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

Percentage Accurate: 99.8% → 99.9%

Time: 6.0s

Alternatives: 7

Speedup: 2.1×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))

C

double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * ((((x * 3.0d0) * x) - (x * 4.0d0)) + 1.0d0)
end function

Java

public static double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Python

def code(x):
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0)

Julia

function code(x)
	return Float64(3.0 * Float64(Float64(Float64(Float64(x * 3.0) * x) - Float64(x * 4.0)) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
end

Wolfram

code[x_] := N[(3.0 * N[(N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))

C

double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * ((((x * 3.0d0) * x) - (x * 4.0d0)) + 1.0d0)
end function

Java

public static double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Python

def code(x):
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0)

Julia

function code(x)
	return Float64(3.0 * Float64(Float64(Float64(Float64(x * 3.0) * x) - Float64(x * 4.0)) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
end

Wolfram

code[x_] := N[(3.0 * N[(N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, 9, \mathsf{fma}\left(-12, x, 3\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* x x) 9.0 (fma -12.0 x 3.0)))

C

double code(double x) {
	return fma((x * x), 9.0, fma(-12.0, x, 3.0));
}

Julia

function code(x)
	return fma(Float64(x * x), 9.0, fma(-12.0, x, 3.0))
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * 9.0 + N[(-12.0 * x + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

      \[\leadsto 3 \cdot \color{blue}{\left(\left(x \cdot 3\right) \cdot x + \left(\mathsf{neg}\left(x \cdot 4\right)\right)\right)} + 3 \]

   7. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(\left(x \cdot 3\right) \cdot x\right) \cdot 3 + \left(\mathsf{neg}\left(x \cdot 4\right)\right) \cdot 3\right)} + 3 \]

   8. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot 3\right) \cdot x\right) \cdot 3 + \left(\left(\mathsf{neg}\left(x \cdot 4\right)\right) \cdot 3 + 3\right)} \]

   9. *-commutative N/A

      \[\leadsto \left(\left(x \cdot 3\right) \cdot x\right) \cdot 3 + \left(\color{blue}{3 \cdot \left(\mathsf{neg}\left(x \cdot 4\right)\right)} + 3\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \color{blue}{\left(\left(x \cdot 3\right) \cdot x\right)} \cdot 3 + \left(3 \cdot \left(\mathsf{neg}\left(x \cdot 4\right)\right) + 3\right) \]

   11. associate-*r* N/A

       \[\leadsto \color{blue}{\left(x \cdot 3\right) \cdot \left(x \cdot 3\right)} + \left(3 \cdot \left(\mathsf{neg}\left(x \cdot 4\right)\right) + 3\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \color{blue}{\left(x \cdot 3\right)} \cdot \left(x \cdot 3\right) + \left(3 \cdot \left(\mathsf{neg}\left(x \cdot 4\right)\right) + 3\right) \]

   13. swap-sqr N/A

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(3 \cdot 3\right)} + \left(3 \cdot \left(\mathsf{neg}\left(x \cdot 4\right)\right) + 3\right) \]

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 9, \mathsf{fma}\left(-12, x, 3\right)\right)} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 \cdot x\right) \cdot x - 4 \cdot x \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(-12, x, 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, x, -12\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (* (* 3.0 x) x) (* 4.0 x)) 0.0001)
   (fma -12.0 x 3.0)
   (* (fma 9.0 x -12.0) x)))

C

double code(double x) {
	double tmp;
	if ((((3.0 * x) * x) - (4.0 * x)) <= 0.0001) {
		tmp = fma(-12.0, x, 3.0);
	} else {
		tmp = fma(9.0, x, -12.0) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 * x) * x) - Float64(4.0 * x)) <= 0.0001)
		tmp = fma(-12.0, x, 3.0);
	else
		tmp = Float64(fma(9.0, x, -12.0) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(3.0 * x), $MachinePrecision] * x), $MachinePrecision] - N[(4.0 * x), $MachinePrecision]), $MachinePrecision], 0.0001], N[(-12.0 * x + 3.0), $MachinePrecision], N[(N[(9.0 * x + -12.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 x #s(literal 3 binary64)) x) (*.f64 x #s(literal 4 binary64))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{3 + -12 \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-12 \cdot x + 3} \]

      2. lower-fma.f64 98.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(-12, x, 3\right)} \]

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-12, x, 3\right)} \]

   if 1.00000000000000005e-4 < (-.f64 (*.f64 (*.f64 x #s(literal 3 binary64)) x) (*.f64 x #s(literal 4 binary64)))

   1. Initial program 99.7%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(9 - 12 \cdot \frac{1}{x}\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(9 - 12 \cdot \frac{1}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(9 - 12 \cdot \frac{1}{x}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(9 - 12 \cdot \frac{1}{x}\right)\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(9 - 12 \cdot \frac{1}{x}\right)\right) \cdot x} \]

