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

Percentage Accurate: 99.7% → 99.8%

Time: 6.5s

Alternatives: 11

Speedup: 1.2×

• 24.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. distribute-lft-out-- 99.9%

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

   3. fma-def 99.9%

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

   4. *-commutative 99.9%

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

   5. fma-neg 99.9%

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

   6. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.56\right):\\ \;\;\;\;3 \cdot \left(x \cdot \left(-4 + 3 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(1 + x \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.58) (not (<= x 0.56)))
   (* 3.0 (* x (+ -4.0 (* 3.0 x))))
   (* 3.0 (+ 1.0 (* x -4.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.56)) {
		tmp = 3.0 * (x * (-4.0 + (3.0 * x)));
	} else {
		tmp = 3.0 * (1.0 + (x * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.58d0)) .or. (.not. (x <= 0.56d0))) then
        tmp = 3.0d0 * (x * ((-4.0d0) + (3.0d0 * x)))
    else
        tmp = 3.0d0 * (1.0d0 + (x * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.56)) {
		tmp = 3.0 * (x * (-4.0 + (3.0 * x)));
	} else {
		tmp = 3.0 * (1.0 + (x * -4.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.58) or not (x <= 0.56):
		tmp = 3.0 * (x * (-4.0 + (3.0 * x)))
	else:
		tmp = 3.0 * (1.0 + (x * -4.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.58) || !(x <= 0.56))
		tmp = Float64(3.0 * Float64(x * Float64(-4.0 + Float64(3.0 * x))));
	else
		tmp = Float64(3.0 * Float64(1.0 + Float64(x * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.58) || ~((x <= 0.56)))
		tmp = 3.0 * (x * (-4.0 + (3.0 * x)));
	else
		tmp = 3.0 * (1.0 + (x * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.58], N[Not[LessEqual[x, 0.56]], $MachinePrecision]], N[(3.0 * N[(x * N[(-4.0 + N[(3.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(1.0 + N[(x * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.57999999999999996 or 0.56000000000000005 < x

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-+r- 99.7%

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

      3. *-commutative 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around inf 97.6%

      \[\leadsto 3 \cdot \color{blue}{\left(3 \cdot {x}^{2} + -4 \cdot x\right)} \]
   5. Step-by-step derivation

      1. +-commutative 97.6%

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

      2. *-commutative 97.6%

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

      3. unpow2 97.6%

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

      4. *-commutative 97.6%

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

      5. associate-*r* 97.6%

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

      6. distribute-lft-out 97.6%

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

      7. *-commutative 97.6%

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

   6. Simplified 97.6%

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

   if -0.57999999999999996 < x < 0.56000000000000005

   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 99.1%

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

      1. *-commutative 99.1%

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

   4. Simplified 99.1%

      \[\leadsto 3 \cdot \left(\color{blue}{x \cdot -4} + 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.56\right):\\ \;\;\;\;3 \cdot \left(x \cdot \left(-4 + 3 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(1 + x \cdot -4\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. associate-+r- 99.9%

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

   3. *-commutative 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 4: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.56\right):\\ \;\;\;\;x \cdot \left(-12 + x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(1 + x \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.58) (not (<= x 0.56)))
   (* x (+ -12.0 (* x 9.0)))
   (* 3.0 (+ 1.0 (* x -4.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.56)) {
		tmp = x * (-12.0 + (x * 9.0));
	} else {
		tmp = 3.0 * (1.0 + (x * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.58d0)) .or. (.not. (x <= 0.56d0))) then
        tmp = x * ((-12.0d0) + (x * 9.0d0))
    else
        tmp = 3.0d0 * (1.0d0 + (x * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.56)) {
		tmp = x * (-12.0 + (x * 9.0));
	} else {
		tmp = 3.0 * (1.0 + (x * -4.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.58) or not (x <= 0.56):
		tmp = x * (-12.0 + (x * 9.0))
	else:
		tmp = 3.0 * (1.0 + (x * -4.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.58) || !(x <= 0.56))
		tmp = Float64(x * Float64(-12.0 + Float64(x * 9.0)));
	else
		tmp = Float64(3.0 * Float64(1.0 + Float64(x * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.58) || ~((x <= 0.56)))
		tmp = x * (-12.0 + (x * 9.0));
	else
		tmp = 3.0 * (1.0 + (x * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.58], N[Not[LessEqual[x, 0.56]], $MachinePrecision]], N[(x * N[(-12.0 + N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(1.0 + N[(x * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.57999999999999996 or 0.56000000000000005 < x

