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

Percentage Accurate: 99.8% → 99.8%

Time: 6.7s

Alternatives: 8

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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left(9 \cdot {x}^{2} + x \cdot -12\right) + 3 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (+ (* 9.0 (pow x 2.0)) (* x -12.0)) 3.0))

C

double code(double x) {
	return ((9.0 * pow(x, 2.0)) + (x * -12.0)) + 3.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((9.0d0 * (x ** 2.0d0)) + (x * (-12.0d0))) + 3.0d0
end function

Java

public static double code(double x) {
	return ((9.0 * Math.pow(x, 2.0)) + (x * -12.0)) + 3.0;
}

Python

def code(x):
	return ((9.0 * math.pow(x, 2.0)) + (x * -12.0)) + 3.0

Julia

function code(x)
	return Float64(Float64(Float64(9.0 * (x ^ 2.0)) + Float64(x * -12.0)) + 3.0)
end

MATLAB

function tmp = code(x)
	tmp = ((9.0 * (x ^ 2.0)) + (x * -12.0)) + 3.0;
end

Wolfram

code[x_] := N[(N[(N[(9.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + 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. 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-define 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-define 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. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.9%

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

6. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

   2. *-commutative 99.9%

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

   3. distribute-lft-in 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-*l* 99.9%

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

   6. unpow2 99.9%

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

   7. *-commutative 99.9%

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

8. Applied egg-rr 99.9%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(3.0 + N[(x * N[(x * 9.0 + -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. 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-define 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-define 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. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x * N[(9.0 * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * -12.0), $MachinePrecision] + 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 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. distribute-rgt-in 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-lft-out-- 99.6%

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

      5. associate-*l* 99.7%

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

      6. fma-define 99.7%

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

      7. *-commutative 99.7%

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

      8. sub-neg 99.7%

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

      9. distribute-lft-in 99.7%

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

      10. *-commutative 99.7%

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

      11. associate-*l* 99.7%

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

      12. fma-define 99.8%

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

      13. metadata-eval 99.8%

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

      14. metadata-eval 99.8%

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

      15. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 98.7%

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

      1. *-commutative 98.7%

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

      2. unpow2 98.7%

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

      3. metadata-eval 98.7%

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

      4. swap-sqr 98.5%

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

      5. pow2 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. unpow-prod-down 98.7%

         \[\leadsto \color{blue}{{x}^{2} \cdot {3}^{2}} \]

      2. metadata-eval 98.7%

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

      3. *-commutative 98.7%

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

      4. unpow2 98.7%

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

      5. associate-*r* 98.7%

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

   9. Applied egg-rr 98.7%

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

   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. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft-out-- 100.0%

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

      5. associate-*l* 100.0%

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

      6. fma-define 100.0%

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

      7. *-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l* 100.0%

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

      12. fma-define 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.2%

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

4. Final simplification 98.9%

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

Alternative 4: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 1.7\right):\\ \;\;\;\;x \cdot \left(9 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.58) (not (<= x 1.7))) (* x (* 9.0 x)) 3.0))

C

double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 1.7)) {
		tmp = x * (9.0 * x);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.58d0)) .or. (.not. (x <= 1.7d0))) then
        tmp = x * (9.0d0 * x)
    else
        tmp = 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 1.7)) {
		tmp = x * (9.0 * x);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.58) or not (x <= 1.7):
		tmp = x * (9.0 * x)
	else:
		tmp = 3.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.58) || !(x <= 1.7))
		tmp = Float64(x * Float64(9.0 * x));
	else
		tmp = 3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.58) || ~((x <= 1.7)))
		tmp = x * (9.0 * x);
	else
		tmp = 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.58], N[Not[LessEqual[x, 1.7]], $MachinePrecision]], N[(x * N[(9.0 * x), $MachinePrecision]), $MachinePrecision], 3.0]

Derivation

1. Split input into 2 regimes

2. if x < -0.57999999999999996 or 1.69999999999999996 < 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. distribute-rgt-in 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-lft-out-- 99.6%

