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

Percentage Accurate: 99.9% → 100.0%

Time: 9.0s

Alternatives: 14

Speedup: 1.2×

• 26.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% 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

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 100.0

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

   8. lift-+.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. associate-+r+ N/A

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

   11. lift-+.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. associate-+r+ N/A

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

   14. lift-+.f64 N/A

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

   15. associate-+l+ N/A

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

   16. +-commutative N/A

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

   17. lift-+.f64 N/A

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

   18. count-2 N/A

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

   19. lift-fma.f64 100.0

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

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: 46.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(2 \cdot z\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+262}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \left(y \cdot 2\right)\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 10^{-12}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+105}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (* 2.0 z))))
   (if (<= x -3.4e+262)
     (* t x)
     (if (<= x -3.5e+31)
       (* x (* y 2.0))
       (if (<= x -4.8e-67)
         t_1
         (if (<= x 1e-12) (* y 5.0) (if (<= x 1.9e+105) (* t x) t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (2.0 * z);
	double tmp;
	if (x <= -3.4e+262) {
		tmp = t * x;
	} else if (x <= -3.5e+31) {
		tmp = x * (y * 2.0);
	} else if (x <= -4.8e-67) {
		tmp = t_1;
	} else if (x <= 1e-12) {
		tmp = y * 5.0;
	} else if (x <= 1.9e+105) {
		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 = x * (2.0d0 * z)
    if (x <= (-3.4d+262)) then
        tmp = t * x
    else if (x <= (-3.5d+31)) then
        tmp = x * (y * 2.0d0)
    else if (x <= (-4.8d-67)) then
        tmp = t_1
    else if (x <= 1d-12) then
        tmp = y * 5.0d0
    else if (x <= 1.9d+105) 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 = x * (2.0 * z);
	double tmp;
	if (x <= -3.4e+262) {
		tmp = t * x;
	} else if (x <= -3.5e+31) {
		tmp = x * (y * 2.0);
	} else if (x <= -4.8e-67) {
		tmp = t_1;
	} else if (x <= 1e-12) {
		tmp = y * 5.0;
	} else if (x <= 1.9e+105) {
		tmp = t * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (2.0 * z)
	tmp = 0
	if x <= -3.4e+262:
		tmp = t * x
	elif x <= -3.5e+31:
		tmp = x * (y * 2.0)
	elif x <= -4.8e-67:
		tmp = t_1
	elif x <= 1e-12:
		tmp = y * 5.0
	elif x <= 1.9e+105:
		tmp = t * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(2.0 * z))
	tmp = 0.0
	if (x <= -3.4e+262)
		tmp = Float64(t * x);
	elseif (x <= -3.5e+31)
		tmp = Float64(x * Float64(y * 2.0));
	elseif (x <= -4.8e-67)
		tmp = t_1;
	elseif (x <= 1e-12)
		tmp = Float64(y * 5.0);
	elseif (x <= 1.9e+105)
		tmp = Float64(t * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (2.0 * z);
	tmp = 0.0;
	if (x <= -3.4e+262)
		tmp = t * x;
	elseif (x <= -3.5e+31)
		tmp = x * (y * 2.0);
	elseif (x <= -4.8e-67)
		tmp = t_1;
	elseif (x <= 1e-12)
		tmp = y * 5.0;
	elseif (x <= 1.9e+105)
		tmp = t * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e+262], N[(t * x), $MachinePrecision], If[LessEqual[x, -3.5e+31], N[(x * N[(y * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.8e-67], t$95$1, If[LessEqual[x, 1e-12], N[(y * 5.0), $MachinePrecision], If[LessEqual[x, 1.9e+105], N[(t * x), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.4000000000000002e262 or 9.9999999999999998e-13 < x < 1.9e105

   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. *-commutative N/A

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

      2. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   if -3.4000000000000002e262 < x < -3.5e31

   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 inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 50.4%

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

   if -3.5e31 < x < -4.8e-67 or 1.9e105 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 47.6

