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

Percentage Accurate: 99.7% → 99.7%

Time: 7.6s

Alternatives: 8

Speedup: 1.3×

• 24.5% 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

\[\begin{array}{l} \\ 3 \cdot \left(x \cdot \left(2 - 3 \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 (* x (- 2.0 (* 3.0 x)))))

C

double code(double x) {
	return 3.0 * (x * (2.0 - (3.0 * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * (x * (2.0d0 - (3.0d0 * x)))
end function

Java

public static double code(double x) {
	return 3.0 * (x * (2.0 - (3.0 * x)));
}

Python

def code(x):
	return 3.0 * (x * (2.0 - (3.0 * x)))

Julia

function code(x)
	return Float64(3.0 * Float64(x * Float64(2.0 - Float64(3.0 * x))))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * (x * (2.0 - (3.0 * x)));
end

Wolfram

code[x_] := N[(3.0 * N[(x * N[(2.0 - N[(3.0 * x), $MachinePrecision]), $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. associate-*l* 99.8%

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

   2. *-commutative 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \frac{x}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.65)
   (* -9.0 (* x x))
   (if (<= x 0.65) (* x 6.0) (* -9.0 (/ x (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = -9.0 * (x * x);
	} else if (x <= 0.65) {
		tmp = x * 6.0;
	} else {
		tmp = -9.0 * (x / (1.0 / x));
	}
	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 = x * 6.0d0
    else
        tmp = (-9.0d0) * (x / (1.0d0 / x))
    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 = x * 6.0;
	} else {
		tmp = -9.0 * (x / (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.65:
		tmp = -9.0 * (x * x)
	elif x <= 0.65:
		tmp = x * 6.0
	else:
		tmp = -9.0 * (x / (1.0 / x))
	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(x * 6.0);
	else
		tmp = Float64(-9.0 * Float64(x / Float64(1.0 / x)));
	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 = x * 6.0;
	else
		tmp = -9.0 * (x / (1.0 / x));
	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[(x * 6.0), $MachinePrecision], N[(-9.0 * N[(x / N[(1.0 / x), $MachinePrecision]), $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. 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. Step-by-step derivation

      1. flip-+ 98.4%

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

      2. associate-*r/ 80.9%

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

      3. metadata-eval 80.9%

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

      4. swap-sqr 80.8%

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

      5. pow2 80.8%

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

      6. metadata-eval 80.8%

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

      7. *-commutative 80.8%

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

      8. cancel-sign-sub-inv 80.8%

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

      9. metadata-eval 80.8%

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

   6. Applied egg-rr 80.8%

      \[\leadsto \color{blue}{\frac{x \cdot \left(36 - {x}^{2} \cdot 81\right)}{6 + 9 \cdot x}} \]
   7. Step-by-step derivation

      1. associate-/l* 98.3%

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

      2. *-lft-identity 98.3%

         \[\leadsto x \cdot \frac{\color{blue}{1 \cdot \left(36 - {x}^{2} \cdot 81\right)}}{6 + 9 \cdot x} \]

      3. associate-*l/ 98.1%

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{6 + 9 \cdot x} \cdot \left(36 - {x}^{2} \cdot 81\right)\right)} \]

      4. associate-/r/ 98.1%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{6 + 9 \cdot x}{36 - {x}^{2} \cdot 81}}} \]

      5. associate-*r/ 98.2%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{\frac{6 + 9 \cdot x}{36 - {x}^{2} \cdot 81}}} \]

      6. *-rgt-identity 98.2%

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

      7. *-rgt-identity 98.2%

         \[\leadsto \frac{x}{\frac{6 + 9 \cdot x}{\color{blue}{\left(36 - {x}^{2} \cdot 81\right) \cdot 1}}} \]

      8. *-commutative 98.2%

         \[\leadsto \frac{x}{\frac{6 + \color{blue}{x \cdot 9}}{\left(36 - {x}^{2} \cdot 81\right) \cdot 1}} \]