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

      5. associate-*l* 99.7%

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

      6. fma-define 99.7%

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

      7. *-commutative 99.7%

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

      8. sub-neg 99.7%

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

      9. distribute-lft-in 99.7%

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

      10. *-commutative 99.7%

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

      11. associate-*l* 99.7%

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

      12. fma-define 99.8%

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

      13. metadata-eval 99.8%

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

      14. metadata-eval 99.8%

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

      15. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 98.7%

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

      1. *-commutative 98.7%

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

      2. unpow2 98.7%

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

      3. metadata-eval 98.7%

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

      4. swap-sqr 98.5%

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

      5. pow2 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. unpow-prod-down 98.7%

         \[\leadsto \color{blue}{{x}^{2} \cdot {3}^{2}} \]

      2. metadata-eval 98.7%

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

      3. *-commutative 98.7%

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

      4. unpow2 98.7%

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

      5. associate-*r* 98.7%

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

   9. Applied egg-rr 98.7%

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

   if -0.57999999999999996 < x < 1.69999999999999996

   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. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft-out-- 100.0%

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

      5. associate-*l* 100.0%

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

      6. fma-define 100.0%

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

      7. *-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l* 100.0%

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

      12. fma-define 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 97.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 1.7\right):\\ \;\;\;\;x \cdot \left(9 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 99.7%

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

      1. distribute-rgt-in 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-commutative 99.7%

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

      4. distribute-lft-out-- 99.7%

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

      5. associate-*l* 99.7%

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

      6. fma-define 99.7%

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

      7. *-commutative 99.7%

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

      8. sub-neg 99.7%

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

      9. distribute-lft-in 99.7%

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

      10. *-commutative 99.7%

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

      11. associate-*l* 99.7%

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

      12. fma-define 99.7%

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

      13. metadata-eval 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. *-commutative 98.6%

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

      2. unpow2 98.6%

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

      3. metadata-eval 98.6%

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

      4. swap-sqr 98.4%

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

      5. pow2 98.4%

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

   7. Applied egg-rr 98.4%

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

      1. unpow2 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*r* 98.5%

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

      4. *-commutative 98.5%

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

   9. Applied egg-rr 98.5%

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

   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. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft-out-- 100.0%

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

      5. associate-*l* 100.0%

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

      6. fma-define 100.0%

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

      7. *-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l* 100.0%

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

      12. fma-define 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.2%

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

   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. distribute-rgt-in 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-lft-out-- 99.6%

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

      5. associate-*l* 99.7%

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

      6. fma-define 99.7%

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

      7. *-commutative 99.7%

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

      8. sub-neg 99.7%

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

      9. distribute-lft-in 99.7%

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

      10. *-commutative 99.7%

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

      11. associate-*l* 99.8%

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

      12. fma-define 99.8%

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

      13. metadata-eval 99.8%

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

      14. metadata-eval 99.8%

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

      15. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. *-commutative 98.9%

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

      2. unpow2 98.9%

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

      3. metadata-eval 98.9%

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

      4. swap-sqr 98.6%

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

      5. pow2 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. unpow-prod-down 98.9%

         \[\leadsto \color{blue}{{x}^{2} \cdot {3}^{2}} \]

      2. metadata-eval 98.9%

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

      3. *-commutative 98.9%

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

      4. unpow2 98.9%

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

      5. associate-*r* 98.8%

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

   9. Applied egg-rr 98.8%

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

4. Final simplification 98.9%

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

Alternative 6: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(3.0 + N[(x * N[(N[(9.0 * x), $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. 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-define 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-define 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. Add Preprocessing

5. Taylor expanded in x around 0 99.9%

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

Alternative 7: 97.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(3.0 + N[(x * N[(9.0 * 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. 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-define 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-define 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. Add Preprocessing

5. Taylor expanded in x around 0 99.9%

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

6. Taylor expanded in x around inf 98.3%

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

   1. *-commutative 98.3%

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

8. Simplified 98.3%

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

9. Final simplification 98.3%

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

Alternative 8: 51.2% 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. 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-define 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-define 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. Add Preprocessing

5. Taylor expanded in x around 0 49.5%

   \[\leadsto \color{blue}{3} \]
6. 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 2024100 
(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)))