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

Percentage Accurate: 99.9% → 99.9%

Time: 6.6s

Alternatives: 11

Speedup: 1.2×

• 26.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   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. associate-+l+ N/A

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

   5. distribute-rgt-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. associate-+l+ N/A

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

   10. +-commutative N/A

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

   11. count-2 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 97.2

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

4. Applied rewrites 97.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 97.2

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. count-2 N/A

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

   8. associate-+l+ N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-rgt-in N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. associate-+l+ N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. associate-+l+ N/A

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

   18. +-commutative N/A

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

   19. count-2 N/A

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

   20. lower-fma.f64 N/A

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

   21. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 48.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot 2\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+230}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{+128}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{elif}\;x \leq -4.25 \cdot 10^{+43}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-86}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* z x) 2.0)))
   (if (<= x -1.3e+230)
     t_1
     (if (<= x -2.5e+128)
       (* (* x y) 2.0)
       (if (<= x -4.25e+43)
         (* t x)
         (if (<= x -3.4e-48) t_1 (if (<= x 2.9e-86) (* 5.0 y) (* t x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * x) * 2.0;
	double tmp;
	if (x <= -1.3e+230) {
		tmp = t_1;
	} else if (x <= -2.5e+128) {
		tmp = (x * y) * 2.0;
	} else if (x <= -4.25e+43) {
		tmp = t * x;
	} else if (x <= -3.4e-48) {
		tmp = t_1;
	} else if (x <= 2.9e-86) {
		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) :: t_1
    real(8) :: tmp
    t_1 = (z * x) * 2.0d0
    if (x <= (-1.3d+230)) then
        tmp = t_1
    else if (x <= (-2.5d+128)) then
        tmp = (x * y) * 2.0d0
    else if (x <= (-4.25d+43)) then
        tmp = t * x
    else if (x <= (-3.4d-48)) then
        tmp = t_1
    else if (x <= 2.9d-86) 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 t_1 = (z * x) * 2.0;
	double tmp;
	if (x <= -1.3e+230) {
		tmp = t_1;
	} else if (x <= -2.5e+128) {
		tmp = (x * y) * 2.0;
	} else if (x <= -4.25e+43) {
		tmp = t * x;
	} else if (x <= -3.4e-48) {
		tmp = t_1;
	} else if (x <= 2.9e-86) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * x) * 2.0
	tmp = 0
	if x <= -1.3e+230:
		tmp = t_1
	elif x <= -2.5e+128:
		tmp = (x * y) * 2.0
	elif x <= -4.25e+43:
		tmp = t * x
	elif x <= -3.4e-48:
		tmp = t_1
	elif x <= 2.9e-86:
		tmp = 5.0 * y
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * x) * 2.0)
	tmp = 0.0
	if (x <= -1.3e+230)
		tmp = t_1;
	elseif (x <= -2.5e+128)
		tmp = Float64(Float64(x * y) * 2.0);
	elseif (x <= -4.25e+43)
		tmp = Float64(t * x);
	elseif (x <= -3.4e-48)
		tmp = t_1;
	elseif (x <= 2.9e-86)
		tmp = Float64(5.0 * y);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * x) * 2.0;
	tmp = 0.0;
	if (x <= -1.3e+230)
		tmp = t_1;
	elseif (x <= -2.5e+128)
		tmp = (x * y) * 2.0;
	elseif (x <= -4.25e+43)
		tmp = t * x;
	elseif (x <= -3.4e-48)
		tmp = t_1;
	elseif (x <= 2.9e-86)
		tmp = 5.0 * y;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[x, -1.3e+230], t$95$1, If[LessEqual[x, -2.5e+128], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[x, -4.25e+43], N[(t * x), $MachinePrecision], If[LessEqual[x, -3.4e-48], t$95$1, If[LessEqual[x, 2.9e-86], N[(5.0 * y), $MachinePrecision], N[(t * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.2999999999999999e230 or -4.25e43 < x < -3.40000000000000028e-48

   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 50.6

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

   5. Applied rewrites 50.6%

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

   if -1.2999999999999999e230 < x < -2.5e128

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      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. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. count-2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 72.7

