Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B

Percentage Accurate: 99.9% → 100.0%

Time: 9.8s

Alternatives: 11

Speedup: 1.2×

• 26.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * ((((y + z) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * ((((y + z) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, 5, x \cdot \mathsf{fma}\left(2, y + z, t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma y 5.0 (* x (fma 2.0 (+ y z) t))))

C

double code(double x, double y, double z, double t) {
	return fma(y, 5.0, (x * fma(2.0, (y + z), t)));
}

Julia

function code(x, y, z, t)
	return fma(y, 5.0, Float64(x * fma(2.0, Float64(y + z), t)))
end

Wolfram

code[x_, y_, z_, t_] := N[(y * 5.0 + N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{5 \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 26.7

      \[\leadsto \color{blue}{5 \cdot y} \]

5. Applied rewrites 26.7%

   \[\leadsto \color{blue}{5 \cdot y} \]

6. Taylor expanded in t around inf

   \[\leadsto \color{blue}{t \cdot \left(x + \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right)\right)} \]

8. Applied rewrites 88.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(5, \frac{y}{t} \cdot t, x \cdot \mathsf{fma}\left(2, y + z, t\right)\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{5 \cdot y + x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto 5 \cdot y + \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

    2. distribute-lft-in N/A

       \[\leadsto 5 \cdot y + \left(t + \color{blue}{2 \cdot \left(y + z\right)}\right) \cdot x \]

    3. *-commutative N/A

       \[\leadsto \color{blue}{y \cdot 5} + \left(t + 2 \cdot \left(y + z\right)\right) \cdot x \]

    4. *-commutative N/A

       \[\leadsto y \cdot 5 + \color{blue}{x \cdot \left(t + 2 \cdot \left(y + z\right)\right)} \]

    5. distribute-rgt-in N/A

       \[\leadsto y \cdot 5 + \color{blue}{\left(t \cdot x + \left(2 \cdot \left(y + z\right)\right) \cdot x\right)} \]

    6. *-commutative N/A

       \[\leadsto y \cdot 5 + \left(t \cdot x + \color{blue}{x \cdot \left(2 \cdot \left(y + z\right)\right)}\right) \]

    7. distribute-lft-in N/A

       \[\leadsto y \cdot 5 + \left(t \cdot x + x \cdot \color{blue}{\left(2 \cdot y + 2 \cdot z\right)}\right) \]

    8. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, t \cdot x + x \cdot \left(2 \cdot y + 2 \cdot z\right)\right)} \]

    9. distribute-lft-in N/A

       \[\leadsto \mathsf{fma}\left(y, 5, t \cdot x + x \cdot \color{blue}{\left(2 \cdot \left(y + z\right)\right)}\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(y, 5, t \cdot x + \color{blue}{\left(2 \cdot \left(y + z\right)\right) \cdot x}\right) \]

    11. distribute-rgt-in N/A

        \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(t + 2 \cdot \left(y + z\right)\right)}\right) \]

    12. lower-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(t + 2 \cdot \left(y + z\right)\right)}\right) \]

    13. +-commutative N/A

        \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \color{blue}{\left(2 \cdot \left(y + z\right) + t\right)}\right) \]

    14. lower-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)}\right) \]

    15. lower-+.f64 100.0

        \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right)\right) \]

11. Applied rewrites 100.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \mathsf{fma}\left(2, y + z, t\right)\right)} \]
12. Add Preprocessing

Alternative 2: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t + \left(z + z\right), x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -2.5) t_1 (if (<= x 2.5) (fma (+ t (+ z z)) x (* y 5.0)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, (y + z), t);
	double tmp;
	if (x <= -2.5) {
		tmp = t_1;
	} else if (x <= 2.5) {
		tmp = fma((t + (z + z)), x, (y * 5.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, Float64(y + z), t))
	tmp = 0.0
	if (x <= -2.5)
		tmp = t_1;
	elseif (x <= 2.5)
		tmp = fma(Float64(t + Float64(z + z)), x, Float64(y * 5.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5], t$95$1, If[LessEqual[x, 2.5], N[(N[(t + N[(z + z), $MachinePrecision]), $MachinePrecision] * x + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5 or 2.5 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \]

