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

Percentage Accurate: 99.9% → 99.9%

Time: 7.6s

Alternatives: 10

Speedup: 1.2×

• 27.0% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(y * 5.0 + N[(N[(N[(z + y), $MachinePrecision] * 2.0 + t), $MachinePrecision] * x), $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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate-+l+ N/A

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

   8. +-commutative N/A

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

   9. count-2 N/A

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

   10. lower-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

   5. lift-+.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. count-2 N/A

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

   9. associate-+l+ N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. associate-+l+ N/A

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

   13. associate-+l+ N/A

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

   14. +-commutative N/A

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

   15. count-2 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 59.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, x, 5\right) \cdot y\\ t_2 := \mathsf{fma}\left(y, 2, t\right) \cdot x\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-161}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-245}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;t \leq 0.062:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 x 5.0) y)) (t_2 (* (fma y 2.0 t) x)))
   (if (<= t -5.4e+18)
     t_2
     (if (<= t -3.2e-161)
       t_1
       (if (<= t -8e-245) (* (* z x) 2.0) (if (<= t 0.062) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, x, 5.0) * y;
	double t_2 = fma(y, 2.0, t) * x;
	double tmp;
	if (t <= -5.4e+18) {
		tmp = t_2;
	} else if (t <= -3.2e-161) {
		tmp = t_1;
	} else if (t <= -8e-245) {
		tmp = (z * x) * 2.0;
	} else if (t <= 0.062) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, x, 5.0) * y)
	t_2 = Float64(fma(y, 2.0, t) * x)
	tmp = 0.0
	if (t <= -5.4e+18)
		tmp = t_2;
	elseif (t <= -3.2e-161)
		tmp = t_1;
	elseif (t <= -8e-245)
		tmp = Float64(Float64(z * x) * 2.0);
	elseif (t <= 0.062)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * 2.0 + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -5.4e+18], t$95$2, If[LessEqual[t, -3.2e-161], t$95$1, If[LessEqual[t, -8e-245], N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t, 0.062], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.4e18 or 0.062 < 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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 72.6%

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

   if -5.4e18 < t < -3.19999999999999985e-161 or -7.9999999999999994e-245 < t < 0.062

   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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-sub0 N/A

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

      6. associate--r- N/A

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

      7. neg-sub0 N/A

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

      8. lower-*.f64 N/A

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

      9. neg-sub0 N/A

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

      10. associate--r- N/A

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

      11. neg-sub0 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if -3.19999999999999985e-161 < t < -7.9999999999999994e-245

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

Alternative 3: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z + y, 2, t\right) \cdot x\\ \mathbf{if}\;x \leq -135000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.00105:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma (+ z y) 2.0 t) x)))
   (if (<= x -135000000.0)
     t_1
     (if (<= x 0.00105) (fma y 5.0 (* (fma 2.0 z t) x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z + y), 2.0, t) * x;
	double tmp;
	if (x <= -135000000.0) {
		tmp = t_1;
	} else if (x <= 0.00105) {
		tmp = fma(y, 5.0, (fma(2.0, z, t) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(z + y), 2.0, t) * x)
	tmp = 0.0
	if (x <= -135000000.0)
		tmp = t_1;
	elseif (x <= 0.00105)
		tmp = fma(y, 5.0, Float64(fma(2.0, z, t) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z + y), $MachinePrecision] * 2.0 + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -135000000.0], t$95$1, If[LessEqual[x, 0.00105], N[(y * 5.0 + N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35e8 or 0.00104999999999999994 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -1.35e8 < x < 0.00104999999999999994

