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

Percentage Accurate: 99.8% → 99.9%

Time: 6.6s

Alternatives: 7

Speedup: 1.4×

• 24.3% 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, 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 99.8%

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

3. Taylor expanded in x around 0 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 98.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -0.56) || !(x <= 0.56)) {
		tmp = x * ((x * 9.0) - 12.0);
	} 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 <= (-0.56d0)) .or. (.not. (x <= 0.56d0))) then
        tmp = x * ((x * 9.0d0) - 12.0d0)
    else
        tmp = 3.0d0 + (x * (-12.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.56000000000000005 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. Add Preprocessing

   3. Taylor expanded in x around inf 97.9%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(9 - 12 \cdot \frac{1}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 97.9%

         \[\leadsto {x}^{2} \cdot \left(9 - \color{blue}{\frac{12 \cdot 1}{x}}\right) \]

      2. metadata-eval 97.9%

         \[\leadsto {x}^{2} \cdot \left(9 - \frac{\color{blue}{12}}{x}\right) \]

   5. Simplified 97.9%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(9 - \frac{12}{x}\right)} \]

   6. Taylor expanded in x around 0 97.9%

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

   if -0.56000000000000005 < 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. Add Preprocessing

   3. Taylor expanded in x around 0 98.5%

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

      1. *-commutative 98.5%

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

   5. Simplified 98.5%

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

4. Final simplification 98.2%

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

Alternative 3: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.56 \lor \neg \left(x \leq 1.65\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.56) (not (<= x 1.65))) (* x (* x 9.0)) 3.0))

C

double code(double x) {
	double tmp;
	if ((x <= -0.56) || !(x <= 1.65)) {
		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 <= (-0.56d0)) .or. (.not. (x <= 1.65d0))) 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 <= -0.56) || !(x <= 1.65)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.56) or not (x <= 1.65):
		tmp = x * (x * 9.0)
	else:
		tmp = 3.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.56) || !(x <= 1.65))
		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 <= -0.56) || ~((x <= 1.65)))
		tmp = x * (x * 9.0);
	else
		tmp = 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.56], N[Not[LessEqual[x, 1.65]], $MachinePrecision]], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], 3.0]

Derivation

1. Split input into 2 regimes

2. if x < -0.56000000000000005 or 1.6499999999999999 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 97.9%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(9 - 12 \cdot \frac{1}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 97.9%

         \[\leadsto {x}^{2} \cdot \left(9 - \color{blue}{\frac{12 \cdot 1}{x}}\right) \]

      2. metadata-eval 97.9%

         \[\leadsto {x}^{2} \cdot \left(9 - \frac{\color{blue}{12}}{x}\right) \]

   5. Simplified 97.9%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(9 - \frac{12}{x}\right)} \]

   6. Taylor expanded in x around 0 97.9%

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

   7. Taylor expanded in x around inf 96.4%

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

      1. *-commutative 96.4%

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

   9. Simplified 96.4%

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

   if -0.56000000000000005 < x < 1.6499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.56 \lor \neg \left(x \leq 1.65\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 1.0)) {
		tmp = x * (x * 9.0);
	} 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 <= 1.0d0))) then
        tmp = x * (x * 9.0d0)
    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 <= 1.0)) {
		tmp = x * (x * 9.0);
	} 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 = x * (x * 9.0)
	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(x * Float64(x * 9.0));
	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 <= 1.0)))
		tmp = x * (x * 9.0);
	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, 1.0]], $MachinePrecision]], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 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. Add Preprocessing

   3. Taylor expanded in x around inf 97.9%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(9 - 12 \cdot \frac{1}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 97.9%

         \[\leadsto {x}^{2} \cdot \left(9 - \color{blue}{\frac{12 \cdot 1}{x}}\right) \]

      2. metadata-eval 97.9%

         \[\leadsto {x}^{2} \cdot \left(9 - \frac{\color{blue}{12}}{x}\right) \]

