Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E

Percentage Accurate: 99.7% → 99.8%

Time: 8.6s

Alternatives: 4

Speedup: 1.6×

• 24.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Python

def code(x):
	return (3.0 * (2.0 - (x * 3.0))) * x

Julia

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

MATLAB

function tmp = code(x)
	tmp = (3.0 * (2.0 - (x * 3.0))) * x;
end

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Python

def code(x):
	return (3.0 * (2.0 - (x * 3.0))) * x

Julia

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

MATLAB

function tmp = code(x)
	tmp = (3.0 * (2.0 - (x * 3.0))) * x;
end

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 6, \left(x \cdot x\right) \cdot -9\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 6.0 (* (* x x) -9.0)))

C

double code(double x) {
	return fma(x, 6.0, ((x * x) * -9.0));
}

Julia

function code(x)
	return fma(x, 6.0, Float64(Float64(x * x) * -9.0))
end

Wolfram

code[x_] := N[(x * 6.0 + N[(N[(x * x), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. distribute-lft-in N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{6}, \mathsf{neg}\left(\left(x \cdot 3\right) \cdot \left(x \cdot 3\right)\right)\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, 6, \mathsf{neg}\left(\color{blue}{\left(x \cdot 3\right)} \cdot \left(x \cdot 3\right)\right)\right) \]

   15. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, 6, \mathsf{neg}\left(\left(x \cdot 3\right) \cdot \color{blue}{\left(x \cdot 3\right)}\right)\right) \]

   16. swap-sqr N/A

       \[\leadsto \mathsf{fma}\left(x, 6, \mathsf{neg}\left(\color{blue}{\left(x \cdot x\right) \cdot \left(3 \cdot 3\right)}\right)\right) \]

   17. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{fma}\left(x, 6, \color{blue}{\left(x \cdot x\right) \cdot \left(\mathsf{neg}\left(3 \cdot 3\right)\right)}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, 6, \color{blue}{\left(x \cdot x\right) \cdot \left(\mathsf{neg}\left(3 \cdot 3\right)\right)}\right) \]

   19. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, 6, \color{blue}{\left(x \cdot x\right)} \cdot \left(\mathsf{neg}\left(3 \cdot 3\right)\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, 6, \left(x \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{9}\right)\right)\right) \]

   21. metadata-eval 99.8

       \[\leadsto \mathsf{fma}\left(x, 6, \left(x \cdot x\right) \cdot \color{blue}{-9}\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 6, \left(x \cdot x\right) \cdot -9\right)} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right) \leq -500:\\ \;\;\;\;\left(x \cdot x\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x (* 3.0 (- 2.0 (* x 3.0)))) -500.0) (* (* x x) -9.0) (* x 6.0)))

C

double code(double x) {
	double tmp;
	if ((x * (3.0 * (2.0 - (x * 3.0)))) <= -500.0) {
		tmp = (x * x) * -9.0;
	} else {
		tmp = x * 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * (3.0d0 * (2.0d0 - (x * 3.0d0)))) <= (-500.0d0)) then
        tmp = (x * x) * (-9.0d0)
    else
        tmp = x * 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * (3.0 * (2.0 - (x * 3.0)))) <= -500.0) {
		tmp = (x * x) * -9.0;
	} else {
		tmp = x * 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * (3.0 * (2.0 - (x * 3.0)))) <= -500.0:
		tmp = (x * x) * -9.0
	else:
		tmp = x * 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * Float64(3.0 * Float64(2.0 - Float64(x * 3.0)))) <= -500.0)
		tmp = Float64(Float64(x * x) * -9.0);
	else
		tmp = Float64(x * 6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * (3.0 * (2.0 - (x * 3.0)))) <= -500.0)
		tmp = (x * x) * -9.0;
	else
		tmp = x * 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -500.0], N[(N[(x * x), $MachinePrecision] * -9.0), $MachinePrecision], N[(x * 6.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (-.f64 #s(literal 2 binary64) (*.f64 x #s(literal 3 binary64)))) x) < -500

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-9 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-9 \cdot {x}^{2}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -500 < (*.f64 (*.f64 #s(literal 3 binary64) (-.f64 #s(literal 2 binary64) (*.f64 x #s(literal 3 binary64)))) x)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{6} \cdot x \]
   4. Step-by-step derivation

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{6} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right) \leq -500:\\ \;\;\;\;\left(x \cdot x\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 6 \cdot x - 9 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* 6.0 x) (* 9.0 (* x x))))

C

double code(double x) {
	return (6.0 * x) - (9.0 * (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (6.0d0 * x) - (9.0d0 * (x * x))
end function

Java

public static double code(double x) {
	return (6.0 * x) - (9.0 * (x * x));
}

Python

def code(x):
	return (6.0 * x) - (9.0 * (x * x))

Julia

function code(x)
	return Float64(Float64(6.0 * x) - Float64(9.0 * Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * x) - (9.0 * (x * x));
end

Wolfram

code[x_] := N[(N[(6.0 * x), $MachinePrecision] - N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024230 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* 6 x) (* 9 (* x x))))

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