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

   5. Applied rewrites 47.6%

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

   if -4.8e-67 < x < 9.9999999999999998e-13

   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 67.3

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

   5. Applied rewrites 67.3%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+262}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \left(y \cdot 2\right)\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \left(2 \cdot z\right)\\ \mathbf{elif}\;x \leq 10^{-12}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+105}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot 2\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+262}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq -680000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{-24}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 10^{-12}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+99}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (* y 2.0))))
   (if (<= x -3.4e+262)
     (* t x)
     (if (<= x -680000000000.0)
       t_1
       (if (<= x -2.55e-24)
         (* t x)
         (if (<= x 1e-12) (* y 5.0) (if (<= x 1.02e+99) (* t x) t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (y * 2.0);
	double tmp;
	if (x <= -3.4e+262) {
		tmp = t * x;
	} else if (x <= -680000000000.0) {
		tmp = t_1;
	} else if (x <= -2.55e-24) {
		tmp = t * x;
	} else if (x <= 1e-12) {
		tmp = y * 5.0;
	} else if (x <= 1.02e+99) {
		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 = x * (y * 2.0d0)
    if (x <= (-3.4d+262)) then
        tmp = t * x
    else if (x <= (-680000000000.0d0)) then
        tmp = t_1
    else if (x <= (-2.55d-24)) then
        tmp = t * x
    else if (x <= 1d-12) then
        tmp = y * 5.0d0
    else if (x <= 1.02d+99) 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 = x * (y * 2.0);
	double tmp;
	if (x <= -3.4e+262) {
		tmp = t * x;
	} else if (x <= -680000000000.0) {
		tmp = t_1;
	} else if (x <= -2.55e-24) {
		tmp = t * x;
	} else if (x <= 1e-12) {
		tmp = y * 5.0;
	} else if (x <= 1.02e+99) {
		tmp = t * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (y * 2.0)
	tmp = 0
	if x <= -3.4e+262:
		tmp = t * x
	elif x <= -680000000000.0:
		tmp = t_1
	elif x <= -2.55e-24:
		tmp = t * x
	elif x <= 1e-12:
		tmp = y * 5.0
	elif x <= 1.02e+99:
		tmp = t * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(y * 2.0))
	tmp = 0.0
	if (x <= -3.4e+262)
		tmp = Float64(t * x);
	elseif (x <= -680000000000.0)
		tmp = t_1;
	elseif (x <= -2.55e-24)
		tmp = Float64(t * x);
	elseif (x <= 1e-12)
		tmp = Float64(y * 5.0);
	elseif (x <= 1.02e+99)
		tmp = Float64(t * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (y * 2.0);
	tmp = 0.0;
	if (x <= -3.4e+262)
		tmp = t * x;
	elseif (x <= -680000000000.0)
		tmp = t_1;
	elseif (x <= -2.55e-24)
		tmp = t * x;
	elseif (x <= 1e-12)
		tmp = y * 5.0;
	elseif (x <= 1.02e+99)
		tmp = t * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e+262], N[(t * x), $MachinePrecision], If[LessEqual[x, -680000000000.0], t$95$1, If[LessEqual[x, -2.55e-24], N[(t * x), $MachinePrecision], If[LessEqual[x, 1e-12], N[(y * 5.0), $MachinePrecision], If[LessEqual[x, 1.02e+99], N[(t * x), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4000000000000002e262 or -6.8e11 < x < -2.55000000000000013e-24 or 9.9999999999999998e-13 < x < 1.01999999999999998e99

   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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if -3.4000000000000002e262 < x < -6.8e11 or 1.01999999999999998e99 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 47.0%

