Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 5.9s

Alternatives: 5

Speedup: 1.0×

• 33.9% 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* 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right) \]

Alternative 2: 97.6% accurate, 0.8× 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 99.9%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.5%

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

      1. unpow2 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*r* 97.5%

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

   6. Simplified 97.5%

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

   if -1.5 < x < 1.5

   1. Initial program 99.8%

      \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
   2. 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. Taylor expanded in x around 0 98.2%

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

4. Final simplification 97.9%

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

Alternative 3: 62.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 (* x x)))

C

double code(double x) {
	return 3.0 * (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * (x * x)
end function

Java

public static double code(double x) {
	return 3.0 * (x * x);
}

Python

def code(x):
	return 3.0 * (x * x)

Julia

function code(x)
	return Float64(3.0 * Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * (x * x);
end

Wolfram

code[x_] := N[(3.0 * N[(x * x), $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* 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 66.8%

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

   1. unpow2 66.8%

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

6. Simplified 66.8%

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

7. Final simplification 66.8%

   \[\leadsto 3 \cdot \left(x \cdot x\right) \]

Alternative 4: 62.4% 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* 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 66.9%

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

5. Final simplification 66.9%

   \[\leadsto x \cdot \left(x \cdot 3\right) \]

Alternative 5: 1.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{10.125}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 10.125 x))

C

double code(double x) {
	return 10.125 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 10.125d0 / x
end function

Java

public static double code(double x) {
	return 10.125 / x;
}

Python

def code(x):
	return 10.125 / x

Julia

function code(x)
	return Float64(10.125 / x)
end

MATLAB

function tmp = code(x)
	tmp = 10.125 / x;
end

Wolfram

code[x_] := N[(10.125 / x), $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* 99.9%

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

3. Simplified 99.9%

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

   1. associate-*r* 99.8%

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

   2. flip-- 99.8%

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

   3. associate-*r/ 95.4%

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

   4. metadata-eval 95.4%

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

   5. swap-sqr 95.4%

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

   6. metadata-eval 95.4%

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

   7. +-commutative 95.4%

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

   8. fma-def 95.4%

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

5. Applied egg-rr 95.4%

   \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(9 - \left(x \cdot x\right) \cdot 4\right)}{\mathsf{fma}\left(x, 2, 3\right)}} \]
6. Step-by-step derivation

   1. *-commutative 95.4%

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

   2. associate-/l* 99.3%

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

   3. associate-*l* 99.3%

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

7. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{9 - x \cdot \left(x \cdot 4\right)}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x \cdot x}}} \]
8. Step-by-step derivation

   1. div-sub 79.0%

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

   2. sub-neg 79.0%

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

   3. div-inv 78.9%

      \[\leadsto \color{blue}{9 \cdot \frac{1}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x \cdot x}}} + \left(-\frac{x \cdot \left(x \cdot 4\right)}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x \cdot x}}\right) \]

   4. clear-num 79.4%

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

   5. associate-/l* 99.7%

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

   6. div-inv 99.7%

      \[\leadsto 9 \cdot \frac{x}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x}} + \left(-\color{blue}{\left(x \cdot \left(x \cdot 4\right)\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x \cdot x}}}\right) \]

   7. clear-num 99.8%

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

   8. associate-/l* 99.8%

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

9. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{9 \cdot \frac{x}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x}} + \left(-\left(x \cdot \left(x \cdot 4\right)\right) \cdot \frac{x}{\frac{\mathsf{fma}\left(x, 2, 3\right)}{x}}\right)} \]
10. Step-by-step derivation

    1. sub-neg 99.8%

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

    2. distribute-rgt-out-- 99.8%

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

    3. associate-/r/ 99.8%

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

11. Simplified 99.8%

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

12. Taylor expanded in x around inf 44.5%

    \[\leadsto \color{blue}{\left(\left(1.125 \cdot \frac{1}{x} + 0.5 \cdot x\right) - 0.75\right)} \cdot \left(9 - x \cdot \left(x \cdot 4\right)\right) \]

13. Taylor expanded in x around 0 1.9%

    \[\leadsto \color{blue}{\frac{10.125}{x}} \]

14. Final simplification 1.9%

    \[\leadsto \frac{10.125}{x} \]

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 2023182 
(FPCore (x)
  :name "Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2"
  :precision binary64

  :herbie-target
  (* x (* x (- 3.0 (* x 2.0))))

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