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-+r- 99.7%

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

      3. *-commutative 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around inf 97.6%

      \[\leadsto \color{blue}{-12 \cdot x + 9 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 97.6%

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

      2. associate-*r* 97.6%

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

      3. *-commutative 97.6%

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

      4. distribute-rgt-out 97.6%

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

   6. Simplified 97.6%

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

   if -0.57999999999999996 < x < 0.56000000000000005

   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 99.1%

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

      1. *-commutative 99.1%

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

   4. Simplified 99.1%

      \[\leadsto 3 \cdot \left(\color{blue}{x \cdot -4} + 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.56\right):\\ \;\;\;\;x \cdot \left(-12 + x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(1 + x \cdot -4\right)\\ \end{array} \]

Alternative 5: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;3 \cdot \left(x \cdot \left(3 \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;3 \cdot \left(1 + x \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (* 3.0 (* x (* 3.0 x)))
   (if (<= x 1.0) (* 3.0 (+ 1.0 (* x -4.0))) (* 9.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 3.0 * (x * (3.0 * x));
	} else if (x <= 1.0) {
		tmp = 3.0 * (1.0 + (x * -4.0));
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = 3.0d0 * (x * (3.0d0 * x))
    else if (x <= 1.0d0) then
        tmp = 3.0d0 * (1.0d0 + (x * (-4.0d0)))
    else
        tmp = 9.0d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 3.0 * (x * (3.0 * x));
	} else if (x <= 1.0) {
		tmp = 3.0 * (1.0 + (x * -4.0));
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.55:
		tmp = 3.0 * (x * (3.0 * x))
	elif x <= 1.0:
		tmp = 3.0 * (1.0 + (x * -4.0))
	else:
		tmp = 9.0 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(3.0 * Float64(x * Float64(3.0 * x)));
	elseif (x <= 1.0)
		tmp = Float64(3.0 * Float64(1.0 + Float64(x * -4.0)));
	else
		tmp = Float64(9.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.55)
		tmp = 3.0 * (x * (3.0 * x));
	elseif (x <= 1.0)
		tmp = 3.0 * (1.0 + (x * -4.0));
	else
		tmp = 9.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(3.0 * N[(x * N[(3.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(3.0 * N[(1.0 + N[(x * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-+r- 99.7%

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

      3. *-commutative 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around inf 99.2%

      \[\leadsto 3 \cdot \color{blue}{\left(3 \cdot {x}^{2} + -4 \cdot x\right)} \]
   5. Step-by-step derivation

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. unpow2 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.2%

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

      6. distribute-lft-out 99.2%

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

      7. *-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around inf 97.3%

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

      1. *-commutative 97.3%

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

   9. Simplified 97.3%

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

   if -1.55000000000000004 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.5%

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

      1. *-commutative 98.5%

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

   4. Simplified 98.5%

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

   if 1 < 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 95.7%

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

      1. unpow2 95.7%

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

      2. associate-*r* 95.7%

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

   4. Simplified 95.7%

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

      1. add-sqr-sqrt 95.5%

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

      2. sqrt-unprod 76.0%

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

      3. associate-*l* 76.0%

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

      4. unpow2 76.0%

         \[\leadsto \sqrt{\left(9 \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\left(9 \cdot x\right) \cdot x\right)} \]

      5. *-commutative 76.0%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{2} \cdot 9\right)} \cdot \left(\left(9 \cdot x\right) \cdot x\right)} \]

      6. associate-*l* 76.0%

         \[\leadsto \sqrt{\left({x}^{2} \cdot 9\right) \cdot \color{blue}{\left(9 \cdot \left(x \cdot x\right)\right)}} \]

