Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 6

Speedup: 1.0×

• 34.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(x * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 1.5))) (* x (* x (* x -2.0))) (* x (* x 3.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.5)) {
		tmp = x * (x * (x * -2.0));
	} else {
		tmp = x * (x * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.5d0)) .or. (.not. (x <= 1.5d0))) then
        tmp = x * (x * (x * (-2.0d0)))
    else
        tmp = x * (x * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.5)) {
		tmp = x * (x * (x * -2.0));
	} else {
		tmp = x * (x * 3.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.5) or not (x <= 1.5):
		tmp = x * (x * (x * -2.0))
	else:
		tmp = x * (x * 3.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.5) || !(x <= 1.5))
		tmp = Float64(x * Float64(x * Float64(x * -2.0)));
	else
		tmp = Float64(x * Float64(x * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.5) || ~((x <= 1.5)))
		tmp = x * (x * (x * -2.0));
	else
		tmp = x * (x * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.5], N[Not[LessEqual[x, 1.5]], $MachinePrecision]], N[(x * N[(x * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 1.5 < x

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 98.2%

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

      1. unpow2 98.2%

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

      2. *-commutative 98.2%

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

      3. associate-*r* 98.2%

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

   6. Simplified 98.2%

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

   if -1.5 < x < 1.5

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.8%

      \[\leadsto x \cdot \color{blue}{\left(3 \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \end{array} \]

Alternative 3: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.2222222222222222 + \frac{0.3333333333333333}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.15) (not (<= x 1.5)))
   (* x (* x (* x -2.0)))
   (/ x (+ 0.2222222222222222 (/ 0.3333333333333333 x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.15) || !(x <= 1.5)) {
		tmp = x * (x * (x * -2.0));
	} else {
		tmp = x / (0.2222222222222222 + (0.3333333333333333 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.15d0)) .or. (.not. (x <= 1.5d0))) then
        tmp = x * (x * (x * (-2.0d0)))
    else
        tmp = x / (0.2222222222222222d0 + (0.3333333333333333d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.15) || !(x <= 1.5)) {
		tmp = x * (x * (x * -2.0));
	} else {
		tmp = x / (0.2222222222222222 + (0.3333333333333333 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.15) or not (x <= 1.5):
		tmp = x * (x * (x * -2.0))
	else:
		tmp = x / (0.2222222222222222 + (0.3333333333333333 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.15) || !(x <= 1.5))
		tmp = Float64(x * Float64(x * Float64(x * -2.0)));
	else
		tmp = Float64(x / Float64(0.2222222222222222 + Float64(0.3333333333333333 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.15) || ~((x <= 1.5)))
		tmp = x * (x * (x * -2.0));
	else
		tmp = x / (0.2222222222222222 + (0.3333333333333333 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.15], N[Not[LessEqual[x, 1.5]], $MachinePrecision]], N[(x * N[(x * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(0.2222222222222222 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1499999999999999 or 1.5 < x

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 98.2%

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

      1. unpow2 98.2%

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

      2. *-commutative 98.2%

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

      3. associate-*r* 98.2%

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

   6. Simplified 98.2%

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

   if -1.1499999999999999 < x < 1.5

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
   4. Step-by-step derivation

      1. flip-- 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. swap-sqr 99.8%

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

      5. metadata-eval 99.8%

         \[\leadsto x \cdot \frac{x \cdot \left(9 - \left(x \cdot x\right) \cdot \color{blue}{4}\right)}{3 + x \cdot 2} \]

      6. +-commutative 99.8%

         \[\leadsto x \cdot \frac{x \cdot \left(9 - \left(x \cdot x\right) \cdot 4\right)}{\color{blue}{x \cdot 2 + 3}} \]

      7. fma-def 99.8%

         \[\leadsto x \cdot \frac{x \cdot \left(9 - \left(x \cdot x\right) \cdot 4\right)}{\color{blue}{\mathsf{fma}\left(x, 2, 3\right)}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto x \cdot \color{blue}{\frac{x \cdot \left(9 - \left(x \cdot x\right) \cdot 4\right)}{\mathsf{fma}\left(x, 2, 3\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 99.8%

         \[\leadsto x \cdot \frac{\color{blue}{\left(9 - \left(x \cdot x\right) \cdot 4\right) \cdot x}}{\mathsf{fma}\left(x, 2, 3\right)} \]

      2. associate-/l* 99.7%

         \[\leadsto x \cdot \color{blue}{\frac{9 - \left(x \cdot x\right) \cdot 4}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x}}} \]

      3. associate-*l* 99.7%

         \[\leadsto x \cdot \frac{9 - \color{blue}{x \cdot \left(x \cdot 4\right)}}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x}} \]

   7. Simplified 99.7%

      \[\leadsto x \cdot \color{blue}{\frac{9 - x \cdot \left(x \cdot 4\right)}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x}}} \]

   8. Taylor expanded in x around 0 99.7%

      \[\leadsto x \cdot \frac{9 - x \cdot \left(x \cdot 4\right)}{\color{blue}{2 + 3 \cdot \frac{1}{x}}} \]
   9. Step-by-step derivation

      1. clear-num 99.7%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{2 + 3 \cdot \frac{1}{x}}{9 - x \cdot \left(x \cdot 4\right)}}} \]

      2. un-div-inv 99.7%

         \[\leadsto \color{blue}{\frac{x}{\frac{2 + 3 \cdot \frac{1}{x}}{9 - x \cdot \left(x \cdot 4\right)}}} \]