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

   if -2.55000000000000013e-24 < x < 9.9999999999999998e-13

   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 64.8

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

   5. Applied rewrites 64.8%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+262}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq -680000000000:\\ \;\;\;\;x \cdot \left(y \cdot 2\right)\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{-24}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 10^{-12}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+99}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, y + z, t\right) \cdot x\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(z + z\right)\right)\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(t, x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 (+ y z) t) x)))
   (if (<= x -2.5e-24)
     t_1
     (if (<= x 8.4e-156)
       (fma y 5.0 (* x (+ z z)))
       (if (<= x 3.05e-21) (fma t x (* y 5.0)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, (y + z), t) * x;
	double tmp;
	if (x <= -2.5e-24) {
		tmp = t_1;
	} else if (x <= 8.4e-156) {
		tmp = fma(y, 5.0, (x * (z + z)));
	} else if (x <= 3.05e-21) {
		tmp = fma(t, x, (y * 5.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, Float64(y + z), t) * x)
	tmp = 0.0
	if (x <= -2.5e-24)
		tmp = t_1;
	elseif (x <= 8.4e-156)
		tmp = fma(y, 5.0, Float64(x * Float64(z + z)));
	elseif (x <= 3.05e-21)
		tmp = fma(t, x, Float64(y * 5.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.5e-24], t$95$1, If[LessEqual[x, 8.4e-156], N[(y * 5.0 + N[(x * N[(z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.05e-21], N[(t * x + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4999999999999999e-24 or 3.05000000000000007e-21 < 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 x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -2.4999999999999999e-24 < x < 8.40000000000000049e-156

   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. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. flip-+ N/A

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

      15. +-inverses N/A

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

      16. +-inverses N/A

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. flip-+ N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   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. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 86.5

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

   7. Applied rewrites 86.5%

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

   9. Applied rewrites 86.5%

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

   if 8.40000000000000049e-156 < x < 3.05000000000000007e-21

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 100.0

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. flip-+ N/A

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

      15. +-inverses N/A

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

      16. +-inverses N/A

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. flip-+ N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 94.0

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

   7. Applied rewrites 94.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(2, y + z, t\right) \cdot x\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(z + z\right)\right)\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(t, x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y + z, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(z + \left(z + t\right)\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ z (+ z t)))))
   (if (<= x -9e+30)
     (* x (fma 2.0 y t))
     (if (<= x -4.8e-67) t_1 (if (<= x 3.05e-21) (fma y 5.0 (* t x)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (z + (z + t));
	double tmp;
	if (x <= -9e+30) {
		tmp = x * fma(2.0, y, t);
	} else if (x <= -4.8e-67) {
		tmp = t_1;
	} else if (x <= 3.05e-21) {
		tmp = fma(y, 5.0, (t * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(z + Float64(z + t)))
	tmp = 0.0
	if (x <= -9e+30)
		tmp = Float64(x * fma(2.0, y, t));
	elseif (x <= -4.8e-67)
		tmp = t_1;
	elseif (x <= 3.05e-21)
		tmp = fma(y, 5.0, Float64(t * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(z + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+30], N[(x * N[(2.0 * y + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.8e-67], t$95$1, If[LessEqual[x, 3.05e-21], N[(y * 5.0 + N[(t * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.9999999999999999e30

   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 inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.0%

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

   if -8.9999999999999999e30 < x < -4.8e-67 or 3.05000000000000007e-21 < 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 y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 74.8

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

   5. Applied rewrites 74.8%

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

   7. Applied rewrites 74.8%

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

   if -4.8e-67 < x < 3.05000000000000007e-21

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lift-+.f64 N/A

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

      15. associate-+l+ N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. count-2 N/A

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

      19. lift-fma.f64 100.0

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

   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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.7

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

   9. Applied rewrites 86.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(z + \left(z + t\right)\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(t, x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ z (+ z t)))))
   (if (<= x -9e+30)
     (* x (fma 2.0 y t))
     (if (<= x -4.8e-67) t_1 (if (<= x 3.05e-21) (fma t x (* y 5.0)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (z + (z + t));
	double tmp;
	if (x <= -9e+30) {
		tmp = x * fma(2.0, y, t);
	} else if (x <= -4.8e-67) {
		tmp = t_1;
	} else if (x <= 3.05e-21) {
		tmp = fma(t, x, (y * 5.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(z + Float64(z + t)))
	tmp = 0.0
	if (x <= -9e+30)
		tmp = Float64(x * fma(2.0, y, t));
	elseif (x <= -4.8e-67)
		tmp = t_1;
	elseif (x <= 3.05e-21)
		tmp = fma(t, x, Float64(y * 5.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(z + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+30], N[(x * N[(2.0 * y + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.8e-67], t$95$1, If[LessEqual[x, 3.05e-21], N[(t * x + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.9999999999999999e30

   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 inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.0%

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

   if -8.9999999999999999e30 < x < -4.8e-67 or 3.05000000000000007e-21 < 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 y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 74.8

