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

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 11

Speedup: 1.2×

• 27.1% 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: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 z around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(-2 \cdot x + -1 \cdot \frac{5 \cdot y + x \cdot \left(t + 2 \cdot y\right)}{z}\right)\right)} \]

5. Applied rewrites 96.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2 \cdot z, \mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), 5 \cdot y\right) \cdot 1\right)} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 100.0%

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

Alternative 2: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-280}:\\ \;\;\;\;\mathsf{fma}\left(5, y, x \cdot \left(2 \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 1.7:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), 5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -6.5e-7)
     t_1
     (if (<= x -7.4e-280)
       (fma 5.0 y (* x (* 2.0 z)))
       (if (<= x 1.7) (fma x (fma y 2.0 t) (* 5.0 y)) 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 <= -6.5e-7) {
		tmp = t_1;
	} else if (x <= -7.4e-280) {
		tmp = fma(5.0, y, (x * (2.0 * z)));
	} else if (x <= 1.7) {
		tmp = fma(x, fma(y, 2.0, t), (5.0 * y));
	} 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 <= -6.5e-7)
		tmp = t_1;
	elseif (x <= -7.4e-280)
		tmp = fma(5.0, y, Float64(x * Float64(2.0 * z)));
	elseif (x <= 1.7)
		tmp = fma(x, fma(y, 2.0, t), Float64(5.0 * y));
	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, -6.5e-7], t$95$1, If[LessEqual[x, -7.4e-280], N[(5.0 * y + N[(x * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.7], N[(x * N[(y * 2.0 + t), $MachinePrecision] + N[(5.0 * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.50000000000000024e-7 or 1.69999999999999996 < 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 99.4

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

   5. Applied rewrites 99.4%

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

   if -6.50000000000000024e-7 < x < -7.3999999999999996e-280

   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 z around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(-2 \cdot x + -1 \cdot \frac{5 \cdot y + x \cdot \left(t + 2 \cdot y\right)}{z}\right)\right)} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2 \cdot z, \mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), 5 \cdot y\right) \cdot 1\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in z around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 87.1

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

   11. Applied rewrites 87.1%

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

   if -7.3999999999999996e-280 < x < 1.69999999999999996

   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 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 87.5

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

   5. Applied rewrites 87.5%

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

Alternative 3: 88.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-265}:\\ \;\;\;\;\mathsf{fma}\left(5, y, x \cdot \left(2 \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 0.34:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -6.5e-7)
     t_1
     (if (<= x -8.2e-265)
       (fma 5.0 y (* x (* 2.0 z)))
       (if (<= x 0.34) (fma 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 <= -6.5e-7) {
		tmp = t_1;
	} else if (x <= -8.2e-265) {
		tmp = fma(5.0, y, (x * (2.0 * z)));
	} else if (x <= 0.34) {
		tmp = fma(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 <= -6.5e-7)
		tmp = t_1;
	elseif (x <= -8.2e-265)
		tmp = fma(5.0, y, Float64(x * Float64(2.0 * z)));
	elseif (x <= 0.34)
		tmp = fma(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, -6.5e-7], t$95$1, If[LessEqual[x, -8.2e-265], N[(5.0 * y + N[(x * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.34], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.50000000000000024e-7 or 0.340000000000000024 < 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 99.4

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

   5. Applied rewrites 99.4%

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

   if -6.50000000000000024e-7 < x < -8.2e-265

   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 z around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(-2 \cdot x + -1 \cdot \frac{5 \cdot y + x \cdot \left(t + 2 \cdot y\right)}{z}\right)\right)} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2 \cdot z, \mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), 5 \cdot y\right) \cdot 1\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in z around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 86.8