   5. Simplified 97.9%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(9 - \frac{12}{x}\right)} \]

   6. Taylor expanded in x around 0 97.9%

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

   7. Taylor expanded in x around inf 96.4%

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

      1. *-commutative 96.4%

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

   9. Simplified 96.4%

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

   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. Add Preprocessing

   3. Taylor expanded in x around 0 98.5%

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

      1. *-commutative 98.5%

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

   5. Simplified 98.5%

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

4. Final simplification 97.5%

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

Alternative 5: 97.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(3.0 * N[(N[(x * N[(3.0 * x), $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. Add Preprocessing

3. Taylor expanded in x around 0 99.8%

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

4. Taylor expanded in x around inf 96.8%

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

5. Final simplification 96.8%

   \[\leadsto 3 \cdot \left(x \cdot \left(3 \cdot x\right) + 1\right) \]
6. Add Preprocessing

Alternative 6: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 3 + \frac{x}{\frac{0.1111111111111111}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 3.0 (/ x (/ 0.1111111111111111 x))))

C

double code(double x) {
	return 3.0 + (x / (0.1111111111111111 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 + (x / (0.1111111111111111d0 / x))
end function

Java

public static double code(double x) {
	return 3.0 + (x / (0.1111111111111111 / x));
}

Python

def code(x):
	return 3.0 + (x / (0.1111111111111111 / x))

Julia

function code(x)
	return Float64(3.0 + Float64(x / Float64(0.1111111111111111 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 + (x / (0.1111111111111111 / x));
end

Wolfram

code[x_] := N[(3.0 + N[(x / N[(0.1111111111111111 / x), $MachinePrecision]), $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. Add Preprocessing

3. Taylor expanded in x around 0 99.9%

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

   1. sub-neg 99.9%

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

   2. metadata-eval 99.9%

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

   3. flip-+ 99.9%

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

   4. metadata-eval 99.9%

      \[\leadsto 3 + x \cdot \frac{\left(9 \cdot x\right) \cdot \left(9 \cdot x\right) - \color{blue}{144}}{9 \cdot x - -12} \]

   5. metadata-eval 99.9%

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

   6. sub-neg 99.9%

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

   7. *-commutative 99.9%

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

   8. *-commutative 99.9%

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

   9. swap-sqr 99.8%

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

   10. unpow2 99.8%

       \[\leadsto 3 + x \cdot \frac{\color{blue}{{x}^{2}} \cdot \left(9 \cdot 9\right) + \left(-12 \cdot 12\right)}{9 \cdot x - -12} \]

   11. metadata-eval 99.8%

       \[\leadsto 3 + x \cdot \frac{{x}^{2} \cdot \color{blue}{81} + \left(-12 \cdot 12\right)}{9 \cdot x - -12} \]

   12. metadata-eval 99.8%

       \[\leadsto 3 + x \cdot \frac{{x}^{2} \cdot 81 + \left(-\color{blue}{144}\right)}{9 \cdot x - -12} \]

   13. metadata-eval 99.8%

       \[\leadsto 3 + x \cdot \frac{{x}^{2} \cdot 81 + \color{blue}{-144}}{9 \cdot x - -12} \]

   14. *-commutative 99.8%

       \[\leadsto 3 + x \cdot \frac{{x}^{2} \cdot 81 + -144}{\color{blue}{x \cdot 9} - -12} \]

5. Applied egg-rr 99.8%

   \[\leadsto 3 + x \cdot \color{blue}{\frac{{x}^{2} \cdot 81 + -144}{x \cdot 9 - -12}} \]
6. Step-by-step derivation

   1. clear-num 99.8%

      \[\leadsto 3 + x \cdot \color{blue}{\frac{1}{\frac{x \cdot 9 - -12}{{x}^{2} \cdot 81 + -144}}} \]

   2. un-div-inv 99.8%

      \[\leadsto 3 + \color{blue}{\frac{x}{\frac{x \cdot 9 - -12}{{x}^{2} \cdot 81 + -144}}} \]

   3. clear-num 99.8%

      \[\leadsto 3 + \frac{x}{\color{blue}{\frac{1}{\frac{{x}^{2} \cdot 81 + -144}{x \cdot 9 - -12}}}} \]

   4. add-sqr-sqrt 99.8%

      \[\leadsto 3 + \frac{x}{\frac{1}{\frac{\color{blue}{\sqrt{{x}^{2} \cdot 81} \cdot \sqrt{{x}^{2} \cdot 81}} + -144}{x \cdot 9 - -12}}} \]

