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

Percentage Accurate: 99.7% → 99.8%

Time: 6.8s

Alternatives: 6

Speedup: 1.3×

• 25.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 3.0 (- 2.0 (* x 3.0))) x))

C

double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (3.0d0 * (2.0d0 - (x * 3.0d0))) * x
end function

Java

public static double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Python

def code(x):
	return (3.0 * (2.0 - (x * 3.0))) * x

Julia

function code(x)
	return Float64(Float64(3.0 * Float64(2.0 - Float64(x * 3.0))) * x)
end

MATLAB

function tmp = code(x)
	tmp = (3.0 * (2.0 - (x * 3.0))) * x;
end

Wolfram

code[x_] := N[(N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 3.0 (- 2.0 (* x 3.0))) x))

C

double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (3.0d0 * (2.0d0 - (x * 3.0d0))) * x
end function

Java

public static double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Python

def code(x):
	return (3.0 * (2.0 - (x * 3.0))) * x

Julia

function code(x)
	return Float64(Float64(3.0 * Float64(2.0 - Float64(x * 3.0))) * x)
end

MATLAB

function tmp = code(x)
	tmp = (3.0 * (2.0 - (x * 3.0))) * x;
end

Wolfram

code[x_] := N[(N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 6, -9 \cdot {x}^{2}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 6.0 (* -9.0 (pow x 2.0))))

C

double code(double x) {
	return fma(x, 6.0, (-9.0 * pow(x, 2.0)));
}

Julia

function code(x)
	return fma(x, 6.0, Float64(-9.0 * (x ^ 2.0)))
end

Wolfram

code[x_] := N[(x * 6.0 + N[(-9.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. cancel-sign-sub-inv 99.7%

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

   3. distribute-rgt-in 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

   5. distribute-lft-neg-out 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

   6. distribute-rgt-neg-in 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

   7. associate-*l* 99.8%

      \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

   8. metadata-eval 99.8%

      \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

   9. metadata-eval 99.8%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. distribute-rgt-in 99.8%

      \[\leadsto \color{blue}{6 \cdot x + \left(x \cdot -9\right) \cdot x} \]

   2. *-commutative 99.8%

      \[\leadsto \color{blue}{x \cdot 6} + \left(x \cdot -9\right) \cdot x \]

   3. fma-define 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 6, \left(x \cdot -9\right) \cdot x\right)} \]

   4. *-commutative 99.8%

      \[\leadsto \mathsf{fma}\left(x, 6, \color{blue}{\left(-9 \cdot x\right)} \cdot x\right) \]

   5. associate-*l* 99.8%

      \[\leadsto \mathsf{fma}\left(x, 6, \color{blue}{-9 \cdot \left(x \cdot x\right)}\right) \]

   6. pow2 99.8%

      \[\leadsto \mathsf{fma}\left(x, 6, -9 \cdot \color{blue}{{x}^{2}}\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 6, -9 \cdot {x}^{2}\right)} \]
7. Add Preprocessing

Alternative 2: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65 \lor \neg \left(x \leq 0.66\right):\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.65) (not (<= x 0.66))) (* x (* x -9.0)) (* x 6.0)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.65) || !(x <= 0.66)) {
		tmp = x * (x * -9.0);
	} else {
		tmp = x * 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.65d0)) .or. (.not. (x <= 0.66d0))) then
        tmp = x * (x * (-9.0d0))
    else
        tmp = x * 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.65) || !(x <= 0.66)) {
		tmp = x * (x * -9.0);
	} else {
		tmp = x * 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.65) or not (x <= 0.66):
		tmp = x * (x * -9.0)
	else:
		tmp = x * 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.65) || !(x <= 0.66))
		tmp = Float64(x * Float64(x * -9.0));
	else
		tmp = Float64(x * 6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.65) || ~((x <= 0.66)))
		tmp = x * (x * -9.0);
	else
		tmp = x * 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.65], N[Not[LessEqual[x, 0.66]], $MachinePrecision]], N[(x * N[(x * -9.0), $MachinePrecision]), $MachinePrecision], N[(x * 6.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.650000000000000022 or 0.660000000000000031 < x

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. metadata-eval 99.6%

         \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

      5. distribute-lft-neg-out 99.6%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

      6. distribute-rgt-neg-in 99.6%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