      9. *-rgt-identity 98.2%

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

      10. sub-neg 98.2%

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

      11. distribute-rgt-neg-in 98.2%

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

      12. metadata-eval 98.2%

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

   8. Simplified 98.2%

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

   9. Taylor expanded in x around inf 95.9%

      \[\leadsto \frac{x}{\color{blue}{\frac{-0.1111111111111111}{x}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 95.9%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{-0.1111111111111111}{x}} \]

       2. div-inv 95.9%

          \[\leadsto \frac{1 \cdot x}{\color{blue}{-0.1111111111111111 \cdot \frac{1}{x}}} \]

       3. times-frac 95.9%

          \[\leadsto \color{blue}{\frac{1}{-0.1111111111111111} \cdot \frac{x}{\frac{1}{x}}} \]

       4. metadata-eval 95.9%

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

   11. Applied egg-rr 95.9%

       \[\leadsto \color{blue}{-9 \cdot \frac{x}{\frac{1}{x}}} \]
   12. Step-by-step derivation

       1. pow1 95.9%

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

       2. inv-pow 95.9%

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

       3. pow-div 95.9%

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

       4. metadata-eval 95.9%

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

       5. pow2 95.9%

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

   13. Applied egg-rr 95.9%

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

   if -0.650000000000000022 < 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 96.3%

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

   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 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. associate-*l* 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 99.9%

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

      2. associate-*r/ 92.7%

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

      3. metadata-eval 92.7%

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

      4. swap-sqr 92.7%

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

      5. pow2 92.7%

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

      6. metadata-eval 92.7%

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

      7. *-commutative 92.7%

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

      8. cancel-sign-sub-inv 92.7%

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

      9. metadata-eval 92.7%

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

   6. Applied egg-rr 92.7%

      \[\leadsto \color{blue}{\frac{x \cdot \left(36 - {x}^{2} \cdot 81\right)}{6 + 9 \cdot x}} \]
   7. Step-by-step derivation

      1. associate-/l* 99.8%

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

      2. *-lft-identity 99.8%

         \[\leadsto x \cdot \frac{\color{blue}{1 \cdot \left(36 - {x}^{2} \cdot 81\right)}}{6 + 9 \cdot x} \]

      3. associate-*l/ 99.8%

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{6 + 9 \cdot x} \cdot \left(36 - {x}^{2} \cdot 81\right)\right)} \]

      4. associate-/r/ 99.8%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{6 + 9 \cdot x}{36 - {x}^{2} \cdot 81}}} \]

      5. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{\frac{6 + 9 \cdot x}{36 - {x}^{2} \cdot 81}}} \]

      6. *-rgt-identity 99.8%

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

      7. *-rgt-identity 99.8%

         \[\leadsto \frac{x}{\frac{6 + 9 \cdot x}{\color{blue}{\left(36 - {x}^{2} \cdot 81\right) \cdot 1}}} \]

      8. *-commutative 99.8%

         \[\leadsto \frac{x}{\frac{6 + \color{blue}{x \cdot 9}}{\left(36 - {x}^{2} \cdot 81\right) \cdot 1}} \]

      9. *-rgt-identity 99.8%

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

      10. sub-neg 99.8%

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

      11. distribute-rgt-neg-in 99.8%

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

      12. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around inf 98.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{-0.1111111111111111}{x}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 98.0%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{-0.1111111111111111}{x}} \]

       2. div-inv 98.0%

          \[\leadsto \frac{1 \cdot x}{\color{blue}{-0.1111111111111111 \cdot \frac{1}{x}}} \]

       3. times-frac 98.2%

          \[\leadsto \color{blue}{\frac{1}{-0.1111111111111111} \cdot \frac{x}{\frac{1}{x}}} \]

