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

Percentage Accurate: 99.8% → 99.9%

Time: 8.2s

Alternatives: 8

Speedup: 1.0×

• 25.8% 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. *-commutative 99.8%

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

   3. distribute-lft-out-- 99.8%

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

   4. associate-*l* 99.8%

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

   5. metadata-eval 99.8%

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

   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-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-def 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. Step-by-step derivation

   1. fma-udef 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

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: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 0.335:\\ \;\;\;\;3 + x \cdot -12\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-6 + x \cdot 9\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (* 9.0 (* x x))
   (if (<= x 0.335) (+ 3.0 (* x -12.0)) (* x (+ -6.0 (* x 9.0))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 9.0 * (x * x);
	} else if (x <= 0.335) {
		tmp = 3.0 + (x * -12.0);
	} else {
		tmp = x * (-6.0 + (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = 9.0d0 * (x * x)
    else if (x <= 0.335d0) then
        tmp = 3.0d0 + (x * (-12.0d0))
    else
        tmp = x * ((-6.0d0) + (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 9.0 * (x * x);
	} else if (x <= 0.335) {
		tmp = 3.0 + (x * -12.0);
	} else {
		tmp = x * (-6.0 + (x * 9.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.55:
		tmp = 9.0 * (x * x)
	elif x <= 0.335:
		tmp = 3.0 + (x * -12.0)
	else:
		tmp = x * (-6.0 + (x * 9.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(9.0 * Float64(x * x));
	elseif (x <= 0.335)
		tmp = Float64(3.0 + Float64(x * -12.0));
	else
		tmp = Float64(x * Float64(-6.0 + Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.55)
		tmp = 9.0 * (x * x);
	elseif (x <= 0.335)
		tmp = 3.0 + (x * -12.0);
	else
		tmp = x * (-6.0 + (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.335], N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(-6.0 + N[(x * 9.0), $MachinePrecision]), $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. Taylor expanded in x around inf 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   if -1.55000000000000004 < x < 0.33500000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.2%

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

   if 0.33500000000000002 < 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-lft-in 99.7%

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

      2. metadata-eval 99.7%

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

      3. fma-def 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-lft-out-- 99.7%

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

      6. *-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. distribute-lft-out-- 99.7%

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

      2. associate-*l* 99.7%

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

      3. fma-def 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. add-sqr-sqrt 99.7%

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

      7. associate-*r* 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around -inf 0.3%

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

      1. *-commutative 0.3%

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

      2. *-commutative 0.3%

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

   8. Simplified 0.3%

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

   9. Taylor expanded in x around -inf 98.0%

      \[\leadsto \color{blue}{3 \cdot \left({\left(\sqrt{3}\right)}^{2} \cdot {x}^{2}\right) + -6 \cdot x} \]
   10. Step-by-step derivation

       1. associate-*r* 97.8%

          \[\leadsto \color{blue}{\left(3 \cdot {\left(\sqrt{3}\right)}^{2}\right) \cdot {x}^{2}} + -6 \cdot x \]

       2. fma-def 97.8%

          \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot {\left(\sqrt{3}\right)}^{2}, {x}^{2}, -6 \cdot x\right)} \]

       3. unpow2 97.8%

          \[\leadsto \mathsf{fma}\left(3 \cdot \color{blue}{\left(\sqrt{3} \cdot \sqrt{3}\right)}, {x}^{2}, -6 \cdot x\right) \]

       4. rem-square-sqrt 98.5%

          \[\leadsto \mathsf{fma}\left(3 \cdot \color{blue}{3}, {x}^{2}, -6 \cdot x\right) \]

       5. metadata-eval 98.5%

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

       6. fma-udef 98.5%

          \[\leadsto \color{blue}{9 \cdot {x}^{2} + -6 \cdot x} \]

       7. *-commutative 98.5%

          \[\leadsto \color{blue}{{x}^{2} \cdot 9} + -6 \cdot x \]

       8. unpow2 98.5%

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

       9. associate-*r* 98.5%

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

       10. +-commutative 98.5%

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

       11. *-commutative 98.5%

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

       12. distribute-lft-out 98.5%

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

       13. *-commutative 98.5%

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

   11. Simplified 98.5%

       \[\leadsto \color{blue}{x \cdot \left(-6 + 9 \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 0.335:\\ \;\;\;\;3 + x \cdot -12\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-6 + x \cdot 9\right)\\ \end{array} \]

Alternative 4: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 1.65\right):\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.58) (not (<= x 1.65))) (* 9.0 (* x x)) 3.0))

C

double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 1.65)) {
		tmp = 9.0 * (x * 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 <= 1.65d0))) then
        tmp = 9.0d0 * (x * 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 <= 1.65)) {
		tmp = 9.0 * (x * x);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.58) or not (x <= 1.65):
		tmp = 9.0 * (x * x)
	else:
		tmp = 3.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.58) || !(x <= 1.65))
		tmp = Float64(9.0 * Float64(x * x));
	else
		tmp = 3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.58) || ~((x <= 1.65)))
		tmp = 9.0 * (x * x);
	else
		tmp = 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.58], N[Not[LessEqual[x, 1.65]], $MachinePrecision]], N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], 3.0]

