Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 7.7s

Alternatives: 4

Speedup: 1.1×

• 32.8% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{1}{\mathsf{fma}\left(-2, x, 3\right) \cdot x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (/ 1.0 (* (fma -2.0 x 3.0) x))))

C

double code(double x) {
	return x / (1.0 / (fma(-2.0, x, 3.0) * x));
}

Julia

function code(x)
	return Float64(x / Float64(1.0 / Float64(fma(-2.0, x, 3.0) * x)))
end

Wolfram

code[x_] := N[(x / N[(1.0 / N[(N[(-2.0 * x + 3.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-*r/ N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. clear-num N/A

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

   11. associate-*r/ N/A

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

   12. flip-- N/A

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

   13. lift--.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(-2, x, 3\right) \cdot x}}} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 - 2 \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := \left(-2 \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-6}:\\ \;\;\;\;\left(3 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (- 3.0 (* 2.0 x)) (* x x))) (t_1 (* (* -2.0 x) (* x x))))
   (if (<= t_0 -1e+17) t_1 (if (<= t_0 1e-6) (* (* 3.0 x) x) t_1))))

C

double code(double x) {
	double t_0 = (3.0 - (2.0 * x)) * (x * x);
	double t_1 = (-2.0 * x) * (x * x);
	double tmp;
	if (t_0 <= -1e+17) {
		tmp = t_1;
	} else if (t_0 <= 1e-6) {
		tmp = (3.0 * x) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (3.0d0 - (2.0d0 * x)) * (x * x)
    t_1 = ((-2.0d0) * x) * (x * x)
    if (t_0 <= (-1d+17)) then
        tmp = t_1
    else if (t_0 <= 1d-6) then
        tmp = (3.0d0 * x) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (3.0 - (2.0 * x)) * (x * x);
	double t_1 = (-2.0 * x) * (x * x);
	double tmp;
	if (t_0 <= -1e+17) {
		tmp = t_1;
	} else if (t_0 <= 1e-6) {
		tmp = (3.0 * x) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (3.0 - (2.0 * x)) * (x * x)
	t_1 = (-2.0 * x) * (x * x)
	tmp = 0
	if t_0 <= -1e+17:
		tmp = t_1
	elif t_0 <= 1e-6:
		tmp = (3.0 * x) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(3.0 - Float64(2.0 * x)) * Float64(x * x))
	t_1 = Float64(Float64(-2.0 * x) * Float64(x * x))
	tmp = 0.0
	if (t_0 <= -1e+17)
		tmp = t_1;
	elseif (t_0 <= 1e-6)
		tmp = Float64(Float64(3.0 * x) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (3.0 - (2.0 * x)) * (x * x);
	t_1 = (-2.0 * x) * (x * x);
	tmp = 0.0;
	if (t_0 <= -1e+17)
		tmp = t_1;
	elseif (t_0 <= 1e-6)
		tmp = (3.0 * x) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(3.0 - N[(2.0 * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+17], t$95$1, If[LessEqual[t$95$0, 1e-6], N[(N[(3.0 * x), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64)))) < -1e17 or 9.99999999999999955e-7 < (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64))))

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if -1e17 < (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64)))) < 9.99999999999999955e-7

   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 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot 3} \]

      2. metadata-eval N/A

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

      3. lft-mult-inverse N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 97.7%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 - 2 \cdot x\right) \cdot \left(x \cdot x\right) \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\left(-2 \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;\left(3 - 2 \cdot x\right) \cdot \left(x \cdot x\right) \leq 10^{-6}:\\ \;\;\;\;\left(3 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. 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 2024230 
(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))))