   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

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. flip-+ N/A

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

      15. +-inverses N/A

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

      16. +-inverses N/A

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. flip-+ N/A

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

      20. count-2 N/A

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

      21. lower-fma.f64 97.7

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

   4. Applied rewrites 97.7%

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

Alternative 4: 45.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-81}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+91}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.4e+18)
   (* t x)
   (if (<= t -1e-81) (* 5.0 y) (if (<= t 3e+91) (* (* z x) 2.0) (* t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.4e+18) {
		tmp = t * x;
	} else if (t <= -1e-81) {
		tmp = 5.0 * y;
	} else if (t <= 3e+91) {
		tmp = (z * x) * 2.0;
	} else {
		tmp = t * x;
	}
	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 <= (-5.4d+18)) then
        tmp = t * x
    else if (t <= (-1d-81)) then
        tmp = 5.0d0 * y
    else if (t <= 3d+91) then
        tmp = (z * x) * 2.0d0
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.4e+18) {
		tmp = t * x;
	} else if (t <= -1e-81) {
		tmp = 5.0 * y;
	} else if (t <= 3e+91) {
		tmp = (z * x) * 2.0;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5.4e+18:
		tmp = t * x
	elif t <= -1e-81:
		tmp = 5.0 * y
	elif t <= 3e+91:
		tmp = (z * x) * 2.0
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5.4e+18)
		tmp = Float64(t * x);
	elseif (t <= -1e-81)
		tmp = Float64(5.0 * y);
	elseif (t <= 3e+91)
		tmp = Float64(Float64(z * x) * 2.0);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -5.4e+18)
		tmp = t * x;
	elseif (t <= -1e-81)
		tmp = 5.0 * y;
	elseif (t <= 3e+91)
		tmp = (z * x) * 2.0;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5.4e+18], N[(t * x), $MachinePrecision], If[LessEqual[t, -1e-81], N[(5.0 * y), $MachinePrecision], If[LessEqual[t, 3e+91], N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision], N[(t * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.4e18 or 3.00000000000000006e91 < 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. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   if -5.4e18 < t < -9.9999999999999996e-82

   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. *-commutative N/A

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

      2. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

   if -9.9999999999999996e-82 < t < 3.00000000000000006e91

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 43.0

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

   5. Applied rewrites 43.0%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-81}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+91}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z + y, 2, t\right) \cdot x\\ \mathbf{if}\;x \leq -0.56:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma (+ z y) 2.0 t) x)))
   (if (<= x -0.56) t_1 (if (<= x 4e-31) (fma y 5.0 (* t x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z + y), 2.0, t) * x;
	double tmp;
	if (x <= -0.56) {
		tmp = t_1;
	} else if (x <= 4e-31) {
		tmp = fma(y, 5.0, (t * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(z + y), 2.0, t) * x)
	tmp = 0.0
	if (x <= -0.56)
		tmp = t_1;
	elseif (x <= 4e-31)
		tmp = fma(y, 5.0, Float64(t * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z + y), $MachinePrecision] * 2.0 + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -0.56], t$95$1, If[LessEqual[x, 4e-31], N[(y * 5.0 + N[(t * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.56000000000000005 or 4e-31 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -0.56000000000000005 < x < 4e-31

   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

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. count-2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. count-2 N/A

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

      9. associate-+l+ N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. associate-+l+ N/A

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

      14. +-commutative N/A

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

      15. count-2 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in t around inf

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

      1. lower-*.f64 76.1

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

   9. Applied rewrites 76.1%

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

Alternative 6: 77.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(z, 2, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 x 5.0) y)))
   (if (<= y -2.8e+76) t_1 (if (<= y 6.5e-35) (* (fma z 2.0 t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, x, 5.0) * y;
	double tmp;
	if (y <= -2.8e+76) {
		tmp = t_1;
	} else if (y <= 6.5e-35) {
		tmp = fma(z, 2.0, t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, x, 5.0) * y)
	tmp = 0.0
	if (y <= -2.8e+76)
		tmp = t_1;
	elseif (y <= 6.5e-35)
		tmp = Float64(fma(z, 2.0, t) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.8e+76], t$95$1, If[LessEqual[y, 6.5e-35], N[(N[(z * 2.0 + t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7999999999999999e76 or 6.4999999999999999e-35 < 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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-sub0 N/A

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

      6. associate--r- N/A

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

      7. neg-sub0 N/A

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

      8. lower-*.f64 N/A

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

      9. neg-sub0 N/A

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

      10. associate--r- N/A

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

      11. neg-sub0 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -2.7999999999999999e76 < y < 6.4999999999999999e-35

