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

Percentage Accurate: 99.8% → 99.9%

Time: 9.6s

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, 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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

      \[\leadsto x \cdot \left(9 \cdot x + -12\right) + 3 \]

   4. metadata-eval N/A

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

   5. lft-mult-inverse N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

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

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

   9. distribute-rgt-in N/A

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

   10. sub-neg N/A

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

   11. *-commutative N/A

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

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(\left(9 - 12 \cdot \frac{1}{x}\right) \cdot x\right)}, 3\right) \]

5. Simplified 99.9%

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

Alternative 2: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, -12, 3\right)\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (* x (* x 9.0))
   (if (<= x 1.0) (fma x -12.0 3.0) (* 9.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = x * (x * 9.0);
	} else if (x <= 1.0) {
		tmp = fma(x, -12.0, 3.0);
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(x * Float64(x * 9.0));
	elseif (x <= 1.0)
		tmp = fma(x, -12.0, 3.0);
	else
		tmp = Float64(9.0 * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x * -12.0 + 3.0), $MachinePrecision], N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   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. +-rgt-identity N/A

         \[\leadsto 9 \cdot {x}^{2} + \color{blue}{0} \]

      2. *-commutative N/A

         \[\leadsto {x}^{2} \cdot 9 + 0 \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{9}, 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(\left({x}^{2} + 0\right), 9, 0\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(x \cdot x + 0\right), 9, 0\right) \]

      6. accelerator-lowering-fma.f64 98.2%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), 9, 0\right) \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 9, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \left(x \cdot x\right) \cdot 9 + 0 \]

      2. +-rgt-identity N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot 9\right), \color{blue}{x}\right) \]

      6. *-lowering-*.f64 98.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, 9\right), x\right) \]

   7. Applied egg-rr 98.4%

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

   if -1.55000000000000004 < x < 1

   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

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

      2. *-commutative N/A

         \[\leadsto x \cdot -12 + 3 \]

      3. accelerator-lowering-fma.f64 98.5%

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{-12}, 3\right) \]

   5. Simplified 98.5%

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

   if 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto 9 \cdot {x}^{2} + \color{blue}{0} \]

      2. *-commutative N/A

         \[\leadsto {x}^{2} \cdot 9 + 0 \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{9}, 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(\left({x}^{2} + 0\right), 9, 0\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(x \cdot x + 0\right), 9, 0\right) \]

      6. accelerator-lowering-fma.f64 96.7%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), 9, 0\right) \]

   5. Simplified 96.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 9, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \left(x \cdot x\right) \cdot 9 + 0 \]

      2. +-rgt-identity N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot 9\right), \color{blue}{x}\right) \]

      6. *-lowering-*.f64 96.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, 9\right), x\right) \]

   7. Applied egg-rr 96.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \color{blue}{9}\right) \]

      4. *-lowering-*.f64 96.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 9\right) \]

   9. Applied egg-rr 96.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, -12, 3\right)\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot 9\right)\\ \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, -12, 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x 9.0))))
   (if (<= x -1.55) t_0 (if (<= x 1.0) (fma x -12.0 3.0) t_0))))

C

double code(double x) {
	double t_0 = x * (x * 9.0);
	double tmp;
	if (x <= -1.55) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = fma(x, -12.0, 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * 9.0))
	tmp = 0.0
	if (x <= -1.55)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = fma(x, -12.0, 3.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.55], t$95$0, If[LessEqual[x, 1.0], N[(x * -12.0 + 3.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto 9 \cdot {x}^{2} + \color{blue}{0} \]

      2. *-commutative N/A

         \[\leadsto {x}^{2} \cdot 9 + 0 \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{9}, 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(\left({x}^{2} + 0\right), 9, 0\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(x \cdot x + 0\right), 9, 0\right) \]

      6. accelerator-lowering-fma.f64 97.4%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), 9, 0\right) \]

   5. Simplified 97.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 9, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \left(x \cdot x\right) \cdot 9 + 0 \]

      2. +-rgt-identity N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot 9\right), \color{blue}{x}\right) \]

      6. *-lowering-*.f64 97.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, 9\right), x\right) \]

   7. Applied egg-rr 97.4%

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

   if -1.55000000000000004 < x < 1

   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

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

      2. *-commutative N/A

         \[\leadsto x \cdot -12 + 3 \]

      3. accelerator-lowering-fma.f64 98.5%

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{-12}, 3\right) \]

   5. Simplified 98.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, -12, 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * 9.0 + 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. Step-by-step derivation

