Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Percentage Accurate: 99.8% → 99.7%

Time: 4.9s

Alternatives: 3

Speedup: 1.0×

• 32.4% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 9 + {x}^{3} \cdot -4}{2 + \frac{3}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (* x 9.0) (* (pow x 3.0) -4.0)) (+ 2.0 (/ 3.0 x))))

C

double code(double x) {
	return ((x * 9.0) + (pow(x, 3.0) * -4.0)) / (2.0 + (3.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * 9.0d0) + ((x ** 3.0d0) * (-4.0d0))) / (2.0d0 + (3.0d0 / x))
end function

Java

public static double code(double x) {
	return ((x * 9.0) + (Math.pow(x, 3.0) * -4.0)) / (2.0 + (3.0 / x));
}

Python

def code(x):
	return ((x * 9.0) + (math.pow(x, 3.0) * -4.0)) / (2.0 + (3.0 / x))

Julia

function code(x)
	return Float64(Float64(Float64(x * 9.0) + Float64((x ^ 3.0) * -4.0)) / Float64(2.0 + Float64(3.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = ((x * 9.0) + ((x ^ 3.0) * -4.0)) / (2.0 + (3.0 / x));
end

Wolfram

code[x_] := N[(N[(N[(x * 9.0), $MachinePrecision] + N[(N[Power[x, 3.0], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(3.0 / x), $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 inf 99.7%

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

6. Taylor expanded in x around 0 99.7%

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

   1. associate-*r* 91.1%

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

   2. unpow2 91.1%

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

   3. cube-mult 91.2%

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

   4. flip-- 65.2%

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

   5. associate-*r/ 65.2%

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

   6. sub-neg 65.2%

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

   7. frac-times 65.2%

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

   8. metadata-eval 65.2%

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

   9. unpow2 65.2%

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

   10. metadata-eval 65.2%

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

   11. metadata-eval 65.2%

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

   12. +-commutative 65.2%

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

8. Applied egg-rr 65.2%

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

   1. distribute-lft-in 30.1%

      \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \frac{9}{{x}^{2}} + {x}^{3} \cdot -4}}{2 + \frac{3}{x}} \]

   2. cube-mult 30.1%

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

   3. unpow2 30.1%

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

   4. associate-*l* 46.0%

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

   5. associate-*r/ 49.1%

      \[\leadsto \frac{x \cdot \color{blue}{\frac{{x}^{2} \cdot 9}{{x}^{2}}} + {x}^{3} \cdot -4}{2 + \frac{3}{x}} \]

   6. *-rgt-identity 49.1%

      \[\leadsto \frac{x \cdot \frac{{x}^{2} \cdot 9}{\color{blue}{{x}^{2} \cdot 1}} + {x}^{3} \cdot -4}{2 + \frac{3}{x}} \]

   7. times-frac 49.1%

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

   8. *-inverses 99.9%

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

   9. metadata-eval 99.9%

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

   10. metadata-eval 99.9%

       \[\leadsto \frac{x \cdot \color{blue}{9} + {x}^{3} \cdot -4}{2 + \frac{3}{x}} \]

10. Simplified 99.9%

    \[\leadsto \color{blue}{\frac{x \cdot 9 + {x}^{3} \cdot -4}{2 + \frac{3}{x}}} \]

11. Final simplification 99.9%

    \[\leadsto \frac{x \cdot 9 + {x}^{3} \cdot -4}{2 + \frac{3}{x}} \]
12. 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: 63.5% 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 60.8%

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

6. Final simplification 60.8%

   \[\leadsto x \cdot \left(x \cdot 3\right) \]
7. 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 2024053 
(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))))