      7. associate-*l* 99.8%

         \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

      8. metadata-eval 99.8%

         \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

      9. metadata-eval 99.8%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.6%

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

      1. *-commutative 97.6%

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

   7. Simplified 97.6%

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

   if -0.650000000000000022 < x < 0.660000000000000031

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. distribute-rgt-in 99.7%

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

      4. metadata-eval 99.7%

         \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

      5. distribute-lft-neg-out 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

      6. distribute-rgt-neg-in 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

      7. associate-*l* 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

      8. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

      9. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto x \cdot \color{blue}{6} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.65 \lor \neg \left(x \leq 0.66\right):\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;\frac{x}{\frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.65)
   (/ x (/ -0.1111111111111111 x))
   (if (<= x 0.66) (* x 6.0) (* x (* x -9.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = x / (-0.1111111111111111 / x);
	} else if (x <= 0.66) {
		tmp = x * 6.0;
	} else {
		tmp = x * (x * -9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.65d0)) then
        tmp = x / ((-0.1111111111111111d0) / x)
    else if (x <= 0.66d0) then
        tmp = x * 6.0d0
    else
        tmp = x * (x * (-9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = x / (-0.1111111111111111 / x);
	} else if (x <= 0.66) {
		tmp = x * 6.0;
	} else {
		tmp = x * (x * -9.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.65:
		tmp = x / (-0.1111111111111111 / x)
	elif x <= 0.66:
		tmp = x * 6.0
	else:
		tmp = x * (x * -9.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.65)
		tmp = Float64(x / Float64(-0.1111111111111111 / x));
	elseif (x <= 0.66)
		tmp = Float64(x * 6.0);
	else
		tmp = Float64(x * Float64(x * -9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.65)
		tmp = x / (-0.1111111111111111 / x);
	elseif (x <= 0.66)
		tmp = x * 6.0;
	else
		tmp = x * (x * -9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.65], N[(x / N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.66], N[(x * 6.0), $MachinePrecision], N[(x * N[(x * -9.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.650000000000000022

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. cancel-sign-sub-inv 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. metadata-eval 99.5%

         \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

      5. distribute-lft-neg-out 99.5%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

      6. distribute-rgt-neg-in 99.5%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

      7. associate-*l* 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

      8. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

      9. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.7%

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(6 \cdot \frac{1}{x} - 9\right)\right)} \]
   6. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(6 \cdot \frac{1}{x} + \left(-9\right)\right)}\right) \]

      2. associate-*r/ 99.7%

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\frac{6 \cdot 1}{x}} + \left(-9\right)\right)\right) \]

      3. metadata-eval 99.7%

         \[\leadsto x \cdot \left(x \cdot \left(\frac{\color{blue}{6}}{x} + \left(-9\right)\right)\right) \]

      4. metadata-eval 99.7%

         \[\leadsto x \cdot \left(x \cdot \left(\frac{6}{x} + \color{blue}{-9}\right)\right) \]

   7. Simplified 99.7%

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{6}{x} + -9\right)\right)} \]
   8. Step-by-step derivation

      1. flip-+ 99.7%

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{\frac{6}{x} \cdot \frac{6}{x} - -9 \cdot -9}{\frac{6}{x} - -9}}\right) \]

      2. associate-*r/ 99.6%

         \[\leadsto x \cdot \color{blue}{\frac{x \cdot \left(\frac{6}{x} \cdot \frac{6}{x} - -9 \cdot -9\right)}{\frac{6}{x} - -9}} \]

      3. sub-neg 99.6%

         \[\leadsto x \cdot \frac{x \cdot \color{blue}{\left(\frac{6}{x} \cdot \frac{6}{x} + \left(--9 \cdot -9\right)\right)}}{\frac{6}{x} - -9} \]

      4. frac-times 99.6%

         \[\leadsto x \cdot \frac{x \cdot \left(\color{blue}{\frac{6 \cdot 6}{x \cdot x}} + \left(--9 \cdot -9\right)\right)}{\frac{6}{x} - -9} \]

      5. metadata-eval 99.6%

         \[\leadsto x \cdot \frac{x \cdot \left(\frac{\color{blue}{36}}{x \cdot x} + \left(--9 \cdot -9\right)\right)}{\frac{6}{x} - -9} \]

      6. unpow2 99.6%

         \[\leadsto x \cdot \frac{x \cdot \left(\frac{36}{\color{blue}{{x}^{2}}} + \left(--9 \cdot -9\right)\right)}{\frac{6}{x} - -9} \]