       4. metadata-eval 98.2%

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

   11. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{-9 \cdot \frac{x}{\frac{1}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \frac{x}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;3 \cdot \left(x \cdot \left(x \cdot -3\right)\right)\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \frac{x}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.65)
   (* 3.0 (* x (* x -3.0)))
   (if (<= x 0.65) (* x 6.0) (* -9.0 (/ x (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = 3.0 * (x * (x * -3.0));
	} else if (x <= 0.65) {
		tmp = x * 6.0;
	} else {
		tmp = -9.0 * (x / (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.65d0)) then
        tmp = 3.0d0 * (x * (x * (-3.0d0)))
    else if (x <= 0.65d0) then
        tmp = x * 6.0d0
    else
        tmp = (-9.0d0) * (x / (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = 3.0 * (x * (x * -3.0));
	} else if (x <= 0.65) {
		tmp = x * 6.0;
	} else {
		tmp = -9.0 * (x / (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.65:
		tmp = 3.0 * (x * (x * -3.0))
	elif x <= 0.65:
		tmp = x * 6.0
	else:
		tmp = -9.0 * (x / (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.65)
		tmp = Float64(3.0 * Float64(x * Float64(x * -3.0)));
	elseif (x <= 0.65)
		tmp = Float64(x * 6.0);
	else
		tmp = Float64(-9.0 * Float64(x / Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.65)
		tmp = 3.0 * (x * (x * -3.0));
	elseif (x <= 0.65)
		tmp = x * 6.0;
	else
		tmp = -9.0 * (x / (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.65], N[(3.0 * N[(x * N[(x * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.65], N[(x * 6.0), $MachinePrecision], N[(-9.0 * N[(x / N[(1.0 / x), $MachinePrecision]), $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. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 96.0%

      \[\leadsto 3 \cdot \left(\color{blue}{\left(-3 \cdot x\right)} \cdot x\right) \]
   6. Step-by-step derivation

      1. *-commutative 96.0%

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

   7. Simplified 96.0%

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

   if -0.650000000000000022 < 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 96.3%

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

   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 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. associate-*l* 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 99.9%

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

      2. associate-*r/ 92.7%

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

      3. metadata-eval 92.7%

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

      4. swap-sqr 92.7%

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

      5. pow2 92.7%

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

      6. metadata-eval 92.7%

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

      7. *-commutative 92.7%

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

      8. cancel-sign-sub-inv 92.7%

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

      9. metadata-eval 92.7%

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

   6. Applied egg-rr 92.7%

      \[\leadsto \color{blue}{\frac{x \cdot \left(36 - {x}^{2} \cdot 81\right)}{6 + 9 \cdot x}} \]
   7. Step-by-step derivation

      1. associate-/l* 99.8%

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

      2. *-lft-identity 99.8%

         \[\leadsto x \cdot \frac{\color{blue}{1 \cdot \left(36 - {x}^{2} \cdot 81\right)}}{6 + 9 \cdot x} \]

      3. associate-*l/ 99.8%

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{6 + 9 \cdot x} \cdot \left(36 - {x}^{2} \cdot 81\right)\right)} \]

      4. associate-/r/ 99.8%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{6 + 9 \cdot x}{36 - {x}^{2} \cdot 81}}} \]

      5. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{\frac{6 + 9 \cdot x}{36 - {x}^{2} \cdot 81}}} \]

      6. *-rgt-identity 99.8%

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

      7. *-rgt-identity 99.8%

         \[\leadsto \frac{x}{\frac{6 + 9 \cdot x}{\color{blue}{\left(36 - {x}^{2} \cdot 81\right) \cdot 1}}} \]

      8. *-commutative 99.8%

         \[\leadsto \frac{x}{\frac{6 + \color{blue}{x \cdot 9}}{\left(36 - {x}^{2} \cdot 81\right) \cdot 1}} \]

      9. *-rgt-identity 99.8%

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

      10. sub-neg 99.8%

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

      11. distribute-rgt-neg-in 99.8%

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

      12. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around inf 98.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{-0.1111111111111111}{x}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 98.0%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{-0.1111111111111111}{x}} \]

       2. div-inv 98.0%

          \[\leadsto \frac{1 \cdot x}{\color{blue}{-0.1111111111111111 \cdot \frac{1}{x}}} \]