      5. lower-+.f64 99.7

         \[\leadsto x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, y + z, t\right)} \]

   if -2.5 < x < 2.5

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x} + y \cdot 5 \]

      4. lower-fma.f64 99.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t, x, y \cdot 5\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t, x, y \cdot 5\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t, x, y \cdot 5\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t, x, y \cdot 5\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t, x, y \cdot 5\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t, x, y \cdot 5\right) \]

      10. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t, x, y \cdot 5\right) \]

      11. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t, x, y \cdot 5\right) \]

      12. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t, x, y \cdot 5\right) \]

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t, x, y \cdot 5\right) \]

      14. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t, x, y \cdot 5\right) \]

      15. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z + z\right)} + t, x, y \cdot 5\right) \]

      16. lower-+.f64 99.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z + z\right)} + t, x, y \cdot 5\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z + z\right) + t, x, y \cdot 5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t + \left(z + z\right), x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -2.5) t_1 (if (<= x 2.5) (fma y 5.0 (* x (+ t (+ z z)))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, (y + z), t);
	double tmp;
	if (x <= -2.5) {
		tmp = t_1;
	} else if (x <= 2.5) {
		tmp = fma(y, 5.0, (x * (t + (z + z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, Float64(y + z), t))
	tmp = 0.0
	if (x <= -2.5)
		tmp = t_1;
	elseif (x <= 2.5)
		tmp = fma(y, 5.0, Float64(x * Float64(t + Float64(z + z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5], t$95$1, If[LessEqual[x, 2.5], N[(y * 5.0 + N[(x * N[(t + N[(z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5 or 2.5 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \]

      5. lower-+.f64 99.7

         \[\leadsto x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, y + z, t\right)} \]

   if -2.5 < x < 2.5

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right)\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right)\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right)\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right)\right) \]

      10. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t\right)\right) \]

      11. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right)\right) \]

      12. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t\right)\right) \]

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right)\right) \]

      14. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t\right)\right) \]

      15. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(z + z\right)} + t\right)\right) \]

      16. lower-+.f64 99.1

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(z + z\right)} + t\right)\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(z + z\right) + t\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(z + z\right)\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, x, y \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.45e+47)
   (fma y 5.0 (* x (+ z z)))
   (if (<= z 1.4e+83)
     (fma x (fma y 2.0 t) (* y 5.0))
     (fma (+ z z) x (* y 5.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.45e+47) {
		tmp = fma(y, 5.0, (x * (z + z)));
	} else if (z <= 1.4e+83) {
		tmp = fma(x, fma(y, 2.0, t), (y * 5.0));
	} else {
		tmp = fma((z + z), x, (y * 5.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.45e+47)
		tmp = fma(y, 5.0, Float64(x * Float64(z + z)));
	elseif (z <= 1.4e+83)
		tmp = fma(x, fma(y, 2.0, t), Float64(y * 5.0));
	else
		tmp = fma(Float64(z + z), x, Float64(y * 5.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.45e+47], N[(y * 5.0 + N[(x * N[(z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.4e+83], N[(x * N[(y * 2.0 + t), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], N[(N[(z + z), $MachinePrecision] * x + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4499999999999999e47

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x} + y \cdot 5 \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t, x, y \cdot 5\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t, x, y \cdot 5\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t, x, y \cdot 5\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t, x, y \cdot 5\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t, x, y \cdot 5\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t, x, y \cdot 5\right) \]

      10. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t, x, y \cdot 5\right) \]

      11. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t, x, y \cdot 5\right) \]

      12. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t, x, y \cdot 5\right) \]

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t, x, y \cdot 5\right) \]

      14. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t, x, y \cdot 5\right) \]

      15. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z + z\right)} + t, x, y \cdot 5\right) \]

      16. lower-+.f64 96.0

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z + z\right)} + t, x, y \cdot 5\right) \]