   1. distribute-lft-in N/A

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

   2. sub-neg N/A

      \[\leadsto 3 \cdot \left(\left(x \cdot 3\right) \cdot x + \left(\mathsf{neg}\left(x \cdot 4\right)\right)\right) + 3 \cdot 1 \]

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. swap-sqr N/A

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

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(x \cdot x\right), \color{blue}{\left(3 \cdot 3\right)}, \left(3 \cdot \left(\mathsf{neg}\left(x \cdot 4\right)\right) + 3\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{3} \cdot 3\right), \left(3 \cdot \left(\mathsf{neg}\left(x \cdot 4\right)\right) + 3\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(3 \cdot \left(\mathsf{neg}\left(x \cdot 4\right)\right) + 3\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(\left(\mathsf{neg}\left(x \cdot 4\right)\right) \cdot 3 + 3\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(\left(x \cdot \left(\mathsf{neg}\left(4\right)\right)\right) \cdot 3 + 3\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(x \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot 3\right) + 3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(x \cdot \left(-4 \cdot 3\right) + 3\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(x \cdot -12 + 3\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(x \cdot \left(3 \cdot -4\right) + 3\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(x \cdot \left(3 \cdot \left(\mathsf{neg}\left(4\right)\right)\right) + 3\right)\right) \]

   19. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \mathsf{fma.f64}\left(x, \left(3 \cdot \left(\mathsf{neg}\left(4\right)\right)\right), 3\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \mathsf{fma.f64}\left(x, \left(3 \cdot -4\right), 3\right)\right) \]

   21. metadata-eval 99.5%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \mathsf{fma.f64}\left(x, -12, 3\right)\right) \]

4. Applied egg-rr 99.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \color{blue}{3}\right) \]
6. Step-by-step derivation

7. Simplified 97.7%

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

Alternative 5: 97.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * 9.0), $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. Step-by-step derivation

   1. distribute-lft-in N/A

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

   2. sub-neg N/A

      \[\leadsto 3 \cdot \left(\left(x \cdot 3\right) \cdot x + \left(\mathsf{neg}\left(x \cdot 4\right)\right)\right) + 3 \cdot 1 \]

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. swap-sqr N/A

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

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(x \cdot x\right), \color{blue}{\left(3 \cdot 3\right)}, \left(3 \cdot \left(\mathsf{neg}\left(x \cdot 4\right)\right) + 3\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{3} \cdot 3\right), \left(3 \cdot \left(\mathsf{neg}\left(x \cdot 4\right)\right) + 3\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(3 \cdot \left(\mathsf{neg}\left(x \cdot 4\right)\right) + 3\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(\left(\mathsf{neg}\left(x \cdot 4\right)\right) \cdot 3 + 3\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(\left(x \cdot \left(\mathsf{neg}\left(4\right)\right)\right) \cdot 3 + 3\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(x \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot 3\right) + 3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(x \cdot \left(-4 \cdot 3\right) + 3\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(x \cdot -12 + 3\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(x \cdot \left(3 \cdot -4\right) + 3\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \left(x \cdot \left(3 \cdot \left(\mathsf{neg}\left(4\right)\right)\right) + 3\right)\right) \]

   19. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \mathsf{fma.f64}\left(x, \left(3 \cdot \left(\mathsf{neg}\left(4\right)\right)\right), 3\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \mathsf{fma.f64}\left(x, \left(3 \cdot -4\right), 3\right)\right) \]

   21. metadata-eval 99.5%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \mathsf{fma.f64}\left(x, -12, 3\right)\right) \]

4. Applied egg-rr 99.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), 9, \color{blue}{3}\right) \]
6. Step-by-step derivation

7. Simplified 97.7%

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

   1. associate-*l* N/A

      \[\leadsto x \cdot \left(x \cdot 9\right) + 3 \]

   2. *-commutative N/A

      \[\leadsto \left(x \cdot 9\right) \cdot x + 3 \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(x \cdot 9\right), \color{blue}{x}, 3\right) \]

   4. *-lowering-*.f64 97.7%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, 9\right), x, 3\right) \]

9. Applied egg-rr 97.7%

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

Alternative 6: 51.1% 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

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

   2. *-commutative N/A

      \[\leadsto x \cdot -12 + 3 \]

   3. accelerator-lowering-fma.f64 52.8%

      \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{-12}, 3\right) \]

5. Simplified 52.8%

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

Alternative 7: 50.7% 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. Simplified 52.6%

   \[\leadsto \color{blue}{3} \]
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 2024193 
(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)))