Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.8%

Time: 6.5s

Alternatives: 4

Speedup: 1.0×

• 33.2% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x * N[(3.0 * x + N[(N[Power[x, 2.0], $MachinePrecision] * -2.0), $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. Taylor expanded in x around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. fma-define 99.9%

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

   3. *-commutative 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 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 3: 63.1% 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* 99.8%

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

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 61.8%

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

6. Final simplification 61.8%

   \[\leadsto x \cdot \left(x \cdot 3\right) \]
7. Add Preprocessing

Alternative 4: 3.2% 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. flip-- 99.8%

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

   2. *-commutative 99.8%

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

   3. metadata-eval 99.8%

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

   4. cancel-sign-sub-inv 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*r/ 99.8%

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

6. Applied egg-rr 46.8%

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

   1. *-commutative 46.8%

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

   2. associate-/l* 46.8%

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

8. Simplified 46.8%

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

   1. associate-*r* 46.8%

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

   2. associate-*r/ 46.8%

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

   3. +-commutative 46.8%

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

   4. fma-define 46.8%

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

10. Applied egg-rr 46.8%

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

11. Taylor expanded in x around 0 46.9%

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

    1. *-commutative 46.9%

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

13. Simplified 46.9%

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

14. Taylor expanded in x around inf 3.0%

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

    1. *-commutative 3.0%

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

16. Simplified 3.0%

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

17. Final simplification 3.0%

    \[\leadsto x \cdot 4.5 \]
18. 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 2024044 
(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))))