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

   11. Applied rewrites 86.8%

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

   if -8.2e-265 < x < 0.340000000000000024

   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 85.9

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

   5. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 85.9

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

   7. Applied rewrites 85.9%

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

Alternative 4: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \left(y + \left(y + t\right)\right)\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.34:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 z t))))
   (if (<= x -4.3e+21)
     (* x (+ y (+ y t)))
     (if (<= x -2.1e-144) t_1 (if (<= x 0.34) (* 5.0 y) 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 (x <= -4.3e+21) {
		tmp = x * (y + (y + t));
	} else if (x <= -2.1e-144) {
		tmp = t_1;
	} else if (x <= 0.34) {
		tmp = 5.0 * y;
	} 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 (x <= -4.3e+21)
		tmp = Float64(x * Float64(y + Float64(y + t)));
	elseif (x <= -2.1e-144)
		tmp = t_1;
	elseif (x <= 0.34)
		tmp = Float64(5.0 * y);
	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[x, -4.3e+21], N[(x * N[(y + N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.1e-144], t$95$1, If[LessEqual[x, 0.34], N[(5.0 * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e21

   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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. count-2 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 77.5

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

   8. Applied rewrites 77.5%

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

   if -4.3e21 < x < -2.1000000000000001e-144 or 0.340000000000000024 < 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 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 70.6

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

   5. Applied rewrites 70.6%

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

   if -2.1000000000000001e-144 < x < 0.340000000000000024

   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 68.3

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

   5. Applied rewrites 68.3%

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

Alternative 5: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -12600:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -12600.0)
     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 <= -12600.0) {
		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 <= -12600.0)
		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, -12600.0], 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 < -12600 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 99.4

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

   5. Applied rewrites 99.4%

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

   if -12600 < 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

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 100.0

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

   4. Applied rewrites 98.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -12600:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.34:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -3.4e-129) t_1 (if (<= x 0.34) (fma 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 <= -3.4e-129) {
		tmp = t_1;
	} else if (x <= 0.34) {
		tmp = fma(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 <= -3.4e-129)
		tmp = t_1;
	elseif (x <= 0.34)
		tmp = fma(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, -3.4e-129], t$95$1, If[LessEqual[x, 0.34], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000013e-129 or 0.340000000000000024 < 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 92.0

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

   5. Applied rewrites 92.0%

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

   if -3.40000000000000013e-129 < x < 0.340000000000000024

   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 85.1

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

   5. Applied rewrites 85.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 85.1

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

   7. Applied rewrites 85.1%

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

Alternative 7: 78.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -4.4e+74) t_1 (if (<= y 3.3e+28) (* 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 <= -4.4e+74) {
		tmp = t_1;
	} else if (y <= 3.3e+28) {
		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 <= -4.4e+74)
		tmp = t_1;
	elseif (y <= 3.3e+28)
		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, -4.4e+74], t$95$1, If[LessEqual[y, 3.3e+28], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4000000000000002e74 or 3.3e28 < 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 83.5

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

   5. Applied rewrites 83.5%

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

   if -4.4000000000000002e74 < y < 3.3e28

   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 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 76.5

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

   5. Applied rewrites 76.5%

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

Alternative 8: 63.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y + \left(y + t\right)\right)\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.34:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ y (+ y t)))))
   (if (<= x -6.5e-7) t_1 (if (<= x 0.34) (* 5.0 y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (y + (y + t));
	double tmp;
	if (x <= -6.5e-7) {
		tmp = t_1;
	} else if (x <= 0.34) {
		tmp = 5.0 * y;
	} 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 * (y + (y + t))
    if (x <= (-6.5d-7)) then
        tmp = t_1
    else if (x <= 0.34d0) then
        tmp = 5.0d0 * y
    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 * (y + (y + t));
	double tmp;
	if (x <= -6.5e-7) {
		tmp = t_1;
	} else if (x <= 0.34) {
		tmp = 5.0 * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (y + (y + t))
	tmp = 0
	if x <= -6.5e-7:
		tmp = t_1
	elif x <= 0.34:
		tmp = 5.0 * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(y + Float64(y + t)))
	tmp = 0.0
	if (x <= -6.5e-7)
		tmp = t_1;
	elseif (x <= 0.34)
		tmp = Float64(5.0 * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (y + (y + t));
	tmp = 0.0;
	if (x <= -6.5e-7)
		tmp = t_1;
	elseif (x <= 0.34)
		tmp = 5.0 * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y + N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.5e-7], t$95$1, If[LessEqual[x, 0.34], N[(5.0 * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.50000000000000024e-7 or 0.340000000000000024 < 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 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. count-2 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 72.3