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z + z\right) + t, x, y \cdot 5\right)} \]

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 12.7

         \[\leadsto \color{blue}{t \cdot x} \]

   7. Applied rewrites 12.7%

      \[\leadsto \color{blue}{t \cdot x} \]

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot z\right) + 5 \cdot y} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{5 \cdot y + 2 \cdot \left(x \cdot z\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5} + 2 \cdot \left(x \cdot z\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, 2 \cdot \left(x \cdot z\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{2 \cdot \left(x \cdot z\right)}\right) \]

      5. lower-*.f64 87.2

         \[\leadsto \mathsf{fma}\left(y, 5, 2 \cdot \color{blue}{\left(x \cdot z\right)}\right) \]

   10. Applied rewrites 87.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, 2 \cdot \left(x \cdot z\right)\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 87.2%

       \[\leadsto \mathsf{fma}\left(y, 5, \left(z + z\right) \cdot x\right) \]

   if -1.4499999999999999e47 < z < 1.4e83

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{5 \cdot y + x \cdot \left(t + 2 \cdot y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot y\right) + 5 \cdot y} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, t + 2 \cdot y, 5 \cdot y\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{2 \cdot y + t}, 5 \cdot y\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot 2} + t, 5 \cdot y\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, 2, t\right)}, 5 \cdot y\right) \]

      6. lower-*.f64 94.1

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), \color{blue}{5 \cdot y}\right) \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), 5 \cdot y\right)} \]

   if 1.4e83 < z

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} + y \cdot 5 \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x} + y \cdot 5 \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t, x, y \cdot 5\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t, x, y \cdot 5\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t, x, y \cdot 5\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t, x, y \cdot 5\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t, x, y \cdot 5\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t, x, y \cdot 5\right) \]

      10. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t, x, y \cdot 5\right) \]

      11. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t, x, y \cdot 5\right) \]

      12. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t, x, y \cdot 5\right) \]

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t, x, y \cdot 5\right) \]

      14. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t, x, y \cdot 5\right) \]

      15. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z + z\right)} + t, x, y \cdot 5\right) \]

      16. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z + z\right)} + t, x, y \cdot 5\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z + z\right) + t, x, y \cdot 5\right)} \]

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 15.8

         \[\leadsto \color{blue}{t \cdot x} \]

   7. Applied rewrites 15.8%

      \[\leadsto \color{blue}{t \cdot x} \]

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot z\right) + 5 \cdot y} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{5 \cdot y + 2 \cdot \left(x \cdot z\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5} + 2 \cdot \left(x \cdot z\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, 2 \cdot \left(x \cdot z\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{2 \cdot \left(x \cdot z\right)}\right) \]

      5. lower-*.f64 91.3

         \[\leadsto \mathsf{fma}\left(y, 5, 2 \cdot \color{blue}{\left(x \cdot z\right)}\right) \]

   10. Applied rewrites 91.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, 2 \cdot \left(x \cdot z\right)\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 91.3%

       \[\leadsto \mathsf{fma}\left(z + z, \color{blue}{x}, y \cdot 5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(z + z\right)\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, x, y \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -2.5e-148) t_1 (if (<= x 4.8e-5) (+ (* y 5.0) (* x t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, (y + z), t);
	double tmp;
	if (x <= -2.5e-148) {
		tmp = t_1;
	} else if (x <= 4.8e-5) {
		tmp = (y * 5.0) + (x * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, Float64(y + z), t))
	tmp = 0.0
	if (x <= -2.5e-148)
		tmp = t_1;
	elseif (x <= 4.8e-5)
		tmp = Float64(Float64(y * 5.0) + Float64(x * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-148], t$95$1, If[LessEqual[x, 4.8e-5], N[(N[(y * 5.0), $MachinePrecision] + N[(x * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4999999999999999e-148 or 4.8000000000000001e-5 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \]

      5. lower-+.f64 95.1

         \[\leadsto x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \]

