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

Percentage Accurate: 99.8% → 99.8%

Time: 6.7s

Alternatives: 7

Speedup: 1.4×

• 24.2% 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: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. distribute-lft-out-- 99.8%

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

   3. fma-define 99.8%

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

   4. *-commutative 99.8%

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

   5. fma-neg 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.9%

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

6. Taylor expanded in x around inf 99.9%

   \[\leadsto 3 + x \cdot \color{blue}{\left(x \cdot \left(9 - 12 \cdot \frac{1}{x}\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r/ 99.9%

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

   2. metadata-eval 99.9%

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

8. Simplified 99.9%

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

Alternative 2: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.57 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.57) (not (<= x 0.2))) (* x (* x 9.0)) 3.0))

C

double code(double x) {
	double tmp;
	if ((x <= -0.57) || !(x <= 0.2)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.57d0)) .or. (.not. (x <= 0.2d0))) then
        tmp = x * (x * 9.0d0)
    else
        tmp = 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.57) || !(x <= 0.2)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.57) or not (x <= 0.2):
		tmp = x * (x * 9.0)
	else:
		tmp = 3.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.57) || !(x <= 0.2))
		tmp = Float64(x * Float64(x * 9.0));
	else
		tmp = 3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.57) || ~((x <= 0.2)))
		tmp = x * (x * 9.0);
	else
		tmp = 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.57], N[Not[LessEqual[x, 0.2]], $MachinePrecision]], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], 3.0]

Derivation

1. Split input into 2 regimes

2. if x < -0.569999999999999951 or 0.20000000000000001 < x

   1. Initial program 98.8%

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

      1. *-commutative 98.8%

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

      2. distribute-lft-out-- 99.6%

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

      3. fma-define 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-neg 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 97.7%

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

      1. *-commutative 97.7%

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

   7. Simplified 97.7%

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

      1. *-commutative 97.7%

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

      2. unpow2 97.7%

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

      3. metadata-eval 97.7%

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

      4. swap-sqr 97.5%

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

      5. pow2 97.5%

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

      6. *-commutative 97.5%

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

   9. Applied egg-rr 97.5%

      \[\leadsto \color{blue}{{\left(x \cdot 3\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow-prod-down 97.7%

          \[\leadsto \color{blue}{{x}^{2} \cdot {3}^{2}} \]

       2. metadata-eval 97.7%

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

       3. *-commutative 97.7%

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

       4. pow2 97.7%

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

       5. associate-*r* 97.7%

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

   11. Applied egg-rr 97.7%

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

   if -0.569999999999999951 < x < 0.20000000000000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-out-- 100.0%

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

      3. fma-define 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-neg 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.57 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;3 + x \cdot -12\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.52)
   (* x (* x 9.0))
   (if (<= x 1.0) (+ 3.0 (* x -12.0)) (* 9.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.52) {
		tmp = x * (x * 9.0);
	} else if (x <= 1.0) {
		tmp = 3.0 + (x * -12.0);
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.52d0)) then
        tmp = x * (x * 9.0d0)
    else if (x <= 1.0d0) then
        tmp = 3.0d0 + (x * (-12.0d0))
    else
        tmp = 9.0d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.52) {
		tmp = x * (x * 9.0);
	} else if (x <= 1.0) {
		tmp = 3.0 + (x * -12.0);
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.52:
		tmp = x * (x * 9.0)
	elif x <= 1.0:
		tmp = 3.0 + (x * -12.0)
	else:
		tmp = 9.0 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.52)
		tmp = Float64(x * Float64(x * 9.0));
	elseif (x <= 1.0)
		tmp = Float64(3.0 + Float64(x * -12.0));
	else
		tmp = Float64(9.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.52)
		tmp = x * (x * 9.0);
	elseif (x <= 1.0)
		tmp = 3.0 + (x * -12.0);
	else
		tmp = 9.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.52], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision], N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.52

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-lft-out-- 99.7%

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

      3. fma-define 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-neg 99.7%

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

      6. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 97.5%

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

      1. *-commutative 97.5%

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

   7. Simplified 97.5%

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

      1. *-commutative 97.5%

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

      2. unpow2 97.5%

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

      3. metadata-eval 97.5%

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

      4. swap-sqr 97.3%

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

      5. pow2 97.3%

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

      6. *-commutative 97.3%

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

   9. Applied egg-rr 97.3%

      \[\leadsto \color{blue}{{\left(x \cdot 3\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow-prod-down 97.5%

          \[\leadsto \color{blue}{{x}^{2} \cdot {3}^{2}} \]

