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

Percentage Accurate: 99.8% → 99.9%

Time: 5.9s

Alternatives: 7

Speedup: 1.0×

• 25.8% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. distribute-rgt-in 99.8%

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

   2. metadata-eval 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-lft-out-- 99.8%

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

   5. associate-*l* 99.8%

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

   6. fma-def 99.8%

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

   7. *-commutative 99.8%

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

   8. sub-neg 99.8%

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

   9. distribute-rgt-in 99.8%

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

   10. associate-*l* 99.9%

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

   11. metadata-eval 99.9%

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

   12. metadata-eval 99.9%

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

   13. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

2. Final simplification 99.8%

   \[\leadsto 3 \cdot \left(\left(x \cdot \left(x \cdot 3\right) - x \cdot 4\right) + 1\right) \]

Alternative 3: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 0.215\right):\\ \;\;\;\;\left(x \cdot 3\right) \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 0.215)))
   (* (* x 3.0) (* x 3.0))
   (+ 3.0 (* x -12.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 0.215)) {
		tmp = (x * 3.0) * (x * 3.0);
	} else {
		tmp = 3.0 + (x * -12.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 0.215d0))) then
        tmp = (x * 3.0d0) * (x * 3.0d0)
    else
        tmp = 3.0d0 + (x * (-12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 0.215)) {
		tmp = (x * 3.0) * (x * 3.0);
	} else {
		tmp = 3.0 + (x * -12.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.55) or not (x <= 0.215):
		tmp = (x * 3.0) * (x * 3.0)
	else:
		tmp = 3.0 + (x * -12.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 0.215))
		tmp = Float64(Float64(x * 3.0) * Float64(x * 3.0));
	else
		tmp = Float64(3.0 + Float64(x * -12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 0.215)))
		tmp = (x * 3.0) * (x * 3.0);
	else
		tmp = 3.0 + (x * -12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 0.215]], $MachinePrecision]], N[(N[(x * 3.0), $MachinePrecision] * N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 0.214999999999999997 < x

   1. Initial program 99.7%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

   2. Taylor expanded in x around inf 96.6%

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

      1. *-commutative 96.6%

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

   4. Simplified 96.6%

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

      1. *-commutative 96.6%

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

      2. unpow2 96.6%

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

      3. metadata-eval 96.6%

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

      4. swap-sqr 96.4%

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

      5. pow2 96.4%

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

      6. *-commutative 96.4%

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

   6. Applied egg-rr 96.4%

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

      1. unpow2 96.4%

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

   8. Applied egg-rr 96.4%

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

   if -1.55000000000000004 < x < 0.214999999999999997

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

   2. Taylor expanded in x around 0 98.8%

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

      1. *-commutative 98.8%

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

   4. Simplified 98.8%

      \[\leadsto \color{blue}{3 + x \cdot -12} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 0.215\right):\\ \;\;\;\;\left(x \cdot 3\right) \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \]

Alternative 4: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 0.215\right):\\ \;\;\;\;x \cdot \left(3 \cdot \left(x \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 0.215)))
   (* x (* 3.0 (* x 3.0)))
   (+ 3.0 (* x -12.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 0.215)) {
		tmp = x * (3.0 * (x * 3.0));
	} else {
		tmp = 3.0 + (x * -12.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 0.215d0))) then
        tmp = x * (3.0d0 * (x * 3.0d0))
    else
        tmp = 3.0d0 + (x * (-12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 0.215)) {
		tmp = x * (3.0 * (x * 3.0));
	} else {
		tmp = 3.0 + (x * -12.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.55) or not (x <= 0.215):
		tmp = x * (3.0 * (x * 3.0))
	else:
		tmp = 3.0 + (x * -12.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 0.215))
		tmp = Float64(x * Float64(3.0 * Float64(x * 3.0)));
	else
		tmp = Float64(3.0 + Float64(x * -12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 0.215)))
		tmp = x * (3.0 * (x * 3.0));
	else
		tmp = 3.0 + (x * -12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 0.215]], $MachinePrecision]], N[(x * N[(3.0 * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 0.214999999999999997 < x

