left(x \cdot 6 - -9 \cdot {x}^{2}\right)}{x \cdot 6 - -9 \cdot {x}^{2}} \]

   3. *-commutative 42.2%

      \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot -9} + x \cdot 6\right) \cdot \left(x \cdot 6 - -9 \cdot {x}^{2}\right)}{x \cdot 6 - -9 \cdot {x}^{2}} \]

   4. unpow2 42.2%

      \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot -9 + x \cdot 6\right) \cdot \left(x \cdot 6 - -9 \cdot {x}^{2}\right)}{x \cdot 6 - -9 \cdot {x}^{2}} \]

   5. associate-*r* 42.2%

      \[\leadsto \frac{\left(\color{blue}{x \cdot \left(x \cdot -9\right)} + x \cdot 6\right) \cdot \left(x \cdot 6 - -9 \cdot {x}^{2}\right)}{x \cdot 6 - -9 \cdot {x}^{2}} \]

   6. distribute-lft-in 42.2%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot -9 + 6\right)\right)} \cdot \left(x \cdot 6 - -9 \cdot {x}^{2}\right)}{x \cdot 6 - -9 \cdot {x}^{2}} \]

   7. fma-udef 42.2%

      \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left(x, -9, 6\right)}\right) \cdot \left(x \cdot 6 - -9 \cdot {x}^{2}\right)}{x \cdot 6 - -9 \cdot {x}^{2}} \]

   8. associate-/l* 75.9%

      \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(x, -9, 6\right)}{\frac{x \cdot 6 - -9 \cdot {x}^{2}}{x \cdot 6 - -9 \cdot {x}^{2}}}} \]

   9. *-inverses 99.8%

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

   10. associate-/l* 99.6%

       \[\leadsto \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x, -9, 6\right)}}} \]

8. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x, -9, 6\right)}}} \]
9. Step-by-step derivation

   1. div-inv 99.8%

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

   2. remove-double-div 99.8%

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

   3. fma-udef 99.8%

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

   4. metadata-eval 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. distribute-rgt-in 99.8%

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

   7. distribute-rgt-neg-in 99.8%

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

   8. metadata-eval 99.8%

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

   9. metadata-eval 99.8%

      \[\leadsto \left(x \cdot -9\right) \cdot x + \color{blue}{\frac{1}{0.16666666666666666}} \cdot x \]

   10. associate-/r/ 99.6%

       \[\leadsto \left(x \cdot -9\right) \cdot x + \color{blue}{\frac{1}{\frac{0.16666666666666666}{x}}} \]

   11. clear-num 99.6%

       \[\leadsto \left(x \cdot -9\right) \cdot x + \color{blue}{\frac{x}{0.16666666666666666}} \]

   12. div-inv 99.8%

       \[\leadsto \left(x \cdot -9\right) \cdot x + \color{blue}{x \cdot \frac{1}{0.16666666666666666}} \]

   13. metadata-eval 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.68 \lor \neg \left(x \leq 0.65\right):\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.68) (not (<= x 0.65))) (* x (* x -9.0)) (* x 6.0)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.68) || !(x <= 0.65)) {
		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 <= (-0.68d0)) .or. (.not. (x <= 0.65d0))) 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 <= -0.68) || !(x <= 0.65)) {
		tmp = x * (x * -9.0);
	} else {
		tmp = x * 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.68) or not (x <= 0.65):
		tmp = x * (x * -9.0)
	else:
		tmp = x * 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.68) || !(x <= 0.65))
		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 <= -0.68) || ~((x <= 0.65)))
		tmp = x * (x * -9.0);
	else
		tmp = x * 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.68], N[Not[LessEqual[x, 0.65]], $MachinePrecision]], N[(x * N[(x * -9.0), $MachinePrecision]), $MachinePrecision], N[(x * 6.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049 or 0.650000000000000022 < x

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. distribute-rgt-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-lft-neg-out 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. associate-*l* 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 96.8%

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

      1. *-commutative 96.8%

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

   7. Simplified 96.8%

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

   if -0.680000000000000049 < 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 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. distribute-rgt-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-lft-neg-out 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. associate-*l* 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 98.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.68 \lor \neg \left(x \leq 0.65\right):\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. cancel-sign-sub-inv 99.7%

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

   3. distribute-rgt-in 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-lft-neg-out 99.7%

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

   6. distribute-rgt-neg-in 99.7%

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

   7. associate-*l* 99.8%

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

   8. metadata-eval 99.8%

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

   9. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 4: 51.1% 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 99.7%

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

   1. *-commutative 99.7%

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

   2. cancel-sign-sub-inv 99.7%

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

   3. distribute-rgt-in 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-lft-neg-out 99.7%

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

   6. distribute-rgt-neg-in 99.7%

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

   7. associate-*l* 99.8%

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

   8. metadata-eval 99.8%

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

   9. metadata-eval 99.8%

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

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 52.6%

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

6. Final simplification 52.6%

   \[\leadsto x \cdot 6 \]
7. 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 99.7%

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

   1. *-commutative 99.7%

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

   2. cancel-sign-sub-inv 99.7%

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

   3. distribute-rgt-in 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-lft-neg-out 99.7%

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

   6. distribute-rgt-neg-in 99.7%

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

   7. associate-*l* 99.8%

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

   8. metadata-eval 99.8%

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

   9. metadata-eval 99.8%

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

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 99.8%

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

   2. flip-+ 99.4%

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

   3. associate-*l/ 91.4%

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

   4. metadata-eval 91.4%

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

   5. swap-sqr 91.4%

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

   6. pow2 91.4%

      \[\leadsto \frac{\left(36 - \color{blue}{{x}^{2}} \cdot \left(-9 \cdot -9\right)\right) \cdot x}{6 - x \cdot -9} \]

   7. metadata-eval 91.4%

      \[\leadsto \frac{\left(36 - {x}^{2} \cdot \color{blue}{81}\right) \cdot x}{6 - x \cdot -9} \]

   8. *-commutative 91.4%

      \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 - \color{blue}{-9 \cdot x}} \]

   9. cancel-sign-sub-inv 91.4%

      \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{\color{blue}{6 + \left(--9\right) \cdot x}} \]

   10. metadata-eval 91.4%

       \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 + \color{blue}{9} \cdot x} \]

6. Applied egg-rr 91.4%

   \[\leadsto \color{blue}{\frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 + 9 \cdot x}} \]

7. Taylor expanded in x around 0 51.2%

   \[\leadsto \frac{\color{blue}{36 \cdot x}}{6 + 9 \cdot x} \]
8. Step-by-step derivation

   1. *-commutative 51.2%

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

9. Simplified 51.2%

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

10. Taylor expanded in x around inf 2.3%

    \[\leadsto \color{blue}{4} \]

11. Final simplification 2.3%

    \[\leadsto 4 \]
12. Add Preprocessing

Developer target: 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 2024026 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :herbie-target
  (- (* 6.0 x) (* 9.0 (* x x)))

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