      7. metadata-eval 99.6%

         \[\leadsto x \cdot \frac{x \cdot \left(\frac{36}{{x}^{2}} + \left(-\color{blue}{81}\right)\right)}{\frac{6}{x} - -9} \]

      8. metadata-eval 99.6%

         \[\leadsto x \cdot \frac{x \cdot \left(\frac{36}{{x}^{2}} + \color{blue}{-81}\right)}{\frac{6}{x} - -9} \]

      9. sub-neg 99.6%

         \[\leadsto x \cdot \frac{x \cdot \left(\frac{36}{{x}^{2}} + -81\right)}{\color{blue}{\frac{6}{x} + \left(--9\right)}} \]

      10. metadata-eval 99.6%

          \[\leadsto x \cdot \frac{x \cdot \left(\frac{36}{{x}^{2}} + -81\right)}{\frac{6}{x} + \color{blue}{9}} \]

   9. Applied egg-rr 99.6%

      \[\leadsto x \cdot \color{blue}{\frac{x \cdot \left(\frac{36}{{x}^{2}} + -81\right)}{\frac{6}{x} + 9}} \]

   11. Simplified 99.6%

       \[\leadsto x \cdot \color{blue}{\frac{x \cdot -81 + \frac{36}{x}}{9 + \frac{6}{x}}} \]
   12. Step-by-step derivation

       1. clear-num 99.6%

          \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{9 + \frac{6}{x}}{x \cdot -81 + \frac{36}{x}}}} \]

       2. un-div-inv 99.7%

          \[\leadsto \color{blue}{\frac{x}{\frac{9 + \frac{6}{x}}{x \cdot -81 + \frac{36}{x}}}} \]

       3. fma-define 99.7%

          \[\leadsto \frac{x}{\frac{9 + \frac{6}{x}}{\color{blue}{\mathsf{fma}\left(x, -81, \frac{36}{x}\right)}}} \]

   13. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\frac{x}{\frac{9 + \frac{6}{x}}{\mathsf{fma}\left(x, -81, \frac{36}{x}\right)}}} \]

   14. Taylor expanded in x around inf 96.7%

       \[\leadsto \frac{x}{\color{blue}{\frac{-0.1111111111111111}{x}}} \]

   if -0.650000000000000022 < x < 0.660000000000000031

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. distribute-rgt-in 99.7%

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

      4. metadata-eval 99.7%

         \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

      5. distribute-lft-neg-out 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

      6. distribute-rgt-neg-in 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

      7. associate-*l* 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

      8. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

      9. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto x \cdot \color{blue}{6} \]

   if 0.660000000000000031 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. metadata-eval 99.8%

         \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

      5. distribute-lft-neg-out 99.8%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

      6. distribute-rgt-neg-in 99.8%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

      7. associate-*l* 99.9%

         \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

      8. metadata-eval 99.9%

         \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

      9. metadata-eval 99.9%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.4%

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

      1. *-commutative 98.4%

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

   7. Simplified 98.4%

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

Alternative 4: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(6 + x \cdot -9\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ 6.0 (* x -9.0))))

C

double code(double x) {
	return x * (6.0 + (x * -9.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (6.0d0 + (x * (-9.0d0)))
end function

Java

public static double code(double x) {
	return x * (6.0 + (x * -9.0));
}

Python

def code(x):
	return x * (6.0 + (x * -9.0))

Julia

function code(x)
	return Float64(x * Float64(6.0 + Float64(x * -9.0)))
end

MATLAB

function tmp = code(x)
	tmp = x * (6.0 + (x * -9.0));
end

Wolfram

code[x_] := N[(x * N[(6.0 + N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. cancel-sign-sub-inv 99.7%

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

   3. distribute-rgt-in 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

   5. distribute-lft-neg-out 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

   6. distribute-rgt-neg-in 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

   7. associate-*l* 99.8%

      \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

   8. metadata-eval 99.8%

      \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

   9. metadata-eval 99.8%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 5: 51.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 6 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 6.0))

C

double code(double x) {
	return x * 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 6.0d0
end function

Java

public static double code(double x) {
	return x * 6.0;
}

Python

def code(x):
	return x * 6.0

Julia

function code(x)
	return Float64(x * 6.0)
end

MATLAB

function tmp = code(x)
	tmp = x * 6.0;
end

Wolfram

code[x_] := N[(x * 6.0), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. cancel-sign-sub-inv 99.7%

