Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 5.1s

Alternatives: 6

Speedup: 1.0×

• 33.8% 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

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

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]

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 97.6% accurate, 0.8× speedup

Math

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

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(Float64(x * 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[(N[(x * x), $MachinePrecision] * 3.0), $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 96.9%

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

      1. unpow2 96.9%

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

      2. *-commutative 96.9%

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

      3. associate-*r* 96.9%

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

   6. Simplified 96.9%

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

   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* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 97.9%

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

      1. unpow2 97.9%

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

   6. Simplified 97.9%

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

4. Final simplification 97.4%

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

Alternative 3: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{0.3333333333333333}{x} + 0.2222222222222222}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.1) (not (<= x 1.5)))
   (* x (* x (* x -2.0)))
   (/ x (+ (/ 0.3333333333333333 x) 0.2222222222222222))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 1.5)) {
		tmp = x * (x * (x * -2.0));
	} else {
		tmp = x / ((0.3333333333333333 / x) + 0.2222222222222222);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.1d0)) .or. (.not. (x <= 1.5d0))) then
        tmp = x * (x * (x * (-2.0d0)))
    else
        tmp = x / ((0.3333333333333333d0 / x) + 0.2222222222222222d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 1.5)) {
		tmp = x * (x * (x * -2.0));
	} else {
		tmp = x / ((0.3333333333333333 / x) + 0.2222222222222222);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.1) or not (x <= 1.5):
		tmp = x * (x * (x * -2.0))
	else:
		tmp = x / ((0.3333333333333333 / x) + 0.2222222222222222)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.1) || !(x <= 1.5))
		tmp = Float64(x * Float64(x * Float64(x * -2.0)));
	else
		tmp = Float64(x / Float64(Float64(0.3333333333333333 / x) + 0.2222222222222222));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.1) || ~((x <= 1.5)))
		tmp = x * (x * (x * -2.0));
	else
		tmp = x / ((0.3333333333333333 / x) + 0.2222222222222222);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.1], N[Not[LessEqual[x, 1.5]], $MachinePrecision]], N[(x * N[(x * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(0.3333333333333333 / x), $MachinePrecision] + 0.2222222222222222), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001 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 96.9%

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

      1. unpow2 96.9%

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

      2. *-commutative 96.9%

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

      3. associate-*r* 96.9%

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

   6. Simplified 96.9%

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

   if -1.1000000000000001 < 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* 99.7%

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

   3. Simplified 99.7%

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

      1. sub-neg 99.7%

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

      2. distribute-lft-in 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. flip-+ 99.7%

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

      2. clear-num 99.6%

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

      3. distribute-lft-out-- 99.6%

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

      4. swap-sqr 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*r* 99.5%

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

      7. associate-*r* 99.5%

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

      8. swap-sqr 99.5%

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

      9. pow2 99.5%

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

      10. pow2 99.5%

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

      11. pow-prod-up 99.5%

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

      12. metadata-eval 99.5%

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

      13. metadata-eval 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in x around 0 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   10. Simplified 99.4%

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

       1. expm1-log1p-u 99.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{0.2222222222222222 + \frac{0.3333333333333333}{x}}\right)\right)} \]

       2. expm1-udef 51.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{1}{0.2222222222222222 + \frac{0.3333333333333333}{x}}\right)} - 1} \]

       3. un-div-inv 51.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{0.2222222222222222 + \frac{0.3333333333333333}{x}}}\right)} - 1 \]

       4. +-commutative 51.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\frac{0.3333333333333333}{x} + 0.2222222222222222}}\right)} - 1 \]

   12. Applied egg-rr 51.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\frac{0.3333333333333333}{x} + 0.2222222222222222}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 99.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\frac{0.3333333333333333}{x} + 0.2222222222222222}\right)\right)} \]

       2. expm1-log1p 99.5%

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

   14. Simplified 99.5%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{0.3333333333333333}{x} + 0.2222222222222222}\\ \end{array} \]

Alternative 4: 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 99.9%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
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. Final simplification 99.8%

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

Alternative 5: 61.2% accurate, 1.8× speedup

Math

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

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(Float64(x * x) * 3.0)
end

MATLAB

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

Wolfram

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

Derivation

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* 99.8%

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

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 63.7%

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

   1. unpow2 63.7%

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

6. Simplified 63.7%

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

7. Final simplification 63.7%

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

Alternative 6: 3.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 6.75 \end{array} \]

FPCore

(FPCore (x) :precision binary64 6.75)

C

double code(double x) {
	return 6.75;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.75d0
end function

Java

public static double code(double x) {
	return 6.75;
}

Python

def code(x):
	return 6.75

Julia

function code(x)
	return 6.75
end

MATLAB

function tmp = code(x)
	tmp = 6.75;
end

Wolfram

code[x_] := 6.75

Derivation

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* 99.8%

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

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. associate-*r* 99.9%

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

   2. *-commutative 99.9%

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

   3. flip3-- 74.5%

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

   4. associate-*l/ 69.3%

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

   5. metadata-eval 69.3%

      \[\leadsto \frac{\left(\color{blue}{27} - {\left(x \cdot 2\right)}^{3}\right) \cdot \left(x \cdot x\right)}{3 \cdot 3 + \left(\left(x \cdot 2\right) \cdot \left(x \cdot 2\right) + 3 \cdot \left(x \cdot 2\right)\right)} \]