       3. times-frac 98.2%

          \[\leadsto \color{blue}{\frac{1}{-0.1111111111111111} \cdot \frac{x}{\frac{1}{x}}} \]

       4. metadata-eval 98.2%

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

   11. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{-9 \cdot \frac{x}{\frac{1}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;3 \cdot \left(x \cdot \left(x \cdot -3\right)\right)\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \frac{x}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.65d0)) .or. (.not. (x <= 0.65d0))) then
        tmp = (-9.0d0) * (x * x)
    else
        tmp = x * 6.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. Step-by-step derivation

      1. *-commutative 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      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. flip-+ 99.1%

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

      2. associate-*r/ 86.6%

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

      3. metadata-eval 86.6%

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

      4. swap-sqr 86.6%

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

      5. pow2 86.6%

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

      6. metadata-eval 86.6%

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

      7. *-commutative 86.6%

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

      8. cancel-sign-sub-inv 86.6%

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

      9. metadata-eval 86.6%

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

   6. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\frac{x \cdot \left(36 - {x}^{2} \cdot 81\right)}{6 + 9 \cdot x}} \]
   7. Step-by-step derivation

      1. associate-/l* 99.0%

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

      2. *-lft-identity 99.0%

         \[\leadsto x \cdot \frac{\color{blue}{1 \cdot \left(36 - {x}^{2} \cdot 81\right)}}{6 + 9 \cdot x} \]

      3. associate-*l/ 98.9%

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{6 + 9 \cdot x} \cdot \left(36 - {x}^{2} \cdot 81\right)\right)} \]

      4. associate-/r/ 98.9%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{6 + 9 \cdot x}{36 - {x}^{2} \cdot 81}}} \]

      5. associate-*r/ 99.0%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{\frac{6 + 9 \cdot x}{36 - {x}^{2} \cdot 81}}} \]

      6. *-rgt-identity 99.0%

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

      7. *-rgt-identity 99.0%

         \[\leadsto \frac{x}{\frac{6 + 9 \cdot x}{\color{blue}{\left(36 - {x}^{2} \cdot 81\right) \cdot 1}}} \]

      8. *-commutative 99.0%

         \[\leadsto \frac{x}{\frac{6 + \color{blue}{x \cdot 9}}{\left(36 - {x}^{2} \cdot 81\right) \cdot 1}} \]

      9. *-rgt-identity 99.0%

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

      10. sub-neg 99.0%

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

      11. distribute-rgt-neg-in 99.0%

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

      12. metadata-eval 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in x around inf 96.9%

      \[\leadsto \frac{x}{\color{blue}{\frac{-0.1111111111111111}{x}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 96.9%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{-0.1111111111111111}{x}} \]

       2. div-inv 96.9%

          \[\leadsto \frac{1 \cdot x}{\color{blue}{-0.1111111111111111 \cdot \frac{1}{x}}} \]

       3. times-frac 97.0%

          \[\leadsto \color{blue}{\frac{1}{-0.1111111111111111} \cdot \frac{x}{\frac{1}{x}}} \]

       4. metadata-eval 97.0%

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

   11. Applied egg-rr 97.0%

       \[\leadsto \color{blue}{-9 \cdot \frac{x}{\frac{1}{x}}} \]
   12. Step-by-step derivation

       1. pow1 97.0%

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

       2. inv-pow 97.0%

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

       3. pow-div 97.0%

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

       4. metadata-eval 97.0%

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

       5. pow2 97.0%

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

   13. Applied egg-rr 97.0%

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

   if -0.650000000000000022 < 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 96.3%

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

4. Final simplification 96.6%

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

Alternative 5: 52.3% 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 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 64.7%

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

   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 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. associate-*l* 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 99.9%

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

      2. associate-*r/ 92.7%

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

      3. metadata-eval 92.7%

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

      4. swap-sqr 92.7%

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

      5. pow2 92.7%

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

      6. metadata-eval 92.7%

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

      7. *-commutative 92.7%

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

      8. cancel-sign-sub-inv 92.7%

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

      9. metadata-eval 92.7%

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

   6. Applied egg-rr 92.7%

      \[\leadsto \color{blue}{\frac{x \cdot \left(36 - {x}^{2} \cdot 81\right)}{6 + 9 \cdot x}} \]
   7. Step-by-step derivation