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

   8. Applied rewrites 72.3%

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

   if -6.50000000000000024e-7 < x < 0.340000000000000024

   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 62.0

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

   5. Applied rewrites 62.0%

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

Alternative 9: 46.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0135:\\ \;\;\;\;x \cdot \left(y + y\right)\\ \mathbf{elif}\;x \leq 0.34:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -0.0135) (* x (+ y y)) (if (<= x 0.34) (* 5.0 y) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.0135) {
		tmp = x * (y + y);
	} else if (x <= 0.34) {
		tmp = 5.0 * y;
	} 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 (x <= (-0.0135d0)) then
        tmp = x * (y + y)
    else if (x <= 0.34d0) then
        tmp = 5.0d0 * y
    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 (x <= -0.0135) {
		tmp = x * (y + y);
	} else if (x <= 0.34) {
		tmp = 5.0 * y;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -0.0135:
		tmp = x * (y + y)
	elif x <= 0.34:
		tmp = 5.0 * y
	else:
		tmp = x * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -0.0135)
		tmp = Float64(x * Float64(y + y));
	elseif (x <= 0.34)
		tmp = Float64(5.0 * y);
	else
		tmp = Float64(x * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -0.0135)
		tmp = x * (y + y);
	elseif (x <= 0.34)
		tmp = 5.0 * y;
	else
		tmp = x * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -0.0135], N[(x * N[(y + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.34], N[(5.0 * y), $MachinePrecision], N[(x * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0134999999999999998

   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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. count-2 N/A

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

      6. lower-+.f64 41.1

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

   8. Applied rewrites 41.1%

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

   if -0.0134999999999999998 < x < 0.340000000000000024

   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 61.5

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

   5. Applied rewrites 61.5%

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

   if 0.340000000000000024 < 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 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 44.4

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

   5. Applied rewrites 44.4%

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

Alternative 10: 47.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-7}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 0.34:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -6.5e-7) (* x t) (if (<= x 0.34) (* 5.0 y) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.5e-7) {
		tmp = x * t;
	} else if (x <= 0.34) {
		tmp = 5.0 * y;
	} 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 (x <= (-6.5d-7)) then
        tmp = x * t
    else if (x <= 0.34d0) then
        tmp = 5.0d0 * y
    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 (x <= -6.5e-7) {
		tmp = x * t;
	} else if (x <= 0.34) {
		tmp = 5.0 * y;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -6.5e-7:
		tmp = x * t
	elif x <= 0.34:
		tmp = 5.0 * y
	else:
		tmp = x * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -6.5e-7)
		tmp = Float64(x * t);
	elseif (x <= 0.34)
		tmp = Float64(5.0 * y);
	else
		tmp = Float64(x * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -6.5e-7)
		tmp = x * t;
	elseif (x <= 0.34)
		tmp = 5.0 * y;
	else
		tmp = x * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -6.5e-7], N[(x * t), $MachinePrecision], If[LessEqual[x, 0.34], N[(5.0 * y), $MachinePrecision], N[(x * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.50000000000000024e-7 or 0.340000000000000024 < 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 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 41.0

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

   5. Applied rewrites 41.0%

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

   if -6.50000000000000024e-7 < x < 0.340000000000000024

   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 62.0

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

   5. Applied rewrites 62.0%

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

Alternative 11: 29.4% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

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 = 5.0d0 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 32.1

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

5. Applied rewrites 32.1%

   \[\leadsto \color{blue}{5 \cdot y} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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)))