   5. Applied rewrites 95.1%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, y + z, t\right)} \]

   if -2.4999999999999999e-148 < x < 4.8000000000000001e-5

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot x} + y \cdot 5 \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot t} + y \cdot 5 \]

      2. lower-*.f64 82.4

         \[\leadsto \color{blue}{x \cdot t} + y \cdot 5 \]

   5. Applied rewrites 82.4%

      \[\leadsto \color{blue}{x \cdot t} + y \cdot 5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-148}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -1.1e+56) t_1 (if (<= y 7.5e+24) (* x (fma 2.0 z t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * fma(x, 2.0, 5.0);
	double tmp;
	if (y <= -1.1e+56) {
		tmp = t_1;
	} else if (y <= 7.5e+24) {
		tmp = x * fma(2.0, z, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * fma(x, 2.0, 5.0))
	tmp = 0.0
	if (y <= -1.1e+56)
		tmp = t_1;
	elseif (y <= 7.5e+24)
		tmp = Float64(x * fma(2.0, z, t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x * 2.0 + 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.1e+56], t$95$1, If[LessEqual[y, 7.5e+24], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.10000000000000008e56 or 7.50000000000000014e24 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + 5\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x + 5\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

      4. neg-sub0 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(0 - -2 \cdot x\right)} + 5\right) \]

      5. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      8. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      9. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(0 - -2 \cdot x\right) + 5\right)} \]

      10. neg-sub0 N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right) \cdot x} + 5\right) \]

      12. metadata-eval N/A

          \[\leadsto y \cdot \left(\color{blue}{2} \cdot x + 5\right) \]

      13. *-commutative N/A

          \[\leadsto y \cdot \left(\color{blue}{x \cdot 2} + 5\right) \]

      14. lower-fma.f64 79.9

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, 2, 5\right)} \]

   5. Applied rewrites 79.9%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, 2, 5\right)} \]

   if -1.10000000000000008e56 < y < 7.50000000000000014e24

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot z + t\right)} \]

      3. lower-fma.f64 83.3

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, z, t\right)} \]

   5. Applied rewrites 83.3%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, z, t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 64.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{-45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{+23}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, 2, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 z t))))
   (if (<= z -5.5e-45) t_1 (if (<= z 1.36e+23) (* x (fma y 2.0 t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, z, t);
	double tmp;
	if (z <= -5.5e-45) {
		tmp = t_1;
	} else if (z <= 1.36e+23) {
		tmp = x * fma(y, 2.0, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, z, t))
	tmp = 0.0
	if (z <= -5.5e-45)
		tmp = t_1;
	elseif (z <= 1.36e+23)
		tmp = Float64(x * fma(y, 2.0, t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5e-45], t$95$1, If[LessEqual[z, 1.36e+23], N[(x * N[(y * 2.0 + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5000000000000003e-45 or 1.36e23 < z

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot z + t\right)} \]

      3. lower-fma.f64 76.6

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, z, t\right)} \]

   5. Applied rewrites 76.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, z, t\right)} \]

   if -5.5000000000000003e-45 < z < 1.36e23

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \]

      5. lower-+.f64 71.2

         \[\leadsto x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \]

   5. Applied rewrites 71.2%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, y + z, t\right)} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 69.7%

      \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{2}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 58.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(z + z\right)\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+83}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, 2, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ z z))))
   (if (<= z -3.6e+47) t_1 (if (<= z 1.4e+83) (* x (fma y 2.0 t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (z + z);
	double tmp;
	if (z <= -3.6e+47) {
		tmp = t_1;
	} else if (z <= 1.4e+83) {
		tmp = x * fma(y, 2.0, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(z + z))
	tmp = 0.0
	if (z <= -3.6e+47)
		tmp = t_1;
	elseif (z <= 1.4e+83)
		tmp = Float64(x * fma(y, 2.0, t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(z + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e+47], t$95$1, If[LessEqual[z, 1.4e+83], N[(x * N[(y * 2.0 + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.60000000000000008e47 or 1.4e83 < z

