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

Percentage Accurate: 99.8% → 99.8%

Time: 5.2s

Alternatives: 7

Speedup: 1.4×

• 24.8% 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: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 9.0 + N[(x * -12.0), $MachinePrecision]), $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. *-commutative 99.8%

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

   2. distribute-lft1-in 99.8%

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

   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-lft-in 99.8%

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

   10. *-commutative 99.8%

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

   11. associate-*l* 99.9%

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

   12. fma-def 99.9%

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

   13. metadata-eval 99.9%

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

   14. metadata-eval 99.9%

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

   15. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. fma-udef 99.9%

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

5. Applied egg-rr 99.9%

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

   1. fma-udef 99.9%

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

   2. distribute-lft-in 99.9%

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

   3. associate-*r* 99.9%

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

   4. *-commutative 99.9%

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

   5. fma-def 99.9%

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

   6. *-commutative 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.195\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot 9\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.58) (not (<= x 0.195))) (* (* x x) 9.0) 3.0))

C

double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.195)) {
		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 <= 0.195d0))) 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 <= 0.195)) {
		tmp = (x * x) * 9.0;
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.58) or not (x <= 0.195):
		tmp = (x * x) * 9.0
	else:
		tmp = 3.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.58) || !(x <= 0.195))
		tmp = Float64(Float64(x * 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 <= 0.195)))
		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, 0.195]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] * 9.0), $MachinePrecision], 3.0]

Derivation

1. Split input into 2 regimes

2. if x < -0.57999999999999996 or 0.19500000000000001 < x

   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 96.4%

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

      1. unpow2 96.4%

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

   4. Simplified 96.4%

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

   if -0.57999999999999996 < x < 0.19500000000000001

   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 97.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.195\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot 9\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \]

Alternative 3: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 0.195:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.58) (* x (* x 9.0)) (if (<= x 0.195) 3.0 (* (* x x) 9.0))))

C

double code(double x) {
	double tmp;
	if (x <= -0.58) {
		tmp = x * (x * 9.0);
	} else if (x <= 0.195) {
		tmp = 3.0;
	} else {
		tmp = (x * x) * 9.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.58d0)) then
        tmp = x * (x * 9.0d0)
    else if (x <= 0.195d0) then
        tmp = 3.0d0
    else
        tmp = (x * x) * 9.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.58) {
		tmp = x * (x * 9.0);
	} else if (x <= 0.195) {
		tmp = 3.0;
	} else {
		tmp = (x * x) * 9.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.58:
		tmp = x * (x * 9.0)
	elif x <= 0.195:
		tmp = 3.0
	else:
		tmp = (x * x) * 9.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.58)
		tmp = Float64(x * Float64(x * 9.0));
	elseif (x <= 0.195)
		tmp = 3.0;
	else
		tmp = Float64(Float64(x * x) * 9.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.58)
		tmp = x * (x * 9.0);
	elseif (x <= 0.195)
		tmp = 3.0;
	else
		tmp = (x * x) * 9.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.58], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.195], 3.0, N[(N[(x * x), $MachinePrecision] * 9.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.57999999999999996

   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 94.7%

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

      1. unpow2 94.7%

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

      2. *-commutative 94.7%

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

      3. associate-*l* 94.7%

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

   4. Simplified 94.7%

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

   if -0.57999999999999996 < x < 0.19500000000000001

   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 97.3%

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

   if 0.19500000000000001 < x

   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 97.8%

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

      1. unpow2 97.8%

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

   4. Simplified 97.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 0.195:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 9\\ \end{array} \]

Alternative 4: 98.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = x * (x * 9.0d0)
    else if (x <= 1.0d0) then
        tmp = 3.0d0 + (x * (-12.0d0))
    else
        tmp = (x * x) * 9.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = x * (x * 9.0);
	} else if (x <= 1.0) {
		tmp = 3.0 + (x * -12.0);
	} else {
		tmp = (x * x) * 9.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.55:
		tmp = x * (x * 9.0)
	elif x <= 1.0:
		tmp = 3.0 + (x * -12.0)
	else:
		tmp = (x * x) * 9.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(x * Float64(x * 9.0));
	elseif (x <= 1.0)
		tmp = Float64(3.0 + Float64(x * -12.0));
	else
		tmp = Float64(Float64(x * x) * 9.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.55)
		tmp = x * (x * 9.0);
	elseif (x <= 1.0)
		tmp = 3.0 + (x * -12.0);
	else
		tmp = (x * x) * 9.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 9.0), $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 94.7%

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

      1. unpow2 94.7%

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

      2. *-commutative 94.7%

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

      3. associate-*l* 94.7%

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

   4. Simplified 94.7%

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

   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. Taylor expanded in x around 0 98.6%

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

   if 1 < x

   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 97.8%

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

      1. unpow2 97.8%

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

   4. Simplified 97.8%

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

4. Final simplification 97.5%

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

Alternative 5: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(3.0 + N[(x * N[(-12.0 + N[(x * 9.0), $MachinePrecision]), $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. Step-by-step derivation

   1. +-commutative 99.8%

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

   2. distribute-lft-in 99.8%

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

   3. metadata-eval 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-lft-out-- 99.8%

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

   6. sub-neg 99.8%

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

   7. metadata-eval 99.8%

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

   8. *-commutative 99.8%

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

   9. fma-def 99.8%

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

3. Applied egg-rr 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-*r* 99.8%

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

   3. *-commutative 99.8%

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

   4. fma-def 99.8%

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

   5. *-commutative 99.8%

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

   6. fma-def 99.7%

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

   7. *-commutative 99.7%

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

   8. fma-udef 99.8%

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

5. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot x, \mathsf{fma}\left(3, x, -4\right), 3\right)} \]
6. Step-by-step derivation

   1. fma-udef 99.8%

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

   2. fma-udef 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

   5. fma-def 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{\left(-12 \cdot x + 9 \cdot {x}^{2}\right)} + 3 \]
9. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. unpow2 99.9%

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

   3. associate-*r* 99.9%

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

   4. *-commutative 99.9%

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

   5. metadata-eval 99.9%

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

   6. associate-*l* 99.8%

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

   7. distribute-rgt-out 99.8%

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

   8. associate-*l* 99.9%

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

   9. metadata-eval 99.9%

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

10. Simplified 99.9%

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

11. Final simplification 99.9%

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

Alternative 6: 97.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(3.0 + N[(N[(x * x), $MachinePrecision] * 9.0), $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. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. distribute-lft1-in 99.8%

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

   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-lft-in 99.8%

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

   10. *-commutative 99.8%

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

   11. associate-*l* 99.9%

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

   12. fma-def 99.9%

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

   13. metadata-eval 99.9%

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

   14. metadata-eval 99.9%

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

   15. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. fma-udef 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around inf 96.8%

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

   1. unpow2 96.8%

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

8. Simplified 96.8%

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

9. Final simplification 96.8%

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

Alternative 7: 51.6% 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 50.1%

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

3. Final simplification 50.1%

   \[\leadsto 3 \]

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 2023196 
(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)))