Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 7.8s

Alternatives: 6

Speedup: 1.1×

• 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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. unpow1 N/A

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

   7. metadata-eval N/A

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

   8. sqrt-pow1 N/A

      \[\leadsto \left(3 + \color{blue}{\sqrt{{\left(\mathsf{neg}\left(x\right)\right)}^{2}}} \cdot 2\right) \cdot \left(x \cdot x\right) \]

   9. pow2 N/A

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

   10. sqr-neg-rev N/A

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

   11. sqrt-prod N/A

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

   12. rem-square-sqrt N/A

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

   13. lift-*.f64 N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \leq -2 \cdot 10^{+24}:\\ \;\;\;\;-3 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (* x x) (- 3.0 (* x 2.0))) -2e+24)
   (* -3.0 (* x x))
   (* (* x x) 3.0)))

C

double code(double x) {
	double tmp;
	if (((x * x) * (3.0 - (x * 2.0))) <= -2e+24) {
		tmp = -3.0 * (x * x);
	} else {
		tmp = (x * x) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x * x) * (3.0d0 - (x * 2.0d0))) <= (-2d+24)) then
        tmp = (-3.0d0) * (x * x)
    else
        tmp = (x * x) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x * x) * (3.0 - (x * 2.0))) <= -2e+24) {
		tmp = -3.0 * (x * x);
	} else {
		tmp = (x * x) * 3.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x * x) * (3.0 - (x * 2.0))) <= -2e+24:
		tmp = -3.0 * (x * x)
	else:
		tmp = (x * x) * 3.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x * x) * Float64(3.0 - Float64(x * 2.0))) <= -2e+24)
		tmp = Float64(-3.0 * Float64(x * x));
	else
		tmp = Float64(Float64(x * x) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x * x) * (3.0 - (x * 2.0))) <= -2e+24)
		tmp = -3.0 * (x * x);
	else
		tmp = (x * x) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+24], N[(-3.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64)))) < -2e24

   1. Initial program 99.9%

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 50.1

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

   7. Applied rewrites 50.1%

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

   if -2e24 < (*.f64 (*.f64 x x) (-.f64 #s(literal 3 binary64) (*.f64 x #s(literal 2 binary64))))

   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

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

   5. Applied rewrites 85.7%

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

Alternative 3: 97.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.5)) {
		tmp = (x * x) * (-2.0 * x);
	} 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) * ((-2.0d0) * x)
    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) * (-2.0 * x);
	} 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) * (-2.0 * x)
	else:
		tmp = (x * x) * 3.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.5) || !(x <= 1.5))
		tmp = Float64(Float64(x * x) * Float64(-2.0 * x));
	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) * (-2.0 * x);
	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[(N[(x * x), $MachinePrecision] * N[(-2.0 * x), $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. 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. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if -1.5 < x < 1.5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.6%

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

4. Final simplification 98.4%

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

Alternative 4: 41.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-124}:\\ \;\;\;\;-9 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.4e-124) (* -9.0 x) (* -3.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= -5.4e-124) {
		tmp = -9.0 * x;
	} else {
		tmp = -3.0 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.4d-124)) then
        tmp = (-9.0d0) * x
    else
        tmp = (-3.0d0) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.4e-124) {
		tmp = -9.0 * x;
	} else {
		tmp = -3.0 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.4e-124:
		tmp = -9.0 * x
	else:
		tmp = -3.0 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.4e-124)
		tmp = Float64(-9.0 * x);
	else
		tmp = Float64(-3.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.4e-124)
		tmp = -9.0 * x;
	else
		tmp = -3.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.4e-124], N[(-9.0 * x), $MachinePrecision], N[(-3.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.40000000000000035e-124

   1. Initial program 99.8%

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

   4. Applied rewrites 64.8%

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

   5. Taylor expanded in x around -inf

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

   7. Applied rewrites 69.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 6.7%

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

   if -5.40000000000000035e-124 < x

   1. Initial program 99.9%

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

   4. Applied rewrites 70.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 55.5

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

   7. Applied rewrites 55.5%

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

Alternative 5: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 6: 6.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ -9 \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* -9.0 x))

C

double code(double x) {
	return -9.0 * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-9.0d0) * x
end function

Java

public static double code(double x) {
	return -9.0 * x;
}

Python

def code(x):
	return -9.0 * x

Julia

function code(x)
	return Float64(-9.0 * x)
end

MATLAB

function tmp = code(x)
	tmp = -9.0 * x;
end

Wolfram

code[x_] := N[(-9.0 * x), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 68.4%

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

5. Taylor expanded in x around -inf

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

7. Applied rewrites 50.1%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 5.9%

    \[\leadsto -9 \cdot \color{blue}{x} \]
11. 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 2024337 
(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))))