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

   7. Applied rewrites 72.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 72.7%

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

   if -2.5e128 < x < -4.25e43 or 2.8999999999999999e-86 < x

   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

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

      1. lower-*.f64 43.4

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

   5. Applied rewrites 43.4%

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

   if -3.40000000000000028e-48 < x < 2.8999999999999999e-86

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 66.3

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

   5. Applied rewrites 66.3%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+230}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{+128}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{elif}\;x \leq -4.25 \cdot 10^{+43}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-48}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-86}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\mathsf{fma}\left(5, y, t \cdot x\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 (+ z y) t) x)))
   (if (<= x -4.4e-46)
     t_1
     (if (<= x -2e-294)
       (fma 5.0 y (* t x))
       (if (<= x 2.9e-86) (fma y 5.0 (* (* 2.0 z) x)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, (z + y), t) * x;
	double tmp;
	if (x <= -4.4e-46) {
		tmp = t_1;
	} else if (x <= -2e-294) {
		tmp = fma(5.0, y, (t * x));
	} else if (x <= 2.9e-86) {
		tmp = fma(y, 5.0, ((2.0 * z) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, Float64(z + y), t) * x)
	tmp = 0.0
	if (x <= -4.4e-46)
		tmp = t_1;
	elseif (x <= -2e-294)
		tmp = fma(5.0, y, Float64(t * x));
	elseif (x <= 2.9e-86)
		tmp = fma(y, 5.0, Float64(Float64(2.0 * z) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4.4e-46], t$95$1, If[LessEqual[x, -2e-294], N[(5.0 * y + N[(t * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e-86], N[(y * 5.0 + N[(N[(2.0 * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4000000000000002e-46 or 2.8999999999999999e-86 < x

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      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. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. count-2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 36.4

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \left(t + y\right) + -1 \cdot \left(y + 2 \cdot z\right)\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. associate-+r+ N/A

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

      9. associate-+l+ N/A

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

      10. count-2 N/A

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

      11. associate-+r+ N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 95.6

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

   10. Applied rewrites 95.6%

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

   if -4.4000000000000002e-46 < x < -2.00000000000000003e-294

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 85.0

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

   7. Applied rewrites 85.0%

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

   if -2.00000000000000003e-294 < x < 2.8999999999999999e-86

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      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. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. count-2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. count-2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-in N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. associate-+l+ N/A

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

      18. +-commutative N/A

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

      19. count-2 N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 92.3

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

   9. Applied rewrites 92.3%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\mathsf{fma}\left(5, y, t \cdot x\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\mathsf{fma}\left(5, y, t \cdot x\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot x, z, 5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 (+ z y) t) x)))
   (if (<= x -4.4e-46)
     t_1
     (if (<= x -2e-294)
       (fma 5.0 y (* t x))
       (if (<= x 2.9e-86) (fma (* 2.0 x) z (* 5.0 y)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, (z + y), t) * x;
	double tmp;
	if (x <= -4.4e-46) {
		tmp = t_1;
	} else if (x <= -2e-294) {
		tmp = fma(5.0, y, (t * x));
	} else if (x <= 2.9e-86) {
		tmp = fma((2.0 * x), z, (5.0 * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, Float64(z + y), t) * x)
	tmp = 0.0
	if (x <= -4.4e-46)
		tmp = t_1;
	elseif (x <= -2e-294)
		tmp = fma(5.0, y, Float64(t * x));
	elseif (x <= 2.9e-86)
		tmp = fma(Float64(2.0 * x), z, Float64(5.0 * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4.4e-46], t$95$1, If[LessEqual[x, -2e-294], N[(5.0 * y + N[(t * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e-86], N[(N[(2.0 * x), $MachinePrecision] * z + N[(5.0 * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4000000000000002e-46 or 2.8999999999999999e-86 < x

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      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. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. count-2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 36.4

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \left(t + y\right) + -1 \cdot \left(y + 2 \cdot z\right)\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. associate-+r+ N/A