       2. metadata-eval 97.5%

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

       3. *-commutative 97.5%

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

       4. pow2 97.5%

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

       5. associate-*r* 97.5%

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

   11. Applied egg-rr 97.5%

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

   if -1.52 < 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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. distribute-lft-out-- 100.0%

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

      3. fma-define 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-neg 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

   7. Simplified 99.9%

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

   if 1 < x

   1. Initial program 98.1%

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

      1. *-commutative 98.1%

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

      2. distribute-lft-out-- 99.5%

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

      3. fma-define 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-neg 99.5%

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

      6. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 97.9%

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

      1. *-commutative 97.9%

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

   7. Simplified 97.9%

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

      1. unpow2 97.9%

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

   9. Applied egg-rr 97.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;3 + x \cdot -12\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.57:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.57) (* x (* x 9.0)) (if (<= x 0.2) 3.0 (* 9.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.57) {
		tmp = x * (x * 9.0);
	} else if (x <= 0.2) {
		tmp = 3.0;
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.57d0)) then
        tmp = x * (x * 9.0d0)
    else if (x <= 0.2d0) then
        tmp = 3.0d0
    else
        tmp = 9.0d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.57) {
		tmp = x * (x * 9.0);
	} else if (x <= 0.2) {
		tmp = 3.0;
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.57:
		tmp = x * (x * 9.0)
	elif x <= 0.2:
		tmp = 3.0
	else:
		tmp = 9.0 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.57)
		tmp = Float64(x * Float64(x * 9.0));
	elseif (x <= 0.2)
		tmp = 3.0;
	else
		tmp = Float64(9.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.57)
		tmp = x * (x * 9.0);
	elseif (x <= 0.2)
		tmp = 3.0;
	else
		tmp = 9.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.57], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.2], 3.0, N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.569999999999999951

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-lft-out-- 99.7%

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

      3. fma-define 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-neg 99.7%

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

      6. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 97.5%

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

      1. *-commutative 97.5%

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

   7. Simplified 97.5%

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

      1. *-commutative 97.5%

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

      2. unpow2 97.5%

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

      3. metadata-eval 97.5%

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

      4. swap-sqr 97.3%

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

      5. pow2 97.3%

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

      6. *-commutative 97.3%

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

   9. Applied egg-rr 97.3%

      \[\leadsto \color{blue}{{\left(x \cdot 3\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow-prod-down 97.5%

          \[\leadsto \color{blue}{{x}^{2} \cdot {3}^{2}} \]

       2. metadata-eval 97.5%

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

       3. *-commutative 97.5%

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

       4. pow2 97.5%

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

       5. associate-*r* 97.5%

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

   11. Applied egg-rr 97.5%

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

   if -0.569999999999999951 < x < 0.20000000000000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-out-- 100.0%

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

      3. fma-define 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-neg 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{3} \]

   if 0.20000000000000001 < x

   1. Initial program 98.1%

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

      1. *-commutative 98.1%

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

      2. distribute-lft-out-- 99.5%

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

      3. fma-define 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-neg 99.5%

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

      6. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 97.9%

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

      1. *-commutative 97.9%

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

   7. Simplified 97.9%

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

      1. unpow2 97.9%

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

   9. Applied egg-rr 97.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.57:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. distribute-lft-out-- 99.8%

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

   3. fma-define 99.8%

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

   4. *-commutative 99.8%

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

   5. fma-neg 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.9%

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

6. Final simplification 99.9%

   \[\leadsto 3 + x \cdot \left(x \cdot 9 - 12\right) \]
7. Add Preprocessing

Alternative 6: 97.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. distribute-lft-out-- 99.8%

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

   3. fma-define 99.8%

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

   4. *-commutative 99.8%

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

   5. fma-neg 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.9%

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

6. Taylor expanded in x around inf 98.5%

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

   1. *-commutative 98.5%

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

8. Simplified 98.5%

   \[\leadsto 3 + x \cdot \color{blue}{\left(x \cdot 9\right)} \]
9. Add Preprocessing

Alternative 7: 51.8% accurate, 13.0× 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 99.4%

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

   1. *-commutative 99.4%

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

   2. distribute-lft-out-- 99.8%

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

   3. fma-define 99.8%

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

   4. *-commutative 99.8%

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

   5. fma-neg 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 53.2%

   \[\leadsto \color{blue}{3} \]
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 2024145 
(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)))