Derivation

1. Split input into 2 regimes

2. if x < -0.57999999999999996 or 1.6499999999999999 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   if -0.57999999999999996 < x < 1.6499999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 96.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 1.65\right):\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \]

Alternative 5: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 1.65:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.58) (* 9.0 (* x x)) (if (<= x 1.65) 3.0 (* x (* x 9.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.58) {
		tmp = 9.0 * (x * x);
	} else if (x <= 1.65) {
		tmp = 3.0;
	} else {
		tmp = x * (x * 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.58d0)) then
        tmp = 9.0d0 * (x * x)
    else if (x <= 1.65d0) then
        tmp = 3.0d0
    else
        tmp = x * (x * 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.58) {
		tmp = 9.0 * (x * x);
	} else if (x <= 1.65) {
		tmp = 3.0;
	} else {
		tmp = x * (x * 9.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.58:
		tmp = 9.0 * (x * x)
	elif x <= 1.65:
		tmp = 3.0
	else:
		tmp = x * (x * 9.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.58)
		tmp = Float64(9.0 * Float64(x * x));
	elseif (x <= 1.65)
		tmp = 3.0;
	else
		tmp = Float64(x * Float64(x * 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.58)
		tmp = 9.0 * (x * x);
	elseif (x <= 1.65)
		tmp = 3.0;
	else
		tmp = x * (x * 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.58], N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65], 3.0, N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.57999999999999996

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   if -0.57999999999999996 < x < 1.6499999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 96.8%

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

   if 1.6499999999999999 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. unpow2 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*l* 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 1.65:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \end{array} \]

Alternative 6: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 0.215:\\ \;\;\;\;3 + x \cdot -12\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (* 9.0 (* x x))
   (if (<= x 0.215) (+ 3.0 (* x -12.0)) (* x (* x 9.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 9.0 * (x * x);
	} else if (x <= 0.215) {
		tmp = 3.0 + (x * -12.0);
	} else {
		tmp = x * (x * 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = 9.0d0 * (x * x)
    else if (x <= 0.215d0) then
        tmp = 3.0d0 + (x * (-12.0d0))
    else
        tmp = x * (x * 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 9.0 * (x * x);
	} else if (x <= 0.215) {
		tmp = 3.0 + (x * -12.0);
	} else {
		tmp = x * (x * 9.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.55:
		tmp = 9.0 * (x * x)
	elif x <= 0.215:
		tmp = 3.0 + (x * -12.0)
	else:
		tmp = x * (x * 9.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(9.0 * Float64(x * x));
	elseif (x <= 0.215)
		tmp = Float64(3.0 + Float64(x * -12.0));
	else
		tmp = Float64(x * Float64(x * 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.55)
		tmp = 9.0 * (x * x);
	elseif (x <= 0.215)
		tmp = 3.0 + (x * -12.0);
	else
		tmp = x * (x * 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.215], N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * 9.0), $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. Taylor expanded in x around inf 99.7%

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

      1. unpow2 99.7%

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

   4. Simplified 99.7%

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

   if -1.55000000000000004 < x < 0.214999999999999997

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.2%

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

   if 0.214999999999999997 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.4%

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

      1. unpow2 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*l* 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 0.215:\\ \;\;\;\;3 + x \cdot -12\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \end{array} \]

Alternative 7: 97.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*l* 99.8%

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

   2. distribute-lft-out-- 99.8%

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

   3. *-commutative 99.8%

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

   4. flip3-- 66.6%

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

   5. associate-*l/ 64.0%

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

3. Applied egg-rr 64.0%

   \[\leadsto 3 \cdot \left(\color{blue}{\frac{\mathsf{fma}\left({x}^{3}, 27, -64\right) \cdot x}{9 \cdot \left(x \cdot x\right) + \left(16 + x \cdot 12\right)}} + 1\right) \]

4. Taylor expanded in x around inf 98.2%

   \[\leadsto 3 \cdot \left(\color{blue}{3 \cdot {x}^{2}} + 1\right) \]
5. Step-by-step derivation

   1. *-commutative 98.2%

      \[\leadsto 3 \cdot \left(\color{blue}{{x}^{2} \cdot 3} + 1\right) \]

   2. unpow2 98.2%

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

   3. associate-*l* 98.2%

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

6. Simplified 98.2%

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

7. Final simplification 98.2%

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

Alternative 8: 51.1% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 3.0)

C

double code(double x) {
	return 3.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 3.0

Julia

function code(x)
	return 3.0
end

MATLAB

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

Wolfram

code[x_] := 3.0

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 51.2%

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

3. Final simplification 51.2%

   \[\leadsto 3 \]

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 2023279 
(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)))