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 69.1%

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

   if -3.60000000000000008e47 < z < 1.4e83

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \]

      5. lower-+.f64 72.2

         \[\leadsto x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, y + z, t\right)} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.4%

      \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{2}, t\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+47}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+83}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, 2, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(z + z\right)\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+82}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ z z))))
   (if (<= z -2.2e+40) t_1 (if (<= z 2.15e+82) (* x t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (z + z);
	double tmp;
	if (z <= -2.2e+40) {
		tmp = t_1;
	} else if (z <= 2.15e+82) {
		tmp = x * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (z + z)
    if (z <= (-2.2d+40)) then
        tmp = t_1
    else if (z <= 2.15d+82) then
        tmp = x * t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (z + z);
	double tmp;
	if (z <= -2.2e+40) {
		tmp = t_1;
	} else if (z <= 2.15e+82) {
		tmp = x * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (z + z)
	tmp = 0
	if z <= -2.2e+40:
		tmp = t_1
	elif z <= 2.15e+82:
		tmp = x * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(z + z))
	tmp = 0.0
	if (z <= -2.2e+40)
		tmp = t_1;
	elseif (z <= 2.15e+82)
		tmp = Float64(x * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (z + z);
	tmp = 0.0;
	if (z <= -2.2e+40)
		tmp = t_1;
	elseif (z <= 2.15e+82)
		tmp = x * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(z + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.2e+40], t$95$1, If[LessEqual[z, 2.15e+82], N[(x * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1999999999999999e40 or 2.15000000000000007e82 < z

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 69.1%

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

   if -2.1999999999999999e40 < z < 2.15000000000000007e82

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot t} \]

      2. lower-*.f64 45.8

         \[\leadsto \color{blue}{x \cdot t} \]

   5. Applied rewrites 45.8%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+82}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-10}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+77}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.4e-10) (* x t) (if (<= t 3.3e+77) (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.4e-10) {
		tmp = x * t;
	} else if (t <= 3.3e+77) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.4d-10)) then
        tmp = x * t
    else if (t <= 3.3d+77) then
        tmp = y * 5.0d0
    else
        tmp = x * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.4e-10) {
		tmp = x * t;
	} else if (t <= 3.3e+77) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.4e-10:
		tmp = x * t
	elif t <= 3.3e+77:
		tmp = y * 5.0
	else:
		tmp = x * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.4e-10)
		tmp = Float64(x * t);
	elseif (t <= 3.3e+77)
		tmp = Float64(y * 5.0);
	else
		tmp = Float64(x * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.4e-10)
		tmp = x * t;
	elseif (t <= 3.3e+77)
		tmp = y * 5.0;
	else
		tmp = x * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.4e-10], N[(x * t), $MachinePrecision], If[LessEqual[t, 3.3e+77], N[(y * 5.0), $MachinePrecision], N[(x * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.40000000000000008e-10 or 3.2999999999999998e77 < t

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot t} \]

      2. lower-*.f64 64.4

         \[\leadsto \color{blue}{x \cdot t} \]

   5. Applied rewrites 64.4%

      \[\leadsto \color{blue}{x \cdot t} \]

   if -1.40000000000000008e-10 < t < 3.2999999999999998e77

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 35.8

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 35.8%

      \[\leadsto \color{blue}{5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-10}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+77}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* y 5.0))

C

double code(double x, double y, double z, double t) {
	return y * 5.0;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y * 5.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return y * 5.0;
}

Python

def code(x, y, z, t):
	return y * 5.0

Julia

function code(x, y, z, t)
	return Float64(y * 5.0)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y * 5.0;
end

Wolfram

code[x_, y_, z_, t_] := N[(y * 5.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{5 \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 26.7

      \[\leadsto \color{blue}{5 \cdot y} \]

5. Applied rewrites 26.7%

   \[\leadsto \color{blue}{5 \cdot y} \]

6. Final simplification 26.7%

   \[\leadsto y \cdot 5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  :precision binary64
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))