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

Percentage Accurate: 99.8% → 99.9%

Time: 7.1s

Alternatives: 7

Speedup: 2.1×

• 25.4% 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

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot 3, \mathsf{fma}\left(x, 3, -4\right), 3\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* x 3.0) (fma x 3.0 -4.0) 3.0))

C

double code(double x) {
	return fma((x * 3.0), fma(x, 3.0, -4.0), 3.0);
}

Julia

function code(x)
	return fma(Float64(x * 3.0), fma(x, 3.0, -4.0), 3.0)
end

Wolfram

code[x_] := N[(N[(x * 3.0), $MachinePrecision] * N[(x * 3.0 + -4.0), $MachinePrecision] + 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. distribute-lft-in N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-lft-out-- N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. sub-neg N/A

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

   17. lift-*.f64 N/A

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

   18. lower-fma.f64 N/A

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

   19. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 3, \mathsf{fma}\left(x, 3, -4\right), 3\right)} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(x \cdot 3\right) - x \cdot 4 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, -12, 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, 9, -12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (* x (* x 3.0)) (* x 4.0)) 5e-7)
   (fma x -12.0 3.0)
   (* x (fma x 9.0 -12.0))))

C

double code(double x) {
	double tmp;
	if (((x * (x * 3.0)) - (x * 4.0)) <= 5e-7) {
		tmp = fma(x, -12.0, 3.0);
	} else {
		tmp = x * fma(x, 9.0, -12.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x * Float64(x * 3.0)) - Float64(x * 4.0)) <= 5e-7)
		tmp = fma(x, -12.0, 3.0);
	else
		tmp = Float64(x * fma(x, 9.0, -12.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision], 5e-7], N[(x * -12.0 + 3.0), $MachinePrecision], N[(x * N[(x * 9.0 + -12.0), $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))) < 4.99999999999999977e-7

   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. *-commutative N/A

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

      3. lower-fma.f64 99.1

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

   5. Simplified 99.1%

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

   if 4.99999999999999977e-7 < (-.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

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. rgt-mult-inverse N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 99.0

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

   5. Simplified 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(x \cdot 3\right) - x \cdot 4 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, -12, 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, 9, -12\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(x \cdot 3\right) - x \cdot 4 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, -12, 3\right)\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (* x (* x 3.0)) (* x 4.0)) 5e-7)
   (fma x -12.0 3.0)
   (* 9.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (((x * (x * 3.0)) - (x * 4.0)) <= 5e-7) {
		tmp = fma(x, -12.0, 3.0);
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x * Float64(x * 3.0)) - Float64(x * 4.0)) <= 5e-7)
		tmp = fma(x, -12.0, 3.0);
	else
		tmp = Float64(9.0 * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision], 5e-7], N[(x * -12.0 + 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))) < 4.99999999999999977e-7

   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. *-commutative N/A

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

      3. lower-fma.f64 99.1

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

   5. Simplified 99.1%

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

   if 4.99999999999999977e-7 < (-.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 98.6

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

   5. Simplified 98.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(x \cdot 3\right) - x \cdot 4 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, -12, 3\right)\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(x \cdot 3\right) - x \cdot 4 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, -12, 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (* x (* x 3.0)) (* x 4.0)) 5e-7)
   (fma x -12.0 3.0)
   (* x (* x 9.0))))

C

double code(double x) {
	double tmp;
	if (((x * (x * 3.0)) - (x * 4.0)) <= 5e-7) {
		tmp = fma(x, -12.0, 3.0);
	} else {
		tmp = x * (x * 9.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x * Float64(x * 3.0)) - Float64(x * 4.0)) <= 5e-7)
		tmp = fma(x, -12.0, 3.0);
	else
		tmp = Float64(x * Float64(x * 9.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision], 5e-7], N[(x * -12.0 + 3.0), $MachinePrecision], N[(x * N[(x * 9.0), $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))) < 4.99999999999999977e-7

   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. *-commutative N/A

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

      3. lower-fma.f64 99.1

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

   5. Simplified 99.1%

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

   if 4.99999999999999977e-7 < (-.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 98.6

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

   5. Simplified 98.6%

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

      1. metadata-eval N/A

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

      2. swap-sqr N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 98.4

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

   7. Applied egg-rr 98.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(x \cdot 3\right) - x \cdot 4 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, -12, 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x * N[(x * 9.0 + -12.0), $MachinePrecision] + 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. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. lft-mult-inverse N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

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

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

   9. distribute-rgt-in N/A

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

   10. sub-neg N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

5. Simplified 99.9%

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

Alternative 6: 51.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x * -12.0 + 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. *-commutative N/A

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

   3. lower-fma.f64 53.5

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

5. Simplified 53.5%

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

Alternative 7: 51.6% 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. Simplified 53.1%

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

Developer Target 1: 99.8% 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 2024207 
(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)))