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

Percentage Accurate: 99.8% → 99.9%

Time: 8.3s

Alternatives: 7

Speedup: 1.4×

• 24.2% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. metadata-eval 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-lft-out-- 99.8%

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

   5. associate-*l* 99.8%

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

   6. fma-def 99.8%

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

   7. *-commutative 99.8%

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

   8. sub-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. distribute-rgt-in 99.8%

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

   11. metadata-eval 99.8%

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

   12. metadata-eval 99.8%

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

   13. associate-*l* 99.8%

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

   14. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. flip-+ 99.8%

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

   2. div-inv 99.8%

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

   3. metadata-eval 99.8%

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

   4. swap-sqr 99.8%

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

   5. pow2 99.8%

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

   6. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(144 - {x}^{2} \cdot 81\right) \cdot \frac{1}{-12 - x \cdot 9}}, 3\right) \]
7. Step-by-step derivation

   1. fma-udef 99.8%

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

   2. un-div-inv 99.7%

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

   3. metadata-eval 99.7%

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

   4. unpow2 99.7%

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

   5. metadata-eval 99.7%

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

   6. swap-sqr 99.8%

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

   7. flip-+ 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 98.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 1.0)))
   (* (* x 3.0) (* x 3.0))
   (* 3.0 (+ 1.0 (* x -4.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 1.0)) {
		tmp = (x * 3.0) * (x * 3.0);
	} 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 <= (-1.55d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x * 3.0d0) * (x * 3.0d0)
    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 <= -1.55) || !(x <= 1.0)) {
		tmp = (x * 3.0) * (x * 3.0);
	} else {
		tmp = 3.0 * (1.0 + (x * -4.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.55) or not (x <= 1.0):
		tmp = (x * 3.0) * (x * 3.0)
	else:
		tmp = 3.0 * (1.0 + (x * -4.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 1.0))
		tmp = Float64(Float64(x * 3.0) * Float64(x * 3.0));
	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 <= -1.55) || ~((x <= 1.0)))
		tmp = (x * 3.0) * (x * 3.0);
	else
		tmp = 3.0 * (1.0 + (x * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(x * 3.0), $MachinePrecision] * N[(x * 3.0), $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 < -1.55000000000000004 or 1 < x

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate-*l* 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 97.4%

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

      1. *-commutative 97.4%

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

   7. Simplified 97.4%

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

      1. *-commutative 97.4%

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

      2. add-sqr-sqrt 97.1%

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

      3. pow2 97.1%

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

      4. sqrt-prod 97.2%

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

      5. metadata-eval 97.2%

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

      6. unpow2 97.2%

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

      7. sqrt-prod 48.2%

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

      8. add-sqr-sqrt 97.2%

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

      9. *-commutative 97.2%

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

   9. Applied egg-rr 97.2%

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

       1. unpow2 97.2%

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

   11. Applied egg-rr 97.2%

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

   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. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\left(x \cdot 3\right) \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(1 + x \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate-*l* 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 97.7%

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

      1. *-commutative 97.7%

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

   7. Simplified 97.7%

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

      1. *-commutative 97.7%

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

      2. add-sqr-sqrt 97.6%

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

      3. pow2 97.6%

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

      4. sqrt-prod 97.6%

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

      5. metadata-eval 97.6%

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

      6. unpow2 97.6%

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

      7. sqrt-prod 0.0%

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

      8. add-sqr-sqrt 97.6%

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

      9. *-commutative 97.6%

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

   9. Applied egg-rr 97.6%

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

       1. unpow2 97.6%

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

   11. Applied egg-rr 97.6%

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

   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. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   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. Step-by-step derivation

      1. associate-+l- 99.6%

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

      2. associate-*l* 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 97.0%

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

      1. *-commutative 97.0%

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

   7. Simplified 97.0%

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

      1. *-commutative 97.0%

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

      2. add-sqr-sqrt 96.7%

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

      3. pow2 96.7%

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

      4. sqrt-prod 96.8%

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

      5. metadata-eval 96.8%

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

      6. unpow2 96.8%

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

      7. sqrt-prod 96.4%

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

      8. add-sqr-sqrt 96.8%

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

      9. *-commutative 96.8%

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

   9. Applied egg-rr 96.8%

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

       1. unpow2 96.8%

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

       2. associate-*r* 96.9%

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

   11. Applied egg-rr 96.9%

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

4. Final simplification 98.4%

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

Alternative 4: 51.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(3.0 * N[(1.0 + N[(x * -4.0), $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. Step-by-step derivation

   1. associate-+l- 99.8%

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

   2. associate-*l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 49.5%

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

   1. *-commutative 49.5%

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

7. Simplified 49.5%

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

8. Final simplification 49.5%

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

Alternative 5: 97.8% accurate, 1.9× speedup

Math

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

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(x * Float64(x * 9.0)))
end

MATLAB

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

Wolfram

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

   1. distribute-rgt-in 99.8%

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

   2. metadata-eval 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-lft-out-- 99.8%

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

   5. associate-*l* 99.8%

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

   6. fma-def 99.8%

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

   7. *-commutative 99.8%

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

   8. sub-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. distribute-rgt-in 99.8%

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

   11. metadata-eval 99.8%

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

   12. metadata-eval 99.8%

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

   13. associate-*l* 99.8%

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

   14. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. flip-+ 99.8%

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

   2. div-inv 99.8%

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

   3. metadata-eval 99.8%

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

   4. swap-sqr 99.8%

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

   5. pow2 99.8%

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

   6. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(144 - {x}^{2} \cdot 81\right) \cdot \frac{1}{-12 - x \cdot 9}}, 3\right) \]
7. Step-by-step derivation

   1. fma-udef 99.8%

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

   2. un-div-inv 99.7%

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

   3. metadata-eval 99.7%

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

   4. unpow2 99.7%

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

   5. metadata-eval 99.7%

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

   6. swap-sqr 99.8%

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

   7. flip-+ 99.8%

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in x around inf 98.1%

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

    1. *-commutative 98.1%

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

11. Simplified 98.1%

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

12. Final simplification 98.1%

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

Alternative 6: 51.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

   2. associate-*l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 49.5%

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

   1. *-commutative 49.5%

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

7. Simplified 49.5%

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

8. Final simplification 49.5%

   \[\leadsto 3 + x \cdot -12 \]
9. Add Preprocessing

Alternative 7: 51.1% 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. Step-by-step derivation

   1. associate-+l- 99.8%

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

   2. associate-*l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 49.5%

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

6. Final simplification 49.5%

   \[\leadsto 3 \]
7. Add Preprocessing

Developer target: 99.9% 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 2024040 
(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)))