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

Percentage Accurate: 99.8% → 99.9%

Time: 4.9s

Alternatives: 5

Speedup: 1.0×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x * N[(N[(x * 9.0), $MachinePrecision] + -12.0), $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* 99.9%

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

   6. fma-def 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-rgt-in 99.9%

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

   10. associate-*l* 99.9%

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

   11. metadata-eval 99.9%

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

   12. metadata-eval 99.9%

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

   13. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

2. Final simplification 99.8%

   \[\leadsto 3 \cdot \left(\left(x \cdot \left(x \cdot 3\right) - x \cdot 4\right) + 1\right) \]

Alternative 3: 97.7% accurate, 1.4× 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* 99.8%

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

      6. fma-def 99.8%

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

      7. *-commutative 99.8%

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

      8. sub-neg 99.8%

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

      9. distribute-rgt-in 99.8%

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

      10. associate-*l* 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 98.5%

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

      1. metadata-eval 98.5%

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

      2. unpow2 98.5%

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

      3. swap-sqr 98.3%

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

      4. pow2 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. unpow-prod-down 98.5%

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

      2. metadata-eval 98.5%

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

      3. unpow2 98.5%

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

      4. associate-*r* 98.5%

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

      5. *-commutative 98.5%

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

   8. Applied egg-rr 98.5%

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

   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-def 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-rgt-in 100.0%

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

      10. associate-*l* 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.4%

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

4. Final simplification 98.5%

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

Alternative 4: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 0.22\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 0.22))) (* x (* x 9.0)) (+ 3.0 (* x -12.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 0.22)) {
		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 <= 0.22d0))) 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 <= 0.22)) {
		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 <= 0.22):
		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 <= 0.22))
		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 <= 0.22)))
		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, 0.22]], $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 0.220000000000000001 < 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* 99.8%

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

      6. fma-def 99.8%

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

      7. *-commutative 99.8%

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

      8. sub-neg 99.8%

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

      9. distribute-rgt-in 99.8%

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

      10. associate-*l* 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 98.5%

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

      1. metadata-eval 98.5%

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

      2. unpow2 98.5%

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

      3. swap-sqr 98.3%

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

      4. pow2 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. unpow-prod-down 98.5%

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

      2. metadata-eval 98.5%

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

      3. unpow2 98.5%

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

      4. associate-*r* 98.5%

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

      5. *-commutative 98.5%

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

   8. Applied egg-rr 98.5%

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

   if -1.55000000000000004 < x < 0.220000000000000001

   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-def 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-rgt-in 100.0%

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

      10. associate-*l* 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 0.22\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \]

Alternative 5: 51.6% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 3.0)

C

double code(double x) {
	return 3.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 3.0

Julia

function code(x)
	return 3.0
end

MATLAB

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

Wolfram

code[x_] := 3.0

Derivation

1. Initial program 99.8%

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

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

   6. fma-def 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-rgt-in 99.9%

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

   10. associate-*l* 99.9%

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

   11. metadata-eval 99.9%

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

   12. metadata-eval 99.9%

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

   13. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 50.8%

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

5. Final simplification 50.8%

   \[\leadsto 3 \]

Developer target: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023332 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x)))

  (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))