      5. sub-neg N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(9 + \left(\mathsf{neg}\left(12 \cdot \frac{1}{x}\right)\right)\right)}\right) \cdot x \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(9 \cdot x + \left(\mathsf{neg}\left(12 \cdot \frac{1}{x}\right)\right) \cdot x\right)} \cdot x \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left(9 \cdot x + \color{blue}{\left(\left(\mathsf{neg}\left(12\right)\right) \cdot \frac{1}{x}\right)} \cdot x\right) \cdot x \]

      8. metadata-eval N/A

         \[\leadsto \left(9 \cdot x + \left(\color{blue}{-12} \cdot \frac{1}{x}\right) \cdot x\right) \cdot x \]

      9. associate-*l* N/A

         \[\leadsto \left(9 \cdot x + \color{blue}{-12 \cdot \left(\frac{1}{x} \cdot x\right)}\right) \cdot x \]

      10. lft-mult-inverse N/A

          \[\leadsto \left(9 \cdot x + -12 \cdot \color{blue}{1}\right) \cdot x \]

      11. metadata-eval N/A

          \[\leadsto \left(9 \cdot x + \color{blue}{-12}\right) \cdot x \]

      12. lower-fma.f64 98.4

          \[\leadsto \color{blue}{\mathsf{fma}\left(9, x, -12\right)} \cdot x \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(9, x, -12\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot x\right) \cdot x - 4 \cdot x \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(-12, x, 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, x, -12\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 \cdot x\right) \cdot x - 4 \cdot x \leq 10:\\ \;\;\;\;\mathsf{fma}\left(-12, x, 3\right)\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (* (* 3.0 x) x) (* 4.0 x)) 10.0)
   (fma -12.0 x 3.0)
   (* 9.0 (* x x))))

C

double code(double x) {
	double tmp;
	if ((((3.0 * x) * x) - (4.0 * x)) <= 10.0) {
		tmp = fma(-12.0, x, 3.0);
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 * x) * x) - Float64(4.0 * x)) <= 10.0)
		tmp = fma(-12.0, x, 3.0);
	else
		tmp = Float64(9.0 * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(3.0 * x), $MachinePrecision] * x), $MachinePrecision] - N[(4.0 * x), $MachinePrecision]), $MachinePrecision], 10.0], N[(-12.0 * x + 3.0), $MachinePrecision], N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 x #s(literal 3 binary64)) x) (*.f64 x #s(literal 4 binary64))) < 10

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{3 + -12 \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-12 \cdot x + 3} \]

      2. lower-fma.f64 98.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(-12, x, 3\right)} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-12, x, 3\right)} \]

   if 10 < (-.f64 (*.f64 (*.f64 x #s(literal 3 binary64)) x) (*.f64 x #s(literal 4 binary64)))

   1. Initial program 99.7%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 9} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot x\right) \cdot x - 4 \cdot x \leq 10:\\ \;\;\;\;\mathsf{fma}\left(-12, x, 3\right)\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 \cdot x\right) \cdot x - 4 \cdot x \leq 10:\\ \;\;\;\;\mathsf{fma}\left(-12, x, 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (* (* 3.0 x) x) (* 4.0 x)) 10.0)
   (fma -12.0 x 3.0)
   (* (* 9.0 x) x)))

C

double code(double x) {
	double tmp;
	if ((((3.0 * x) * x) - (4.0 * x)) <= 10.0) {
		tmp = fma(-12.0, x, 3.0);
	} else {
		tmp = (9.0 * x) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 * x) * x) - Float64(4.0 * x)) <= 10.0)
		tmp = fma(-12.0, x, 3.0);
	else
		tmp = Float64(Float64(9.0 * x) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(3.0 * x), $MachinePrecision] * x), $MachinePrecision] - N[(4.0 * x), $MachinePrecision]), $MachinePrecision], 10.0], N[(-12.0 * x + 3.0), $MachinePrecision], N[(N[(9.0 * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 x #s(literal 3 binary64)) x) (*.f64 x #s(literal 4 binary64))) < 10

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{3 + -12 \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-12 \cdot x + 3} \]

      2. lower-fma.f64 98.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(-12, x, 3\right)} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-12, x, 3\right)} \]

   if 10 < (-.f64 (*.f64 (*.f64 x #s(literal 3 binary64)) x) (*.f64 x #s(literal 4 binary64)))

   1. Initial program 99.7%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 9} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.2%

      \[\leadsto \left(9 \cdot x\right) \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot x\right) \cdot x - 4 \cdot x \leq 10:\\ \;\;\;\;\mathsf{fma}\left(-12, x, 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(9, x, -12\right), x, 3\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (fma 9.0 x -12.0) x 3.0))

C

double code(double x) {
	return fma(fma(9.0, x, -12.0), x, 3.0);
}

Julia

function code(x)
	return fma(fma(9.0, x, -12.0), x, 3.0)
end

Wolfram

code[x_] := N[(N[(9.0 * x + -12.0), $MachinePrecision] * x + 3.0), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{3 + x \cdot \left(9 \cdot x - 12\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left(9 \cdot x - 12\right) + 3} \]

   2. distribute-rgt-out-- N/A

      \[\leadsto \color{blue}{\left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right)} + 3 \]

   3. associate-*r* N/A

      \[\leadsto \left(\color{blue}{9 \cdot \left(x \cdot x\right)} - 12 \cdot x\right) + 3 \]

