Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 7.4s

Alternatives: 4

Speedup: 1.0×

• 33.0% 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* 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. Final simplification 99.8%

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

Alternative 2: 64.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.5) (* x (* x 3.0)) (* x -4.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = x * (x * 3.0);
	} else {
		tmp = x * -4.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.5d0) then
        tmp = x * (x * 3.0d0)
    else
        tmp = x * (-4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = x * (x * 3.0);
	} else {
		tmp = x * -4.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.5:
		tmp = x * (x * 3.0)
	else:
		tmp = x * -4.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5)
		tmp = Float64(x * Float64(x * 3.0));
	else
		tmp = Float64(x * -4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.5)
		tmp = x * (x * 3.0);
	else
		tmp = x * -4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.5], N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * -4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if 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 85.5%

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

   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* 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%

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

      2. flip-- 99.8%

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

      3. *-commutative 99.8%

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

      4. metadata-eval 99.8%

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

      5. cancel-sign-sub-inv 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*l/ 99.8%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in x around 0 6.5%

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

      1. *-commutative 6.5%

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

   9. Simplified 6.5%

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

   10. Taylor expanded in x around inf 6.5%

       \[\leadsto \color{blue}{-4.5 \cdot x} \]
   11. Step-by-step derivation

       1. *-commutative 6.5%

          \[\leadsto \color{blue}{x \cdot -4.5} \]

   12. Simplified 6.5%

       \[\leadsto \color{blue}{x \cdot -4.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = x * (x * 3.0);
	} 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) then
        tmp = x * (x * 3.0d0)
    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) {
		tmp = x * (x * 3.0);
	} else {
		tmp = x * (x * -3.0);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5)
		tmp = Float64(x * Float64(x * 3.0));
	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)
		tmp = x * (x * 3.0);
	else
		tmp = x * (x * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if 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 85.5%

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

   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* 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 0.4%

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

      1. add-cube-cbrt 0.4%

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

      2. pow3 0.4%

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

      3. *-commutative 0.4%

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

      4. associate-*r* 0.4%

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

      5. unpow2 0.4%

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

   7. Applied egg-rr 0.4%

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

      1. rem-cube-cbrt 0.4%

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

      2. *-commutative 0.4%

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

      3. unpow2 0.4%

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

      4. associate-*r* 0.4%

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

      5. metadata-eval 0.4%

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

      6. associate-*r* 0.4%

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

      7. neg-mul-1 0.4%

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

      8. *-commutative 0.4%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 50.9%

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

      11. sqr-neg 50.9%

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

      12. sqrt-prod 50.9%

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

      13. add-sqr-sqrt 50.9%

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

   9. Applied egg-rr 50.9%

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

4. Final simplification 75.3%

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

Alternative 4: 6.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot -4.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x -4.5))

C

double code(double x) {
	return x * -4.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (-4.5d0)
end function

Java

public static double code(double x) {
	return x * -4.5;
}

Python

def code(x):
	return x * -4.5

Julia

function code(x)
	return Float64(x * -4.5)
end

MATLAB

function tmp = code(x)
	tmp = x * -4.5;
end

Wolfram

code[x_] := N[(x * -4.5), $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* 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%

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

   2. flip-- 99.8%

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

   3. *-commutative 99.8%

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

   4. metadata-eval 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. *-commutative 99.8%

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

   7. associate-*l/ 99.8%

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

6. Applied egg-rr 97.7%

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

7. Taylor expanded in x around 0 52.3%

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

   1. *-commutative 52.3%

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

9. Simplified 52.3%

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

10. Taylor expanded in x around inf 6.3%

    \[\leadsto \color{blue}{-4.5 \cdot x} \]
11. Step-by-step derivation

    1. *-commutative 6.3%

       \[\leadsto \color{blue}{x \cdot -4.5} \]

12. Simplified 6.3%

    \[\leadsto \color{blue}{x \cdot -4.5} \]

13. Final simplification 6.3%

    \[\leadsto x \cdot -4.5 \]
14. Add Preprocessing

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 2024046 
(FPCore (x)
  :name "Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2"
  :precision binary64

  :alt
  (* x (* x (- 3.0 (* x 2.0))))

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