      7. unpow2 76.0%

         \[\leadsto \sqrt{\left({x}^{2} \cdot 9\right) \cdot \left(9 \cdot \color{blue}{{x}^{2}}\right)} \]

      8. *-commutative 76.0%

         \[\leadsto \sqrt{\left({x}^{2} \cdot 9\right) \cdot \color{blue}{\left({x}^{2} \cdot 9\right)}} \]

      9. swap-sqr 76.0%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(9 \cdot 9\right)}} \]

      10. pow-prod-up 76.0%

          \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(9 \cdot 9\right)} \]

      11. metadata-eval 76.0%

          \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(9 \cdot 9\right)} \]

      12. metadata-eval 76.0%

          \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{81}} \]

   6. Applied egg-rr 76.0%

      \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 81}} \]
   7. Step-by-step derivation

      1. sqrt-prod 75.9%

         \[\leadsto \color{blue}{\sqrt{{x}^{4}} \cdot \sqrt{81}} \]

      2. sqrt-pow1 95.7%

         \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{81} \]

      3. metadata-eval 95.7%

         \[\leadsto {x}^{\color{blue}{2}} \cdot \sqrt{81} \]

      4. pow2 95.7%

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \sqrt{81} \]

      5. metadata-eval 95.7%

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

   8. Applied egg-rr 95.7%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;3 \cdot \left(x \cdot \left(3 \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;3 \cdot \left(1 + x \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 6: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. associate-+r- 99.9%

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

   3. *-commutative 99.9%

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

3. Applied egg-rr 99.9%

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

   1. associate--l+ 99.9%

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

   2. add-sqr-sqrt 99.7%

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

   3. unpow2 99.7%

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

   4. distribute-lft-in 99.7%

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

   5. metadata-eval 99.7%

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

   6. +-commutative 99.7%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 7: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.58) (not (<= x 0.2))) (* x (* x 9.0)) 3.0))

C

double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(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 <= (-0.58d0)) .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 <= -0.58) || !(x <= 0.2)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.58) 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 <= -0.58) || !(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 <= -0.58) || ~((x <= 0.2)))
		tmp = x * (x * 9.0);
	else
		tmp = 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.58], 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 < -0.57999999999999996 or 0.20000000000000001 < 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 95.9%

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

      1. unpow2 95.9%

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

      2. associate-*r* 95.9%

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

   4. Simplified 95.9%

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

   if -0.57999999999999996 < 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. Taylor expanded in x around 0 98.5%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \]

Alternative 8: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.58) (not (<= x 0.2))) (* 9.0 (* x x)) 3.0))

C

double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.2)) {
		tmp = 9.0 * (x * x);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.58d0)) .or. (.not. (x <= 0.2d0))) then
        tmp = 9.0d0 * (x * x)
    else
        tmp = 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.2)) {
		tmp = 9.0 * (x * x);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.58) or not (x <= 0.2):
		tmp = 9.0 * (x * x)
	else:
		tmp = 3.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.58) || !(x <= 0.2))
		tmp = Float64(9.0 * Float64(x * x));
	else
		tmp = 3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.58) || ~((x <= 0.2)))
		tmp = 9.0 * (x * x);
	else
		tmp = 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.58], N[Not[LessEqual[x, 0.2]], $MachinePrecision]], N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], 3.0]

Derivation

1. Split input into 2 regimes

2. if x < -0.57999999999999996 or 0.20000000000000001 < 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 95.9%

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

      1. unpow2 95.9%

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

      2. associate-*r* 95.9%

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

   4. Simplified 95.9%

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

      1. add-sqr-sqrt 95.8%

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

      2. sqrt-unprod 77.2%

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

      3. associate-*l* 77.2%

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

      4. unpow2 77.2%

         \[\leadsto \sqrt{\left(9 \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\left(9 \cdot x\right) \cdot x\right)} \]

      5. *-commutative 77.2%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{2} \cdot 9\right)} \cdot \left(\left(9 \cdot x\right) \cdot x\right)} \]