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

      9. associate-+l+ N/A

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

      10. count-2 N/A

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

      11. associate-+r+ N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 95.6

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

   10. Applied rewrites 95.6%

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

   if -4.4000000000000002e-46 < x < -2.00000000000000003e-294

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 85.0

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

   7. Applied rewrites 85.0%

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

   if -2.00000000000000003e-294 < x < 2.8999999999999999e-86

   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 92.3

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

   7. Applied rewrites 92.3%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\mathsf{fma}\left(5, y, t \cdot x\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot x, z, 5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -210000 \lor \neg \left(x \leq 0.08\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -210000.0) (not (<= x 0.08)))
   (* (fma 2.0 (+ z y) t) x)
   (fma y 5.0 (* (fma 2.0 z t) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -210000.0) || !(x <= 0.08)) {
		tmp = fma(2.0, (z + y), t) * x;
	} else {
		tmp = fma(y, 5.0, (fma(2.0, z, t) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -210000.0) || !(x <= 0.08))
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	else
		tmp = fma(y, 5.0, Float64(fma(2.0, z, t) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -210000.0], N[Not[LessEqual[x, 0.08]], $MachinePrecision]], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision], N[(y * 5.0 + N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e5 or 0.0800000000000000017 < x

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      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. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. count-2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 38.0

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

   7. Applied rewrites 38.0%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \left(t + y\right) + -1 \cdot \left(y + 2 \cdot z\right)\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. associate-+r+ N/A

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

      9. associate-+l+ N/A

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

      10. count-2 N/A

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

      11. associate-+r+ N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

   10. Applied rewrites 99.7%

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

   if -2.1e5 < x < 0.0800000000000000017

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -210000 \lor \neg \left(x \leq 0.08\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot 2\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{+66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-216}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+29}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* z x) 2.0)))
   (if (<= z -1.65e+66)
     t_1
     (if (<= z -4.8e-216) (* 5.0 y) (if (<= z 4e+29) (* t x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * x) * 2.0;
	double tmp;
	if (z <= -1.65e+66) {
		tmp = t_1;
	} else if (z <= -4.8e-216) {
		tmp = 5.0 * y;
	} else if (z <= 4e+29) {
		tmp = t * x;
	} 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 = (z * x) * 2.0d0
    if (z <= (-1.65d+66)) then
        tmp = t_1
    else if (z <= (-4.8d-216)) then
        tmp = 5.0d0 * y
    else if (z <= 4d+29) then
        tmp = t * x
    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 = (z * x) * 2.0;
	double tmp;
	if (z <= -1.65e+66) {
		tmp = t_1;
	} else if (z <= -4.8e-216) {
		tmp = 5.0 * y;
	} else if (z <= 4e+29) {
		tmp = t * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * x) * 2.0
	tmp = 0
	if z <= -1.65e+66:
		tmp = t_1
	elif z <= -4.8e-216:
		tmp = 5.0 * y
	elif z <= 4e+29:
		tmp = t * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * x) * 2.0)
	tmp = 0.0
	if (z <= -1.65e+66)
		tmp = t_1;
	elseif (z <= -4.8e-216)
		tmp = Float64(5.0 * y);
	elseif (z <= 4e+29)
		tmp = Float64(t * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * x) * 2.0;
	tmp = 0.0;
	if (z <= -1.65e+66)
		tmp = t_1;
	elseif (z <= -4.8e-216)
		tmp = 5.0 * y;
	elseif (z <= 4e+29)
		tmp = t * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[z, -1.65e+66], t$95$1, If[LessEqual[z, -4.8e-216], N[(5.0 * y), $MachinePrecision], If[LessEqual[z, 4e+29], N[(t * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6500000000000001e66 or 3.99999999999999966e29 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-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 59.5

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

   5. Applied rewrites 59.5%

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

   if -1.6500000000000001e66 < z < -4.80000000000000007e-216

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 38.0

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

   5. Applied rewrites 38.0%

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

   if -4.80000000000000007e-216 < z < 3.99999999999999966e29

   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

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

      1. lower-*.f64 49.4

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

   5. Applied rewrites 49.4%

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

Alternative 7: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-46} \lor \neg \left(x \leq 8.6 \cdot 10^{-86}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, y, t \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.4e-46) (not (<= x 8.6e-86)))
   (* (fma 2.0 (+ z y) t) x)
   (fma 5.0 y (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.4e-46) || !(x <= 8.6e-86)) {
		tmp = fma(2.0, (z + y), t) * x;
	} else {
		tmp = fma(5.0, y, (t * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.4e-46) || !(x <= 8.6e-86))
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	else
		tmp = fma(5.0, y, Float64(t * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.4e-46], N[Not[LessEqual[x, 8.6e-86]], $MachinePrecision]], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision], N[(5.0 * y + N[(t * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4000000000000002e-46 or 8.60000000000000026e-86 < x