      1. associate-/l* 99.8%

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

      2. *-lft-identity 99.8%

         \[\leadsto x \cdot \frac{\color{blue}{1 \cdot \left(36 - {x}^{2} \cdot 81\right)}}{6 + 9 \cdot x} \]

      3. associate-*l/ 99.8%

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{6 + 9 \cdot x} \cdot \left(36 - {x}^{2} \cdot 81\right)\right)} \]

      4. associate-/r/ 99.8%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{6 + 9 \cdot x}{36 - {x}^{2} \cdot 81}}} \]

      5. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{\frac{6 + 9 \cdot x}{36 - {x}^{2} \cdot 81}}} \]

      6. *-rgt-identity 99.8%

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

      7. *-rgt-identity 99.8%

         \[\leadsto \frac{x}{\frac{6 + 9 \cdot x}{\color{blue}{\left(36 - {x}^{2} \cdot 81\right) \cdot 1}}} \]

      8. *-commutative 99.8%

         \[\leadsto \frac{x}{\frac{6 + \color{blue}{x \cdot 9}}{\left(36 - {x}^{2} \cdot 81\right) \cdot 1}} \]

      9. *-rgt-identity 99.8%

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

      10. sub-neg 99.8%

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

      11. distribute-rgt-neg-in 99.8%

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

      12. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 0.4%

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

       1. clear-num 0.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{0.16666666666666666}{x}}} \]

       2. inv-pow 0.4%

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

   11. Applied egg-rr 0.4%

       \[\leadsto \color{blue}{{\left(\frac{0.16666666666666666}{x}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 0.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{0.16666666666666666}{x}}} \]

       2. clear-num 0.4%

          \[\leadsto \color{blue}{\frac{x}{0.16666666666666666}} \]

       3. frac-2neg 0.4%

          \[\leadsto \color{blue}{\frac{-x}{-0.16666666666666666}} \]

       4. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{-0.16666666666666666} \]

       5. sqrt-unprod 66.6%

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

       6. sqr-neg 66.6%

          \[\leadsto \frac{\sqrt{\color{blue}{x \cdot x}}}{-0.16666666666666666} \]

       7. sqrt-prod 7.0%

          \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-0.16666666666666666} \]

       8. add-sqr-sqrt 7.0%

          \[\leadsto \frac{\color{blue}{x}}{-0.16666666666666666} \]

       9. metadata-eval 7.0%

          \[\leadsto \frac{x}{\color{blue}{-0.16666666666666666}} \]

   13. Applied egg-rr 7.0%

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

4. Final simplification 50.3%

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

Alternative 6: 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 99.7%

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
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. Final simplification 99.7%

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

Alternative 7: 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 99.7%

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
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 48.6%

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

6. Final simplification 48.6%

   \[\leadsto x \cdot 6 \]
7. Add Preprocessing

Alternative 8: 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* 99.7%

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

   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. Step-by-step derivation

   1. *-commutative 99.7%

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

   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/ 92.9%

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

   4. metadata-eval 92.9%

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

   5. swap-sqr 92.9%

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

   6. pow2 92.9%

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

   7. metadata-eval 92.9%

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

   8. *-commutative 92.9%

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

   9. cancel-sign-sub-inv 92.9%

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

   10. metadata-eval 92.9%

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

6. Applied egg-rr 92.9%

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

7. Taylor expanded in x around 0 47.7%

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

   1. *-commutative 47.7%

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

9. Simplified 47.7%

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

10. Taylor expanded in x around inf 2.4%

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

11. Final simplification 2.4%

    \[\leadsto 4 \]
12. Add Preprocessing

Developer target: 99.8% 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 2024048 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :alt
  (- (* 6.0 x) (* 9.0 (* x x)))

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