   6. add-log-exp 62.2%

      \[\leadsto \frac{\left(27 - {\color{blue}{\log \left(e^{x \cdot 2}\right)}}^{3}\right) \cdot \left(x \cdot x\right)}{3 \cdot 3 + \left(\left(x \cdot 2\right) \cdot \left(x \cdot 2\right) + 3 \cdot \left(x \cdot 2\right)\right)} \]

   7. exp-lft-sqr 62.2%

      \[\leadsto \frac{\left(27 - {\log \color{blue}{\left(e^{x} \cdot e^{x}\right)}}^{3}\right) \cdot \left(x \cdot x\right)}{3 \cdot 3 + \left(\left(x \cdot 2\right) \cdot \left(x \cdot 2\right) + 3 \cdot \left(x \cdot 2\right)\right)} \]

   8. log-prod 62.2%

      \[\leadsto \frac{\left(27 - {\color{blue}{\left(\log \left(e^{x}\right) + \log \left(e^{x}\right)\right)}}^{3}\right) \cdot \left(x \cdot x\right)}{3 \cdot 3 + \left(\left(x \cdot 2\right) \cdot \left(x \cdot 2\right) + 3 \cdot \left(x \cdot 2\right)\right)} \]

   9. add-log-exp 62.2%

      \[\leadsto \frac{\left(27 - {\left(\color{blue}{x} + \log \left(e^{x}\right)\right)}^{3}\right) \cdot \left(x \cdot x\right)}{3 \cdot 3 + \left(\left(x \cdot 2\right) \cdot \left(x \cdot 2\right) + 3 \cdot \left(x \cdot 2\right)\right)} \]

   10. add-log-exp 69.3%

       \[\leadsto \frac{\left(27 - {\left(x + \color{blue}{x}\right)}^{3}\right) \cdot \left(x \cdot x\right)}{3 \cdot 3 + \left(\left(x \cdot 2\right) \cdot \left(x \cdot 2\right) + 3 \cdot \left(x \cdot 2\right)\right)} \]

   11. +-commutative 69.3%

       \[\leadsto \frac{\left(27 - {\left(x + x\right)}^{3}\right) \cdot \left(x \cdot x\right)}{\color{blue}{\left(\left(x \cdot 2\right) \cdot \left(x \cdot 2\right) + 3 \cdot \left(x \cdot 2\right)\right) + 3 \cdot 3}} \]

   12. distribute-rgt-out 69.3%

       \[\leadsto \frac{\left(27 - {\left(x + x\right)}^{3}\right) \cdot \left(x \cdot x\right)}{\color{blue}{\left(x \cdot 2\right) \cdot \left(x \cdot 2 + 3\right)} + 3 \cdot 3} \]

   13. +-commutative 69.3%

       \[\leadsto \frac{\left(27 - {\left(x + x\right)}^{3}\right) \cdot \left(x \cdot x\right)}{\left(x \cdot 2\right) \cdot \color{blue}{\left(3 + x \cdot 2\right)} + 3 \cdot 3} \]

   14. fma-def 69.3%

       \[\leadsto \frac{\left(27 - {\left(x + x\right)}^{3}\right) \cdot \left(x \cdot x\right)}{\color{blue}{\mathsf{fma}\left(x \cdot 2, 3 + x \cdot 2, 3 \cdot 3\right)}} \]

5. Applied egg-rr 69.3%

   \[\leadsto \color{blue}{\frac{\left(27 - {\left(x + x\right)}^{3}\right) \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(x + x, \mathsf{fma}\left(x, 2, 3\right), 9\right)}} \]

6. Taylor expanded in x around 0 50.5%

   \[\leadsto \frac{\color{blue}{27 \cdot {x}^{2}}}{\mathsf{fma}\left(x + x, \mathsf{fma}\left(x, 2, 3\right), 9\right)} \]
7. Step-by-step derivation

   1. unpow2 50.5%

      \[\leadsto \frac{27 \cdot \color{blue}{\left(x \cdot x\right)}}{\mathsf{fma}\left(x + x, \mathsf{fma}\left(x, 2, 3\right), 9\right)} \]

   2. *-commutative 50.5%

      \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot 27}}{\mathsf{fma}\left(x + x, \mathsf{fma}\left(x, 2, 3\right), 9\right)} \]

   3. associate-*l* 50.6%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot 27\right)}}{\mathsf{fma}\left(x + x, \mathsf{fma}\left(x, 2, 3\right), 9\right)} \]

8. Simplified 50.6%

   \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot 27\right)}}{\mathsf{fma}\left(x + x, \mathsf{fma}\left(x, 2, 3\right), 9\right)} \]

9. Taylor expanded in x around inf 3.3%

   \[\leadsto \color{blue}{6.75} \]

10. Final simplification 3.3%

    \[\leadsto 6.75 \]

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 2023230 
(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))))