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      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. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. count-2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 36.4

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \left(t + y\right) + -1 \cdot \left(y + 2 \cdot z\right)\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-lft-out N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. associate-+r+ N/A

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

      9. associate-+l+ N/A

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

      10. count-2 N/A

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

      11. associate-+r+ N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 95.6

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

   10. Applied rewrites 95.6%

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

   if -4.4000000000000002e-46 < x < 8.60000000000000026e-86

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 83.3

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

   7. Applied rewrites 83.3%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-46} \lor \neg \left(x \leq 8.6 \cdot 10^{-86}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, y, t \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+65} \lor \neg \left(y \leq 1.2 \cdot 10^{+32}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -8.5e+65) (not (<= y 1.2e+32)))
   (* (fma 2.0 x 5.0) y)
   (* (fma 2.0 z t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -8.5e+65) || !(y <= 1.2e+32)) {
		tmp = fma(2.0, x, 5.0) * y;
	} else {
		tmp = fma(2.0, z, t) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -8.5e+65) || !(y <= 1.2e+32))
		tmp = Float64(fma(2.0, x, 5.0) * y);
	else
		tmp = Float64(fma(2.0, z, t) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -8.5e+65], N[Not[LessEqual[y, 1.2e+32]], $MachinePrecision]], N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.50000000000000075e65 or 1.19999999999999996e32 < 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. *-commutative 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 86.8

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

   5. Applied rewrites 86.8%

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

   if -8.50000000000000075e65 < y < 1.19999999999999996e32

   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. *-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. lower-fma.f64 77.6

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

   5. Applied rewrites 77.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+65} \lor \neg \left(y \leq 1.2 \cdot 10^{+32}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-144} \lor \neg \left(y \leq 360\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -4e-144) (not (<= y 360.0))) (* (fma 2.0 x 5.0) y) (* t x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4e-144) || !(y <= 360.0)) {
		tmp = fma(2.0, x, 5.0) * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4e-144) || !(y <= 360.0))
		tmp = Float64(fma(2.0, x, 5.0) * y);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -4e-144], N[Not[LessEqual[y, 360.0]], $MachinePrecision]], N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision], N[(t * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9999999999999998e-144 or 360 < y

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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. *-commutative 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 68.8

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

   5. Applied rewrites 68.8%

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

   if -3.9999999999999998e-144 < y < 360

   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 54.8

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

   5. Applied rewrites 54.8%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-144} \lor \neg \left(y \leq 360\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-73} \lor \neg \left(x \leq 2.9 \cdot 10^{-86}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.5e-73) (not (<= x 2.9e-86))) (* t x) (* 5.0 y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.5e-73) || !(x <= 2.9e-86)) {
		tmp = t * x;
	} else {
		tmp = 5.0 * y;
	}
	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 <= (-4.5d-73)) .or. (.not. (x <= 2.9d-86))) then
        tmp = t * x
    else
        tmp = 5.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.5e-73) || !(x <= 2.9e-86)) {
		tmp = t * x;
	} else {
		tmp = 5.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.5e-73) or not (x <= 2.9e-86):
		tmp = t * x
	else:
		tmp = 5.0 * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.5e-73) || !(x <= 2.9e-86))
		tmp = Float64(t * x);
	else
		tmp = Float64(5.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4.5e-73) || ~((x <= 2.9e-86)))
		tmp = t * x;
	else
		tmp = 5.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.5e-73], N[Not[LessEqual[x, 2.9e-86]], $MachinePrecision]], N[(t * x), $MachinePrecision], N[(5.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5e-73 or 2.8999999999999999e-86 < x

   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

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

      1. lower-*.f64 37.9

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

   5. Applied rewrites 37.9%

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

   if -4.5e-73 < x < 2.8999999999999999e-86

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-73} \lor \neg \left(x \leq 2.9 \cdot 10^{-86}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.0% 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 29.2

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

5. Applied rewrites 29.2%

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

Reproduce

herbie shell --seed 2024318 
(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)))