   1. Initial program 99.7%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

   2. Taylor expanded in x around inf 96.6%

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

      1. *-commutative 96.6%

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

   4. Simplified 96.6%

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

      1. *-commutative 96.6%

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

      2. unpow2 96.6%

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

      3. metadata-eval 96.6%

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

      4. swap-sqr 96.4%

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

      5. pow2 96.4%

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

      6. *-commutative 96.4%

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

   6. Applied egg-rr 96.4%

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

      1. unpow2 96.4%

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

      2. *-commutative 96.4%

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

      3. associate-*r* 96.4%

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

   8. Applied egg-rr 96.4%

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

   if -1.55000000000000004 < x < 0.214999999999999997

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

   2. Taylor expanded in x around 0 98.8%

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

      1. *-commutative 98.8%

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

   4. Simplified 98.8%

      \[\leadsto \color{blue}{3 + x \cdot -12} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 0.215\right):\\ \;\;\;\;x \cdot \left(3 \cdot \left(x \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \]

Alternative 5: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;3 \cdot \left(x \cdot \left(x \cdot 3\right)\right)\\ \mathbf{elif}\;x \leq 0.215:\\ \;\;\;\;3 + x \cdot -12\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(3 \cdot \left(x \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (* 3.0 (* x (* x 3.0)))
   (if (<= x 0.215) (+ 3.0 (* x -12.0)) (* x (* 3.0 (* x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 3.0 * (x * (x * 3.0));
	} else if (x <= 0.215) {
		tmp = 3.0 + (x * -12.0);
	} else {
		tmp = x * (3.0 * (x * 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = 3.0d0 * (x * (x * 3.0d0))
    else if (x <= 0.215d0) then
        tmp = 3.0d0 + (x * (-12.0d0))
    else
        tmp = x * (3.0d0 * (x * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 3.0 * (x * (x * 3.0));
	} else if (x <= 0.215) {
		tmp = 3.0 + (x * -12.0);
	} else {
		tmp = x * (3.0 * (x * 3.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.55:
		tmp = 3.0 * (x * (x * 3.0))
	elif x <= 0.215:
		tmp = 3.0 + (x * -12.0)
	else:
		tmp = x * (3.0 * (x * 3.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(3.0 * Float64(x * Float64(x * 3.0)));
	elseif (x <= 0.215)
		tmp = Float64(3.0 + Float64(x * -12.0));
	else
		tmp = Float64(x * Float64(3.0 * Float64(x * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.55)
		tmp = 3.0 * (x * (x * 3.0));
	elseif (x <= 0.215)
		tmp = 3.0 + (x * -12.0);
	else
		tmp = x * (3.0 * (x * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(3.0 * N[(x * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.215], N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(3.0 * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 99.6%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

   2. Taylor expanded in x around inf 94.6%

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

      1. *-commutative 94.6%

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

   4. Simplified 94.6%

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

      1. *-commutative 94.6%

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

      2. unpow2 94.6%

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

      3. metadata-eval 94.6%

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

      4. swap-sqr 94.4%

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

      5. pow2 94.4%

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

      6. *-commutative 94.4%

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

   6. Applied egg-rr 94.4%

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

      1. unpow2 94.4%

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

      2. associate-*r* 94.5%

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

   8. Applied egg-rr 94.5%

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

   if -1.55000000000000004 < x < 0.214999999999999997

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

   2. Taylor expanded in x around 0 98.8%

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

      1. *-commutative 98.8%

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

   4. Simplified 98.8%

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

   if 0.214999999999999997 < x

   1. Initial program 99.7%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

   2. Taylor expanded in x around inf 98.5%

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

      1. *-commutative 98.5%

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

   4. Simplified 98.5%

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

      1. *-commutative 98.5%

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

      2. unpow2 98.5%

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

      3. metadata-eval 98.5%

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

      4. swap-sqr 98.3%

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

      5. pow2 98.3%

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

      6. *-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. unpow2 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-*r* 98.4%

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

   8. Applied egg-rr 98.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;3 \cdot \left(x \cdot \left(x \cdot 3\right)\right)\\ \mathbf{elif}\;x \leq 0.215:\\ \;\;\;\;3 + x \cdot -12\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(3 \cdot \left(x \cdot 3\right)\right)\\ \end{array} \]

Alternative 6: 50.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ 3 + x \cdot -12 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

2. Taylor expanded in x around 0 52.5%

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

   1. *-commutative 52.5%

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

4. Simplified 52.5%

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

5. Final simplification 52.5%

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

Alternative 7: 50.3% 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 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

2. Taylor expanded in x around 0 51.8%

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

3. Final simplification 51.8%

   \[\leadsto 3 \]

Developer target: 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 2023322 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x)))

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