   5. fma-define 99.8%

      \[\leadsto 3 + \frac{x}{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\sqrt{{x}^{2} \cdot 81}, \sqrt{{x}^{2} \cdot 81}, -144\right)}}{x \cdot 9 - -12}}} \]

   6. sqrt-prod 99.8%

      \[\leadsto 3 + \frac{x}{\frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\sqrt{{x}^{2}} \cdot \sqrt{81}}, \sqrt{{x}^{2} \cdot 81}, -144\right)}{x \cdot 9 - -12}}} \]

   7. sqrt-pow1 75.8%

      \[\leadsto 3 + \frac{x}{\frac{1}{\frac{\mathsf{fma}\left(\color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{81}, \sqrt{{x}^{2} \cdot 81}, -144\right)}{x \cdot 9 - -12}}} \]

   8. metadata-eval 75.8%

      \[\leadsto 3 + \frac{x}{\frac{1}{\frac{\mathsf{fma}\left({x}^{\color{blue}{1}} \cdot \sqrt{81}, \sqrt{{x}^{2} \cdot 81}, -144\right)}{x \cdot 9 - -12}}} \]

   9. pow1 75.8%

      \[\leadsto 3 + \frac{x}{\frac{1}{\frac{\mathsf{fma}\left(\color{blue}{x} \cdot \sqrt{81}, \sqrt{{x}^{2} \cdot 81}, -144\right)}{x \cdot 9 - -12}}} \]

   10. metadata-eval 75.8%

       \[\leadsto 3 + \frac{x}{\frac{1}{\frac{\mathsf{fma}\left(x \cdot \color{blue}{9}, \sqrt{{x}^{2} \cdot 81}, -144\right)}{x \cdot 9 - -12}}} \]

   11. sqrt-prod 75.8%

       \[\leadsto 3 + \frac{x}{\frac{1}{\frac{\mathsf{fma}\left(x \cdot 9, \color{blue}{\sqrt{{x}^{2}} \cdot \sqrt{81}}, -144\right)}{x \cdot 9 - -12}}} \]

   12. sqrt-pow1 99.9%

       \[\leadsto 3 + \frac{x}{\frac{1}{\frac{\mathsf{fma}\left(x \cdot 9, \color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{81}, -144\right)}{x \cdot 9 - -12}}} \]

   13. metadata-eval 99.9%

       \[\leadsto 3 + \frac{x}{\frac{1}{\frac{\mathsf{fma}\left(x \cdot 9, {x}^{\color{blue}{1}} \cdot \sqrt{81}, -144\right)}{x \cdot 9 - -12}}} \]

   14. pow1 99.9%

       \[\leadsto 3 + \frac{x}{\frac{1}{\frac{\mathsf{fma}\left(x \cdot 9, \color{blue}{x} \cdot \sqrt{81}, -144\right)}{x \cdot 9 - -12}}} \]

   15. metadata-eval 99.9%

       \[\leadsto 3 + \frac{x}{\frac{1}{\frac{\mathsf{fma}\left(x \cdot 9, x \cdot \color{blue}{9}, -144\right)}{x \cdot 9 - -12}}} \]

   16. metadata-eval 99.9%

       \[\leadsto 3 + \frac{x}{\frac{1}{\frac{\mathsf{fma}\left(x \cdot 9, x \cdot 9, \color{blue}{-144}\right)}{x \cdot 9 - -12}}} \]

   17. metadata-eval 99.9%

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

   18. fma-neg 99.9%

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

7. Applied egg-rr 99.9%

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

8. Taylor expanded in x around inf 96.8%

   \[\leadsto 3 + \frac{x}{\color{blue}{\frac{0.1111111111111111}{x}}} \]

9. Final simplification 96.8%

   \[\leadsto 3 + \frac{x}{\frac{0.1111111111111111}{x}} \]
10. Add Preprocessing

Alternative 7: 51.4% 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. Add Preprocessing

3. Taylor expanded in x around 0 52.4%

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

4. Final simplification 52.4%

   \[\leadsto 3 \]
5. Add Preprocessing

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 2024078 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :alt
  (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x)))

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