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

Percentage Accurate: 99.8% → 99.9%

Time: 7.0s

Alternatives: 7

Speedup: 1.4×

• 25.5% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * N[(x * 9.0 + -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-- 99.9%

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

   5. associate-*l* 99.9%

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

   6. fma-define 99.9%

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

   7. *-commutative 99.9%

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

   8. sub-neg 99.9%

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

   9. distribute-lft-in 99.9%

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

   10. *-commutative 99.9%

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

   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%

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

Alternative 2: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 0.21\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot -4 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 0.21)))
   (* x (* x 9.0))
   (* 3.0 (+ (* x -4.0) 1.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 0.21)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0 * ((x * -4.0) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 0.21d0))) then
        tmp = x * (x * 9.0d0)
    else
        tmp = 3.0d0 * ((x * (-4.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 0.21)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0 * ((x * -4.0) + 1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.55) or not (x <= 0.21):
		tmp = x * (x * 9.0)
	else:
		tmp = 3.0 * ((x * -4.0) + 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 0.21))
		tmp = Float64(x * Float64(x * 9.0));
	else
		tmp = Float64(3.0 * Float64(Float64(x * -4.0) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 0.21)))
		tmp = x * (x * 9.0);
	else
		tmp = 3.0 * ((x * -4.0) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 0.21]], $MachinePrecision]], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[(x * -4.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 0.209999999999999992 < x

   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* 99.8%

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

      12. fma-define 99.9%

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

      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 inf 97.5%

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

      1. +-commutative 97.5%

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

      2. unpow2 97.5%

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

      3. associate-*r* 97.5%

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

      4. *-commutative 97.5%

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

      5. distribute-rgt-in 97.5%

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

      6. fma-undefine 97.6%

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

   7. Simplified 97.6%

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

      1. fma-undefine 97.5%

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

      2. distribute-rgt-in 97.5%

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

      3. metadata-eval 97.5%

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

      4. associate-*r* 97.5%

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

      5. *-commutative 97.5%

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

      6. *-commutative 97.5%

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

      7. associate-*l* 97.5%

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

      8. metadata-eval 97.5%

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

   9. Applied egg-rr 97.5%

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

       1. +-commutative 97.5%

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

       2. *-commutative 97.5%

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

       3. distribute-lft-out 97.5%

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

   11. Applied egg-rr 97.5%

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

   12. Taylor expanded in x around inf 95.9%

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

       1. *-commutative 95.9%

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

   14. Simplified 95.9%

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

   if -1.55000000000000004 < x < 0.209999999999999992

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 0.21\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot -4 + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.56\right):\\ \;\;\;\;x \cdot \left(-12 + x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot -4 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.58) (not (<= x 0.56)))
   (* x (+ -12.0 (* x 9.0)))
   (* 3.0 (+ (* x -4.0) 1.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.56)) {
		tmp = x * (-12.0 + (x * 9.0));
	} else {
		tmp = 3.0 * ((x * -4.0) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.58d0)) .or. (.not. (x <= 0.56d0))) then
        tmp = x * ((-12.0d0) + (x * 9.0d0))
    else
        tmp = 3.0d0 * ((x * (-4.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.56)) {
		tmp = x * (-12.0 + (x * 9.0));
	} else {
		tmp = 3.0 * ((x * -4.0) + 1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.58) or not (x <= 0.56):
		tmp = x * (-12.0 + (x * 9.0))
	else:
		tmp = 3.0 * ((x * -4.0) + 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.58) || !(x <= 0.56))
		tmp = Float64(x * Float64(-12.0 + Float64(x * 9.0)));
	else
		tmp = Float64(3.0 * Float64(Float64(x * -4.0) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.58) || ~((x <= 0.56)))
		tmp = x * (-12.0 + (x * 9.0));
	else
		tmp = 3.0 * ((x * -4.0) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.58], N[Not[LessEqual[x, 0.56]], $MachinePrecision]], N[(x * N[(-12.0 + N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[(x * -4.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.57999999999999996 or 0.56000000000000005 < x

   1. Initial program 99.7%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. 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* 99.8%

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

      12. fma-define 99.9%

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

      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 inf 97.5%

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

      1. +-commutative 97.5%

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

      2. unpow2 97.5%

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

      3. associate-*r* 97.5%

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

      4. *-commutative 97.5%

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

      5. distribute-rgt-in 97.5%

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

      6. fma-undefine 97.6%

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

   7. Simplified 97.6%

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

      1. fma-undefine 97.5%

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

      2. distribute-rgt-in 97.5%

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

      3. metadata-eval 97.5%

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

      4. associate-*r* 97.5%

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

      5. *-commutative 97.5%

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

      6. *-commutative 97.5%

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

      7. associate-*l* 97.5%

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

      8. metadata-eval 97.5%

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

   9. Applied egg-rr 97.5%

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

       1. +-commutative 97.5%

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

       2. *-commutative 97.5%

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

       3. distribute-lft-out 97.5%

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

   11. Applied egg-rr 97.5%

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

   if -0.57999999999999996 < x < 0.56000000000000005

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.56\right):\\ \;\;\;\;x \cdot \left(-12 + x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot -4 + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.58) (not (<= x 0.2))) (* x (* x 9.0)) 3.0))