   4. unpow2 N/A

      \[\leadsto \left(9 \cdot \color{blue}{{x}^{2}} - 12 \cdot x\right) + 3 \]

   5. metadata-eval N/A

      \[\leadsto \left(9 \cdot {x}^{2} - \color{blue}{\left(\mathsf{neg}\left(-12\right)\right)} \cdot x\right) + 3 \]

   6. metadata-eval N/A

      \[\leadsto \left(9 \cdot {x}^{2} - \left(\mathsf{neg}\left(\color{blue}{-12 \cdot 1}\right)\right) \cdot x\right) + 3 \]

   7. lft-mult-inverse N/A

      \[\leadsto \left(9 \cdot {x}^{2} - \left(\mathsf{neg}\left(-12 \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right)\right) \cdot x\right) + 3 \]

   8. associate-*l* N/A

      \[\leadsto \left(9 \cdot {x}^{2} - \left(\mathsf{neg}\left(\color{blue}{\left(-12 \cdot \frac{1}{x}\right) \cdot x}\right)\right) \cdot x\right) + 3 \]

   9. metadata-eval N/A

      \[\leadsto \left(9 \cdot {x}^{2} - \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(12\right)\right)} \cdot \frac{1}{x}\right) \cdot x\right)\right) \cdot x\right) + 3 \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \left(9 \cdot {x}^{2} - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(12 \cdot \frac{1}{x}\right)\right)} \cdot x\right)\right) \cdot x\right) + 3 \]

   11. distribute-lft-neg-out N/A

       \[\leadsto \left(9 \cdot {x}^{2} - \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(12 \cdot \frac{1}{x}\right)\right)\right)\right) \cdot x\right)} \cdot x\right) + 3 \]

   12. remove-double-neg N/A

       \[\leadsto \left(9 \cdot {x}^{2} - \left(\color{blue}{\left(12 \cdot \frac{1}{x}\right)} \cdot x\right) \cdot x\right) + 3 \]

   13. associate-*r* N/A

       \[\leadsto \left(9 \cdot {x}^{2} - \color{blue}{\left(12 \cdot \frac{1}{x}\right) \cdot \left(x \cdot x\right)}\right) + 3 \]

   14. unpow2 N/A

       \[\leadsto \left(9 \cdot {x}^{2} - \left(12 \cdot \frac{1}{x}\right) \cdot \color{blue}{{x}^{2}}\right) + 3 \]

   15. distribute-rgt-out-- N/A

       \[\leadsto \color{blue}{{x}^{2} \cdot \left(9 - 12 \cdot \frac{1}{x}\right)} + 3 \]

   16. unpow2 N/A

       \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(9 - 12 \cdot \frac{1}{x}\right) + 3 \]

   17. associate-*l* N/A

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(9 - 12 \cdot \frac{1}{x}\right)\right)} + 3 \]

   18. *-commutative N/A

       \[\leadsto \color{blue}{\left(x \cdot \left(9 - 12 \cdot \frac{1}{x}\right)\right) \cdot x} + 3 \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(9, x, -12\right), x, 3\right)} \]
6. Add Preprocessing

Alternative 6: 52.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-12, x, 3\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma -12.0 x 3.0))

C

double code(double x) {
	return fma(-12.0, x, 3.0);
}

Julia

function code(x)
	return fma(-12.0, x, 3.0)
end

Wolfram

code[x_] := N[(-12.0 * x + 3.0), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{3 + -12 \cdot x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{-12 \cdot x + 3} \]

   2. lower-fma.f64 50.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(-12, x, 3\right)} \]

5. Applied rewrites 50.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-12, x, 3\right)} \]
6. Add Preprocessing

Alternative 7: 51.7% accurate, 27.0× speedup

Math

\[\begin{array}{l} \\ 3 \end{array} \]

FPCore

(FPCore (x) :precision binary64 3.0)

C

double code(double x) {
	return 3.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 3.0

Julia

function code(x)
	return 3.0
end

MATLAB

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

Wolfram

code[x_] := 3.0

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{3} \]
4. Step-by-step derivation

5. Applied rewrites 49.3%

   \[\leadsto \color{blue}{3} \]
6. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x))))

C

double code(double x) {
	return 3.0 + (((9.0 * x) * x) - (12.0 * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 + (((9.0d0 * x) * x) - (12.0d0 * x))
end function

Java

public static double code(double x) {
	return 3.0 + (((9.0 * x) * x) - (12.0 * x));
}

Python

def code(x):
	return 3.0 + (((9.0 * x) * x) - (12.0 * x))

Julia

function code(x)
	return Float64(3.0 + Float64(Float64(Float64(9.0 * x) * x) - Float64(12.0 * x)))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 + (((9.0 * x) * x) - (12.0 * x));
end

Wolfram

code[x_] := N[(3.0 + N[(N[(N[(9.0 * x), $MachinePrecision] * x), $MachinePrecision] - N[(12.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024249 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :alt
  (! :herbie-platform default (+ 3 (- (* (* 9 x) x) (* 12 x))))

  (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))