      3. div-inv 99.7%

         \[\leadsto \frac{x}{\frac{2 + \color{blue}{\frac{3}{x}}}{9 - x \cdot \left(x \cdot 4\right)}} \]

      4. sub-neg 99.7%

         \[\leadsto \frac{x}{\frac{2 + \frac{3}{x}}{\color{blue}{9 + \left(-x \cdot \left(x \cdot 4\right)\right)}}} \]

      5. associate-*r* 99.7%

         \[\leadsto \frac{x}{\frac{2 + \frac{3}{x}}{9 + \left(-\color{blue}{\left(x \cdot x\right) \cdot 4}\right)}} \]

      6. distribute-rgt-neg-in 99.7%

         \[\leadsto \frac{x}{\frac{2 + \frac{3}{x}}{9 + \color{blue}{\left(x \cdot x\right) \cdot \left(-4\right)}}} \]

      7. metadata-eval 99.7%

         \[\leadsto \frac{x}{\frac{2 + \frac{3}{x}}{9 + \left(x \cdot x\right) \cdot \color{blue}{-4}}} \]

   10. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\frac{x}{\frac{2 + \frac{3}{x}}{9 + \left(x \cdot x\right) \cdot -4}}} \]

   11. Taylor expanded in x around 0 98.9%

       \[\leadsto \frac{x}{\color{blue}{0.2222222222222222 + 0.3333333333333333 \cdot \frac{1}{x}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 98.9%

          \[\leadsto \frac{x}{0.2222222222222222 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{x}}} \]

       2. metadata-eval 98.9%

          \[\leadsto \frac{x}{0.2222222222222222 + \frac{\color{blue}{0.3333333333333333}}{x}} \]

   13. Simplified 98.9%

       \[\leadsto \frac{x}{\color{blue}{0.2222222222222222 + \frac{0.3333333333333333}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.2222222222222222 + \frac{0.3333333333333333}{x}}\\ \end{array} \]

Alternative 4: 61.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(3.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 64.2%

   \[\leadsto \color{blue}{3 \cdot {x}^{2}} \]
5. Step-by-step derivation

   1. unpow2 64.2%

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

6. Simplified 64.2%

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

7. Final simplification 64.2%

   \[\leadsto 3 \cdot \left(x \cdot x\right) \]

Alternative 5: 61.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 64.3%

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

5. Final simplification 64.3%

   \[\leadsto x \cdot \left(x \cdot 3\right) \]

Alternative 6: 3.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 4.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 4.5))

C

double code(double x) {
	return x * 4.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 4.5d0
end function

Java

public static double code(double x) {
	return x * 4.5;
}

Python

def code(x):
	return x * 4.5

Julia

function code(x)
	return Float64(x * 4.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 4.5;
end

Wolfram

code[x_] := N[(x * 4.5), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)} \]
4. Step-by-step derivation

   1. flip-- 99.5%

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

   2. associate-*r/ 99.5%

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

   3. metadata-eval 99.5%

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

   4. swap-sqr 99.5%

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

   5. metadata-eval 99.5%

      \[\leadsto x \cdot \frac{x \cdot \left(9 - \left(x \cdot x\right) \cdot \color{blue}{4}\right)}{3 + x \cdot 2} \]

   6. +-commutative 99.5%

      \[\leadsto x \cdot \frac{x \cdot \left(9 - \left(x \cdot x\right) \cdot 4\right)}{\color{blue}{x \cdot 2 + 3}} \]

   7. fma-def 99.5%

      \[\leadsto x \cdot \frac{x \cdot \left(9 - \left(x \cdot x\right) \cdot 4\right)}{\color{blue}{\mathsf{fma}\left(x, 2, 3\right)}} \]

5. Applied egg-rr 99.5%

   \[\leadsto x \cdot \color{blue}{\frac{x \cdot \left(9 - \left(x \cdot x\right) \cdot 4\right)}{\mathsf{fma}\left(x, 2, 3\right)}} \]
6. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto x \cdot \frac{\color{blue}{\left(9 - \left(x \cdot x\right) \cdot 4\right) \cdot x}}{\mathsf{fma}\left(x, 2, 3\right)} \]

   2. associate-/l* 99.4%

      \[\leadsto x \cdot \color{blue}{\frac{9 - \left(x \cdot x\right) \cdot 4}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x}}} \]

   3. associate-*l* 99.4%

      \[\leadsto x \cdot \frac{9 - \color{blue}{x \cdot \left(x \cdot 4\right)}}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x}} \]

7. Simplified 99.4%

   \[\leadsto x \cdot \color{blue}{\frac{9 - x \cdot \left(x \cdot 4\right)}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x}}} \]

8. Taylor expanded in x around 0 99.8%

   \[\leadsto x \cdot \frac{9 - x \cdot \left(x \cdot 4\right)}{\color{blue}{2 + 3 \cdot \frac{1}{x}}} \]

9. Taylor expanded in x around 0 49.0%

   \[\leadsto x \cdot \frac{\color{blue}{9}}{2 + 3 \cdot \frac{1}{x}} \]

10. Taylor expanded in x around inf 2.9%

    \[\leadsto x \cdot \color{blue}{4.5} \]

11. Final simplification 2.9%

    \[\leadsto x \cdot 4.5 \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(x * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023228 
(FPCore (x)
  :name "Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2"
  :precision binary64

  :herbie-target
  (* x (* x (- 3.0 (* x 2.0))))

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