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

   3. distribute-rgt-in 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

   5. distribute-lft-neg-out 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

   6. distribute-rgt-neg-in 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

   7. associate-*l* 99.8%

      \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

   8. metadata-eval 99.8%

      \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

   9. metadata-eval 99.8%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 48.9%

   \[\leadsto x \cdot \color{blue}{6} \]
6. Add Preprocessing

Alternative 6: 2.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x) :precision binary64 4.0)

C

double code(double x) {
	return 4.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 4.0d0
end function

Java

public static double code(double x) {
	return 4.0;
}

Python

def code(x):
	return 4.0

Julia

function code(x)
	return 4.0
end

MATLAB

function tmp = code(x)
	tmp = 4.0;
end

Wolfram

code[x_] := 4.0

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. cancel-sign-sub-inv 99.7%

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

   3. distribute-rgt-in 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

   5. distribute-lft-neg-out 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

   6. distribute-rgt-neg-in 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

   7. associate-*l* 99.8%

      \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

   8. metadata-eval 99.8%

      \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

   9. metadata-eval 99.8%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 99.8%

      \[\leadsto \color{blue}{\left(6 + x \cdot -9\right) \cdot x} \]

   2. flip-+ 99.7%

      \[\leadsto \color{blue}{\frac{6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)}{6 - x \cdot -9}} \cdot x \]

   3. associate-*l/ 90.8%

      \[\leadsto \color{blue}{\frac{\left(6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right) \cdot x}{6 - x \cdot -9}} \]

   4. metadata-eval 90.8%

      \[\leadsto \frac{\left(\color{blue}{36} - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right) \cdot x}{6 - x \cdot -9} \]

   5. swap-sqr 90.7%

      \[\leadsto \frac{\left(36 - \color{blue}{\left(x \cdot x\right) \cdot \left(-9 \cdot -9\right)}\right) \cdot x}{6 - x \cdot -9} \]

   6. pow2 90.7%

      \[\leadsto \frac{\left(36 - \color{blue}{{x}^{2}} \cdot \left(-9 \cdot -9\right)\right) \cdot x}{6 - x \cdot -9} \]

   7. metadata-eval 90.7%

      \[\leadsto \frac{\left(36 - {x}^{2} \cdot \color{blue}{81}\right) \cdot x}{6 - x \cdot -9} \]

   8. *-commutative 90.7%

      \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 - \color{blue}{-9 \cdot x}} \]

   9. cancel-sign-sub-inv 90.7%

      \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{\color{blue}{6 + \left(--9\right) \cdot x}} \]

   10. metadata-eval 90.7%

       \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 + \color{blue}{9} \cdot x} \]

6. Applied egg-rr 90.7%

   \[\leadsto \color{blue}{\frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 + 9 \cdot x}} \]

7. Taylor expanded in x around 0 47.3%

   \[\leadsto \frac{\color{blue}{36 \cdot x}}{6 + 9 \cdot x} \]
8. Step-by-step derivation

   1. *-commutative 47.3%

      \[\leadsto \frac{\color{blue}{x \cdot 36}}{6 + 9 \cdot x} \]

9. Simplified 47.3%

   \[\leadsto \frac{\color{blue}{x \cdot 36}}{6 + 9 \cdot x} \]

10. Taylor expanded in x around inf 2.1%

    \[\leadsto \color{blue}{4} \]
11. Add Preprocessing

Developer target: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 6 \cdot x - 9 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* 6.0 x) (* 9.0 (* x x))))

C

double code(double x) {
	return (6.0 * x) - (9.0 * (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (6.0d0 * x) - (9.0d0 * (x * x))
end function

Java

public static double code(double x) {
	return (6.0 * x) - (9.0 * (x * x));
}

Python

def code(x):
	return (6.0 * x) - (9.0 * (x * x))

Julia

function code(x)
	return Float64(Float64(6.0 * x) - Float64(9.0 * Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * x) - (9.0 * (x * x));
end

Wolfram

code[x_] := N[(N[(6.0 * x), $MachinePrecision] - N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024089 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :alt
  (- (* 6.0 x) (* 9.0 (* x x)))

  (* (* 3.0 (- 2.0 (* x 3.0))) x))