      6. associate-*l* 77.2%

         \[\leadsto \sqrt{\left({x}^{2} \cdot 9\right) \cdot \color{blue}{\left(9 \cdot \left(x \cdot x\right)\right)}} \]

      7. unpow2 77.2%

         \[\leadsto \sqrt{\left({x}^{2} \cdot 9\right) \cdot \left(9 \cdot \color{blue}{{x}^{2}}\right)} \]

      8. *-commutative 77.2%

         \[\leadsto \sqrt{\left({x}^{2} \cdot 9\right) \cdot \color{blue}{\left({x}^{2} \cdot 9\right)}} \]

      9. swap-sqr 77.2%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(9 \cdot 9\right)}} \]

      10. pow-prod-up 77.2%

          \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(9 \cdot 9\right)} \]

      11. metadata-eval 77.2%

          \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(9 \cdot 9\right)} \]

      12. metadata-eval 77.2%

          \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{81}} \]

   6. Applied egg-rr 77.2%

      \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 81}} \]
   7. Step-by-step derivation

      1. sqrt-prod 77.2%

         \[\leadsto \color{blue}{\sqrt{{x}^{4}} \cdot \sqrt{81}} \]

      2. sqrt-pow1 95.9%

         \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{81} \]

      3. metadata-eval 95.9%

         \[\leadsto {x}^{\color{blue}{2}} \cdot \sqrt{81} \]

      4. pow2 95.9%

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \sqrt{81} \]

      5. metadata-eval 95.9%

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

   8. Applied egg-rr 95.9%

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

   if -0.57999999999999996 < 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. Taylor expanded in x around 0 98.5%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \]

Alternative 9: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 1.0))) (* 9.0 (* x x)) (+ 3.0 (* x -12.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 1.0)) {
		tmp = 9.0 * (x * x);
	} 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 <= 1.0d0))) then
        tmp = 9.0d0 * (x * x)
    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 <= 1.0)) {
		tmp = 9.0 * (x * x);
	} else {
		tmp = 3.0 + (x * -12.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.55) or not (x <= 1.0):
		tmp = 9.0 * (x * x)
	else:
		tmp = 3.0 + (x * -12.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 1.0))
		tmp = Float64(9.0 * Float64(x * x));
	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 <= 1.0)))
		tmp = 9.0 * (x * x);
	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, 1.0]], $MachinePrecision]], N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 1 < 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 96.5%

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

      1. unpow2 96.5%

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

      2. associate-*r* 96.5%

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

   4. Simplified 96.5%

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

      1. add-sqr-sqrt 96.4%

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

      2. sqrt-unprod 77.6%

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

      3. associate-*l* 77.6%

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

      4. unpow2 77.6%

         \[\leadsto \sqrt{\left(9 \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\left(9 \cdot x\right) \cdot x\right)} \]

      5. *-commutative 77.6%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{2} \cdot 9\right)} \cdot \left(\left(9 \cdot x\right) \cdot x\right)} \]

      6. associate-*l* 77.6%

         \[\leadsto \sqrt{\left({x}^{2} \cdot 9\right) \cdot \color{blue}{\left(9 \cdot \left(x \cdot x\right)\right)}} \]

      7. unpow2 77.6%

         \[\leadsto \sqrt{\left({x}^{2} \cdot 9\right) \cdot \left(9 \cdot \color{blue}{{x}^{2}}\right)} \]

      8. *-commutative 77.6%

         \[\leadsto \sqrt{\left({x}^{2} \cdot 9\right) \cdot \color{blue}{\left({x}^{2} \cdot 9\right)}} \]

      9. swap-sqr 77.6%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(9 \cdot 9\right)}} \]

      10. pow-prod-up 77.7%

          \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(9 \cdot 9\right)} \]

      11. metadata-eval 77.7%

          \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(9 \cdot 9\right)} \]

      12. metadata-eval 77.7%

          \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{81}} \]

   6. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 81}} \]
   7. Step-by-step derivation

