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

Percentage Accurate: 99.8% → 99.9%

Time: 6.3s

Alternatives: 6

Speedup: 2.1×

• 25.2% 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

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

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. remove-double-neg N/A

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

   2. +-commutative N/A

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

   3. remove-double-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. rgt-mult-inverse N/A

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

   7. metadata-eval N/A

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

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

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

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

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

   13. fp-cancel-sign-sub-inv N/A

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

   14. *-commutative N/A

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

   15. distribute-lft-in N/A

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

   16. fp-cancel-sub-sign-inv N/A

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

   17. lower-fma.f64 N/A

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

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 2: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \leq 5:\\ \;\;\;\;3 \cdot \mathsf{fma}\left(-4, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, x, -12\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)) 5.0)
   (* 3.0 (fma -4.0 x 1.0))
   (* (fma 9.0 x -12.0) x)))

C

double code(double x) {
	double tmp;
	if ((3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0)) <= 5.0) {
		tmp = 3.0 * fma(-4.0, x, 1.0);
	} else {
		tmp = fma(9.0, x, -12.0) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(3.0 * Float64(Float64(Float64(Float64(x * 3.0) * x) - Float64(x * 4.0)) + 1.0)) <= 5.0)
		tmp = Float64(3.0 * fma(-4.0, x, 1.0));
	else
		tmp = Float64(fma(9.0, x, -12.0) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(3.0 * N[(N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 5.0], N[(3.0 * N[(-4.0 * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(9.0 * x + -12.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 3 binary64) (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 3 binary64)) x) (*.f64 x #s(literal 4 binary64))) #s(literal 1 binary64))) < 5

   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

      \[\leadsto 3 \cdot \color{blue}{\left(1 + -4 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. rgt-mult-inverse N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. rgt-mult-inverse N/A

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

      12. associate-*r/ N/A

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

      13. associate-/r* N/A

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

      14. unpow2 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative N/A

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

      18. fp-cancel-sub-sign-inv N/A

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

      19. associate-*l* N/A

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

      20. unpow2 N/A

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

   5. Applied rewrites 99.2%

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

   if 5 < (*.f64 #s(literal 3 binary64) (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 3 binary64)) x) (*.f64 x #s(literal 4 binary64))) #s(literal 1 binary64)))

   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

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(9 - 12 \cdot \frac{1}{x}\right)} \]
   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. fp-cancel-sub-sign-inv N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. rgt-mult-inverse N/A

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

      13. metadata-eval 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 3: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \leq 5:\\ \;\;\;\;\mathsf{fma}\left(-12, x, 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, x, -12\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)) 5.0)
   (fma -12.0 x 3.0)
   (* (fma 9.0 x -12.0) x)))

C

double code(double x) {
	double tmp;
	if ((3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0)) <= 5.0) {
		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(3.0 * Float64(Float64(Float64(Float64(x * 3.0) * x) - Float64(x * 4.0)) + 1.0)) <= 5.0)
		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[(3.0 * N[(N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 5.0], 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 #s(literal 3 binary64) (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 3 binary64)) x) (*.f64 x #s(literal 4 binary64))) #s(literal 1 binary64))) < 5

   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

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. rgt-mult-inverse N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

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

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

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

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

      19. associate-*r/ N/A

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

      20. *-rgt-identity N/A

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

      21. distribute-frac-neg N/A

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

      22. distribute-frac-neg2 N/A

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

   5. Applied rewrites 99.2%

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

   if 5 < (*.f64 #s(literal 3 binary64) (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 3 binary64)) x) (*.f64 x #s(literal 4 binary64))) #s(literal 1 binary64)))

   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

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(9 - 12 \cdot \frac{1}{x}\right)} \]
   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. fp-cancel-sub-sign-inv N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. rgt-mult-inverse N/A

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

      13. metadata-eval 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 4: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \leq 5:\\ \;\;\;\;\mathsf{fma}\left(-12, x, 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)) 5.0)
   (fma -12.0 x 3.0)
   (* (* 9.0 x) x)))

C

double code(double x) {
	double tmp;
	if ((3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0)) <= 5.0) {
		tmp = fma(-12.0, x, 3.0);
	} else {
		tmp = (9.0 * x) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(3.0 * Float64(Float64(Float64(Float64(x * 3.0) * x) - Float64(x * 4.0)) + 1.0)) <= 5.0)
		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[(3.0 * N[(N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 5.0], 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 #s(literal 3 binary64) (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 3 binary64)) x) (*.f64 x #s(literal 4 binary64))) #s(literal 1 binary64))) < 5

   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

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. rgt-mult-inverse N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

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

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

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

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

      19. associate-*r/ N/A

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

      20. *-rgt-identity N/A

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

      21. distribute-frac-neg N/A

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

      22. distribute-frac-neg2 N/A

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

   5. Applied rewrites 99.2%

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

   if 5 < (*.f64 #s(literal 3 binary64) (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 3 binary64)) x) (*.f64 x #s(literal 4 binary64))) #s(literal 1 binary64)))

   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. *-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 98.3%

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

Alternative 5: 51.2% 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

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. rgt-mult-inverse N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. unpow2 N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

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

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

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

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

   19. associate-*r/ N/A

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

   20. *-rgt-identity N/A

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

   21. distribute-frac-neg N/A

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

   22. distribute-frac-neg2 N/A

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

5. Applied rewrites 53.6%

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

Alternative 6: 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. Applied rewrites 53.4%

   \[\leadsto \color{blue}{3} \]
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 2024339 
(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)))