Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 5.6s

Alternatives: 4

Speedup: 1.0×

• 33.1% 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, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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* 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. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. cancel-sign-sub-inv 99.8%

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

   3. metadata-eval 99.8%

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

   4. *-commutative 99.8%

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

   5. distribute-lft-in 99.8%

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

   6. fma-define 99.8%

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

   7. add-sqr-sqrt 41.7%

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

   8. sqrt-unprod 73.9%

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

   9. swap-sqr 73.9%

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

   10. metadata-eval 73.9%

       \[\leadsto x \cdot \mathsf{fma}\left(x, 3, x \cdot \sqrt{\left(x \cdot x\right) \cdot \color{blue}{4}}\right) \]

   11. metadata-eval 73.9%

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

   12. swap-sqr 73.9%

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

   13. sqrt-unprod 32.2%

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

   14. add-sqr-sqrt 51.0%

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

6. Applied egg-rr 51.0%

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

   1. fma-undefine 51.0%

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

   2. distribute-lft-in 51.0%

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

   3. *-commutative 51.0%

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

   4. *-commutative 51.0%

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

   5. add-sqr-sqrt 32.2%

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

   6. sqrt-unprod 73.9%

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

   7. swap-sqr 73.9%

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

   8. metadata-eval 73.9%

      \[\leadsto \left(x \cdot 3\right) \cdot x + x \cdot \left(\sqrt{\left(x \cdot x\right) \cdot \color{blue}{4}} \cdot x\right) \]

   9. metadata-eval 73.9%

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

   10. swap-sqr 73.9%

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

   11. sqrt-unprod 41.7%

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

   12. add-sqr-sqrt 91.6%

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

   13. *-commutative 91.6%

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

   14. associate-*r* 91.6%

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

   15. associate-*r* 91.6%

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

   16. cube-mult 91.7%

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

   17. *-commutative 91.7%

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

   18. fma-define 99.9%

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

   19. *-commutative 99.9%

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

8. Applied egg-rr 99.9%

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

Alternative 2: 97.7% accurate, 0.5× 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 99.8%

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

   5. Taylor expanded in x around inf 98.5%

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

      1. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   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* 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. Add Preprocessing

   5. Taylor expanded in x around 0 95.9%

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

      1. *-commutative 95.9%

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

   7. Simplified 95.9%

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

4. Final simplification 97.1%

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

Alternative 3: 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 99.7%

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
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. Add Preprocessing
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 99.7%

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

5. Taylor expanded in x around 0 65.9%

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

   1. *-commutative 65.9%

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

7. Simplified 65.9%

   \[\leadsto x \cdot \color{blue}{\left(x \cdot 3\right)} \]
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 2024181 
(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))))