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.5

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

   5. Applied rewrites 81.5%

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

Alternative 7: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 2, t\right) \cdot x\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-40}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma y 2.0 t) x)))
   (if (<= x -2.3e-80) t_1 (if (<= x 7.5e-40) (* 5.0 y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, 2.0, t) * x;
	double tmp;
	if (x <= -2.3e-80) {
		tmp = t_1;
	} else if (x <= 7.5e-40) {
		tmp = 5.0 * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(y, 2.0, t) * x)
	tmp = 0.0
	if (x <= -2.3e-80)
		tmp = t_1;
	elseif (x <= 7.5e-40)
		tmp = Float64(5.0 * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * 2.0 + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.3e-80], t$95$1, If[LessEqual[x, 7.5e-40], N[(5.0 * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2999999999999998e-80 or 7.50000000000000069e-40 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 69.1%

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

   if -2.2999999999999998e-80 < x < 7.50000000000000069e-40

   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. *-commutative N/A

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

      2. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(y, 2, t\right) \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-40}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 2, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -200000000000:\\ \;\;\;\;\left(2 \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-40}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -200000000000.0)
   (* (* 2.0 y) x)
   (if (<= x 7.5e-40) (* 5.0 y) (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -200000000000.0) {
		tmp = (2.0 * y) * x;
	} else if (x <= 7.5e-40) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	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 <= (-200000000000.0d0)) then
        tmp = (2.0d0 * y) * x
    else if (x <= 7.5d-40) then
        tmp = 5.0d0 * y
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -200000000000.0) {
		tmp = (2.0 * y) * x;
	} else if (x <= 7.5e-40) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -200000000000.0:
		tmp = (2.0 * y) * x
	elif x <= 7.5e-40:
		tmp = 5.0 * y
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -200000000000.0)
		tmp = Float64(Float64(2.0 * y) * x);
	elseif (x <= 7.5e-40)
		tmp = Float64(5.0 * y);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -200000000000.0)
		tmp = (2.0 * y) * x;
	elseif (x <= 7.5e-40)
		tmp = 5.0 * y;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -200000000000.0], N[(N[(2.0 * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 7.5e-40], N[(5.0 * y), $MachinePrecision], N[(t * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e11

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 48.6%

      \[\leadsto \left(y \cdot 2\right) \cdot x \]

   if -2e11 < x < 7.50000000000000069e-40

   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. *-commutative N/A

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

      2. lower-*.f64 52.6

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

   5. Applied rewrites 52.6%

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

   if 7.50000000000000069e-40 < 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. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200000000000:\\ \;\;\;\;\left(2 \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-40}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-40}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.3e-80) (* t x) (if (<= x 7.5e-40) (* 5.0 y) (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.3e-80) {
		tmp = t * x;
	} else if (x <= 7.5e-40) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	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 <= (-2.3d-80)) then
        tmp = t * x
    else if (x <= 7.5d-40) then
        tmp = 5.0d0 * y
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.3e-80) {
		tmp = t * x;
	} else if (x <= 7.5e-40) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.3e-80:
		tmp = t * x
	elif x <= 7.5e-40:
		tmp = 5.0 * y
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.3e-80)
		tmp = Float64(t * x);
	elseif (x <= 7.5e-40)
		tmp = Float64(5.0 * y);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.3e-80)
		tmp = t * x;
	elseif (x <= 7.5e-40)
		tmp = 5.0 * y;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.3e-80], N[(t * x), $MachinePrecision], If[LessEqual[x, 7.5e-40], N[(5.0 * y), $MachinePrecision], N[(t * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2999999999999998e-80 or 7.50000000000000069e-40 < 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. lower-*.f64 37.5

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

   5. Applied rewrites 37.5%

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

   if -2.2999999999999998e-80 < x < 7.50000000000000069e-40

   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. *-commutative N/A

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

      2. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-80}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-40}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ t \cdot x \end{array} \]

FPCore

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

C

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

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 = t * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(t * x), $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 t around inf

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

   1. lower-*.f64 31.8

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

5. Applied rewrites 31.8%

   \[\leadsto \color{blue}{t \cdot x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(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)))