C

double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.2)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.58d0)) .or. (.not. (x <= 0.2d0))) then
        tmp = x * (x * 9.0d0)
    else
        tmp = 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.2)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.58) or not (x <= 0.2):
		tmp = x * (x * 9.0)
	else:
		tmp = 3.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.58) || !(x <= 0.2))
		tmp = Float64(x * Float64(x * 9.0));
	else
		tmp = 3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.58) || ~((x <= 0.2)))
		tmp = x * (x * 9.0);
	else
		tmp = 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.58], N[Not[LessEqual[x, 0.2]], $MachinePrecision]], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], 3.0]

Derivation

1. Split input into 2 regimes

2. if x < -0.57999999999999996 or 0.20000000000000001 < x

   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* 99.8%

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

      12. fma-define 99.9%

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

      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 inf 97.5%

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

      1. +-commutative 97.5%

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

      2. unpow2 97.5%

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

      3. associate-*r* 97.5%

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

      4. *-commutative 97.5%

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

      5. distribute-rgt-in 97.5%

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

      6. fma-undefine 97.6%

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

   7. Simplified 97.6%

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

      1. fma-undefine 97.5%

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

      2. distribute-rgt-in 97.5%

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

      3. metadata-eval 97.5%

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

      4. associate-*r* 97.5%

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

      5. *-commutative 97.5%

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

      6. *-commutative 97.5%

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

      7. associate-*l* 97.5%

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

      8. metadata-eval 97.5%

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

   9. Applied egg-rr 97.5%

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

       1. +-commutative 97.5%

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

       2. *-commutative 97.5%

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

       3. distribute-lft-out 97.5%

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

   11. Applied egg-rr 97.5%

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

   12. Taylor expanded in x around inf 95.9%

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

       1. *-commutative 95.9%

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

   14. Simplified 95.9%

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

   if -0.57999999999999996 < x < 0.20000000000000001

   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 97.4%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 0.21\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 0.21))) (* x (* x 9.0)) (+ 3.0 (* x -12.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 0.21)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0 + (x * -12.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 0.21d0))) then
        tmp = x * (x * 9.0d0)
    else
        tmp = 3.0d0 + (x * (-12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 0.21)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0 + (x * -12.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.55) or not (x <= 0.21):
		tmp = x * (x * 9.0)
	else:
		tmp = 3.0 + (x * -12.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 0.21))
		tmp = Float64(x * Float64(x * 9.0));
	else
		tmp = Float64(3.0 + Float64(x * -12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 0.21)))
		tmp = x * (x * 9.0);
	else
		tmp = 3.0 + (x * -12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 0.21]], $MachinePrecision]], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 0.209999999999999992 < x

   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* 99.8%

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

      12. fma-define 99.9%

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

      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 inf 97.5%

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

      1. +-commutative 97.5%

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

      2. unpow2 97.5%

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

      3. associate-*r* 97.5%

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

      4. *-commutative 97.5%

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

      5. distribute-rgt-in 97.5%

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

      6. fma-undefine 97.6%

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

   7. Simplified 97.6%

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

      1. fma-undefine 97.5%

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

      2. distribute-rgt-in 97.5%

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

      3. metadata-eval 97.5%

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

      4. associate-*r* 97.5%

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

      5. *-commutative 97.5%

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

      6. *-commutative 97.5%

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

      7. associate-*l* 97.5%

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

      8. metadata-eval 97.5%

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

   9. Applied egg-rr 97.5%

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

       1. +-commutative 97.5%

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

       2. *-commutative 97.5%

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

       3. distribute-lft-out 97.5%

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

   11. Applied egg-rr 97.5%

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

   12. Taylor expanded in x around inf 95.9%

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

       1. *-commutative 95.9%

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

   14. Simplified 95.9%

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

   if -1.55000000000000004 < x < 0.209999999999999992

   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 99.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 0.21\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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-- 99.9%

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

   5. associate-*l* 99.9%

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

   6. fma-define 99.9%

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

   7. *-commutative 99.9%

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

   8. sub-neg 99.9%

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

   9. distribute-lft-in 99.9%

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

   10. *-commutative 99.9%

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

   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 53.7%

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

   2. distribute-rgt-in 53.7%

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

   3. metadata-eval 53.7%

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

   4. associate-*r* 53.7%

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

   5. *-commutative 53.7%

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

   6. *-commutative 53.7%

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

   7. associate-*l* 53.7%

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

   8. metadata-eval 53.7%

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

8. Applied egg-rr 99.9%

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

   1. +-commutative 53.7%

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

   2. *-commutative 53.7%

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

   3. distribute-lft-out 53.7%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 7: 50.6% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 3 \end{array} \]

FPCore

(FPCore (x) :precision binary64 3.0)

C

double code(double x) {
	return 3.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 3.0

Julia

function code(x)
	return 3.0
end

MATLAB

function tmp = code(x)
	tmp = 3.0;
end

Wolfram

code[x_] := 3.0

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. 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-- 99.9%

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

   5. associate-*l* 99.9%

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

   6. fma-define 99.9%

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

   7. *-commutative 99.9%

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

   8. sub-neg 99.9%

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

   9. distribute-lft-in 99.9%

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

   10. *-commutative 99.9%

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

   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 48.0%

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

6. Final simplification 48.0%

   \[\leadsto 3 \]
7. 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 2024043 
(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)))