      1. sqrt-prod 77.6%

         \[\leadsto \color{blue}{\sqrt{{x}^{4}} \cdot \sqrt{81}} \]

      2. sqrt-pow1 96.5%

         \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{81} \]

      3. metadata-eval 96.5%

         \[\leadsto {x}^{\color{blue}{2}} \cdot \sqrt{81} \]

      4. pow2 96.5%

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \sqrt{81} \]

      5. metadata-eval 96.5%

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

   8. Applied egg-rr 96.5%

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

   if -1.55000000000000004 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \]

Alternative 10: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 3.0 * (x * (3.0 * x));
	} 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 <= (-1.55d0)) then
        tmp = 3.0d0 * (x * (3.0d0 * x))
    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 <= -1.55) {
		tmp = 3.0 * (x * (3.0 * x));
	} 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 <= -1.55:
		tmp = 3.0 * (x * (3.0 * x))
	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 <= -1.55)
		tmp = Float64(3.0 * Float64(x * Float64(3.0 * x)));
	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 <= -1.55)
		tmp = 3.0 * (x * (3.0 * x));
	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, -1.55], N[(3.0 * N[(x * N[(3.0 * x), $MachinePrecision]), $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 < -1.55000000000000004

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-+r- 99.7%

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

      3. *-commutative 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around inf 99.2%

      \[\leadsto 3 \cdot \color{blue}{\left(3 \cdot {x}^{2} + -4 \cdot x\right)} \]
   5. Step-by-step derivation

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. unpow2 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.2%

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

      6. distribute-lft-out 99.2%

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

      7. *-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around inf 97.3%

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

      1. *-commutative 97.3%

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

   9. Simplified 97.3%

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

   if -1.55000000000000004 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.5%

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

   if 1 < 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 95.7%

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

      1. unpow2 95.7%

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

      2. associate-*r* 95.7%

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

   4. Simplified 95.7%

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

      1. add-sqr-sqrt 95.5%

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

      2. sqrt-unprod 76.0%

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

      3. associate-*l* 76.0%

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

      4. unpow2 76.0%

         \[\leadsto \sqrt{\left(9 \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\left(9 \cdot x\right) \cdot x\right)} \]

      5. *-commutative 76.0%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{2} \cdot 9\right)} \cdot \left(\left(9 \cdot x\right) \cdot x\right)} \]

      6. associate-*l* 76.0%

         \[\leadsto \sqrt{\left({x}^{2} \cdot 9\right) \cdot \color{blue}{\left(9 \cdot \left(x \cdot x\right)\right)}} \]

      7. unpow2 76.0%

         \[\leadsto \sqrt{\left({x}^{2} \cdot 9\right) \cdot \left(9 \cdot \color{blue}{{x}^{2}}\right)} \]

      8. *-commutative 76.0%

         \[\leadsto \sqrt{\left({x}^{2} \cdot 9\right) \cdot \color{blue}{\left({x}^{2} \cdot 9\right)}} \]

      9. swap-sqr 76.0%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(9 \cdot 9\right)}} \]

      10. pow-prod-up 76.0%

          \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(9 \cdot 9\right)} \]

      11. metadata-eval 76.0%

          \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(9 \cdot 9\right)} \]

      12. metadata-eval 76.0%

          \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{81}} \]

   6. Applied egg-rr 76.0%

      \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 81}} \]
   7. Step-by-step derivation

      1. sqrt-prod 75.9%

         \[\leadsto \color{blue}{\sqrt{{x}^{4}} \cdot \sqrt{81}} \]

      2. sqrt-pow1 95.7%

         \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{81} \]

      3. metadata-eval 95.7%

         \[\leadsto {x}^{\color{blue}{2}} \cdot \sqrt{81} \]

      4. pow2 95.7%

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \sqrt{81} \]

      5. metadata-eval 95.7%

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

   8. Applied egg-rr 95.7%

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

4. Final simplification 97.5%

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

Alternative 11: 51.9% 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 99.9%

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

2. Taylor expanded in x around 0 50.0%

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

3. Final simplification 50.0%

   \[\leadsto 3 \]

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 2023200 
(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)))