Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 8.3s

Alternatives: 4

Speedup: 1.0×

• 33.7% 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* N/A

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

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

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

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

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

   4. sub-neg N/A

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

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

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

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

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

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

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

   8. metadata-eval 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\left(x \cdot -2\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{\frac{-0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.5)
   (* (* x -2.0) (* x x))
   (if (<= x 1.5) (* x (* x 3.0)) (/ (* x x) (/ -0.5 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.5) {
		tmp = (x * -2.0) * (x * x);
	} else if (x <= 1.5) {
		tmp = x * (x * 3.0);
	} else {
		tmp = (x * x) / (-0.5 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.5d0)) then
        tmp = (x * (-2.0d0)) * (x * x)
    else if (x <= 1.5d0) then
        tmp = x * (x * 3.0d0)
    else
        tmp = (x * x) / ((-0.5d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.5) {
		tmp = (x * -2.0) * (x * x);
	} else if (x <= 1.5) {
		tmp = x * (x * 3.0);
	} else {
		tmp = (x * x) / (-0.5 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.5:
		tmp = (x * -2.0) * (x * x)
	elif x <= 1.5:
		tmp = x * (x * 3.0)
	else:
		tmp = (x * x) / (-0.5 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.5)
		tmp = Float64(Float64(x * -2.0) * Float64(x * x));
	elseif (x <= 1.5)
		tmp = Float64(x * Float64(x * 3.0));
	else
		tmp = Float64(Float64(x * x) / Float64(-0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.5)
		tmp = (x * -2.0) * (x * x);
	elseif (x <= 1.5)
		tmp = x * (x * 3.0);
	else
		tmp = (x * x) / (-0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.5], N[(N[(x * -2.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.5], N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] / N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(-2 \cdot x\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. *-lowering-*.f64 97.1%

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

   5. Simplified 97.1%

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

   if -1.5 < x < 1.5

   1. Initial program 99.7%

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

      1. associate-*l* N/A

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

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

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{3}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 99.0%

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

   if 1.5 < x

   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* N/A

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

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

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-rgt-in N/A

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

      2. fma-define N/A

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

      3. metadata-eval N/A

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

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

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

      5. distribute-lft-neg-out N/A

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

      6. fmm-undef N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-lowering-*.f64 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{\left(x \cdot 3\right) \cdot \left(x \cdot 3\right) - \left(x \cdot \left(x \cdot 2\right)\right) \cdot \left(x \cdot \left(x \cdot 2\right)\right)}{x \cdot 3 + x \cdot \left(x \cdot 2\right)}}}\right)\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{x \cdot 3 - \color{blue}{x \cdot \left(x \cdot 2\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot 3 - x \cdot \left(x \cdot 2\right)\right)}\right)\right) \]

      8. distribute-lft-out-- N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(3 - x \cdot 2\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(3 - x \cdot 2\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(3 - 2 \cdot \color{blue}{x}\right)\right)\right)\right) \]

      11. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(3 + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(3 + -2 \cdot x\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(3 + x \cdot \color{blue}{-2}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \color{blue}{\left(x \cdot -2\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \mathsf{*.f64}\left(x, \color{blue}{-2}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{-1}{2}}{{x}^{2}}\right)}\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\frac{-1}{2}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       3. *-lowering-*.f64 98.7%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   11. Simplified 98.7%

       \[\leadsto \frac{x}{\color{blue}{\frac{-0.5}{x \cdot x}}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{x}{\frac{\frac{\frac{-1}{2}}{x}}{\color{blue}{x}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{x}{\frac{\frac{-1}{2}}{x}} \cdot \color{blue}{x} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{\frac{-1}{2}}{x}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\color{blue}{\frac{-1}{2}}}{x}\right)\right) \]

       6. /-lowering-/.f64 98.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{x}\right)\right) \]

   13. Applied egg-rr 98.7%

       \[\leadsto \color{blue}{\frac{x \cdot x}{\frac{-0.5}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\left(x \cdot -2\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{\frac{-0.5}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot -2\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x -2.0) (* x x))))
   (if (<= x -1.5) t_0 (if (<= x 1.5) (* x (* x 3.0)) t_0))))

C

double code(double x) {
	double t_0 = (x * -2.0) * (x * x);
	double tmp;
	if (x <= -1.5) {
		tmp = t_0;
	} else if (x <= 1.5) {
		tmp = x * (x * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (-2.0d0)) * (x * x)
    if (x <= (-1.5d0)) then
        tmp = t_0
    else if (x <= 1.5d0) then
        tmp = x * (x * 3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * -2.0) * (x * x);
	double tmp;
	if (x <= -1.5) {
		tmp = t_0;
	} else if (x <= 1.5) {
		tmp = x * (x * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * -2.0) * (x * x)
	tmp = 0
	if x <= -1.5:
		tmp = t_0
	elif x <= 1.5:
		tmp = x * (x * 3.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * -2.0) * Float64(x * x))
	tmp = 0.0
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 1.5)
		tmp = Float64(x * Float64(x * 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * -2.0) * (x * x);
	tmp = 0.0;
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 1.5)
		tmp = x * (x * 3.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * -2.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5], t$95$0, If[LessEqual[x, 1.5], N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 1.5 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(-2 \cdot x\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. *-lowering-*.f64 97.9%

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

   5. Simplified 97.9%

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

   if -1.5 < x < 1.5

   1. Initial program 99.7%

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

      1. associate-*l* N/A

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

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

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{3}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 99.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\left(x \cdot -2\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot -2\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.6% 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* N/A

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

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

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

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

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

   4. sub-neg N/A

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

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

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

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

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

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

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

   8. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{3}\right)\right) \]
6. Step-by-step derivation

7. Simplified 63.1%

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

Developer Target 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]

Reproduce

herbie shell --seed 2024161 
(FPCore (x)
  :name "Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (* x (- 3 (* x 2)))))

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