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

   5. Applied rewrites 74.8%

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

   7. Applied rewrites 74.8%

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

   if -4.8e-67 < x < 3.05000000000000007e-21

   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. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. flip-+ N/A

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

      15. +-inverses N/A

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

      16. +-inverses N/A

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. flip-+ N/A

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

      20. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 86.7

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

   7. Applied rewrites 86.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(t, x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{elif}\;x \leq -7.3 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-62}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -9e+30)
   (* x (fma 2.0 y t))
   (if (<= x -7.3e-180)
     (* x (fma 2.0 z t))
     (if (<= x 2.4e-62) (* y 5.0) (* x (+ z (+ z t)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -9e+30) {
		tmp = x * fma(2.0, y, t);
	} else if (x <= -7.3e-180) {
		tmp = x * fma(2.0, z, t);
	} else if (x <= 2.4e-62) {
		tmp = y * 5.0;
	} else {
		tmp = x * (z + (z + t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -9e+30)
		tmp = Float64(x * fma(2.0, y, t));
	elseif (x <= -7.3e-180)
		tmp = Float64(x * fma(2.0, z, t));
	elseif (x <= 2.4e-62)
		tmp = Float64(y * 5.0);
	else
		tmp = Float64(x * Float64(z + Float64(z + t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -9e+30], N[(x * N[(2.0 * y + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.3e-180], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-62], N[(y * 5.0), $MachinePrecision], N[(x * N[(z + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.9999999999999999e30

   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 inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.0%

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

   if -8.9999999999999999e30 < x < -7.2999999999999996e-180

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if -7.2999999999999996e-180 < x < 2.39999999999999984e-62

   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 79.0

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

   5. Applied rewrites 79.0%

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

   if 2.39999999999999984e-62 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 72.9

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

   5. Applied rewrites 72.9%

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

   7. Applied rewrites 72.9%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{elif}\;x \leq -7.3 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-62}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(z + \left(z + t\right)\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{elif}\;x \leq -7.3 \cdot 10^{-180}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-62}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ z (+ z t)))))
   (if (<= x -9e+30)
     (* x (fma 2.0 y t))
     (if (<= x -7.3e-180) t_1 (if (<= x 2.4e-62) (* y 5.0) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (z + (z + t));
	double tmp;
	if (x <= -9e+30) {
		tmp = x * fma(2.0, y, t);
	} else if (x <= -7.3e-180) {
		tmp = t_1;
	} else if (x <= 2.4e-62) {
		tmp = y * 5.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(z + Float64(z + t)))
	tmp = 0.0
	if (x <= -9e+30)
		tmp = Float64(x * fma(2.0, y, t));
	elseif (x <= -7.3e-180)
		tmp = t_1;
	elseif (x <= 2.4e-62)
		tmp = Float64(y * 5.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(z + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+30], N[(x * N[(2.0 * y + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.3e-180], t$95$1, If[LessEqual[x, 2.4e-62], N[(y * 5.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.9999999999999999e30

   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 inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.0%

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

   if -8.9999999999999999e30 < x < -7.2999999999999996e-180 or 2.39999999999999984e-62 < 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 y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 70.0

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

   5. Applied rewrites 70.0%

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

   7. Applied rewrites 70.0%

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

   if -7.2999999999999996e-180 < x < 2.39999999999999984e-62

   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 79.0

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

   5. Applied rewrites 79.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{elif}\;x \leq -7.3 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-62}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, y + z, t\right) \cdot x\\ \mathbf{if}\;x \leq -80:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 (+ y z) t) x)))
   (if (<= x -80.0)
     t_1
     (if (<= x 6.2e-10) (fma y 5.0 (* x (+ t (+ z z)))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, (y + z), t) * x;
	double tmp;
	if (x <= -80.0) {
		tmp = t_1;
	} else if (x <= 6.2e-10) {
		tmp = fma(y, 5.0, (x * (t + (z + z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, Float64(y + z), t) * x)
	tmp = 0.0
	if (x <= -80.0)
		tmp = t_1;
	elseif (x <= 6.2e-10)
		tmp = fma(y, 5.0, Float64(x * Float64(t + Float64(z + z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -80.0], t$95$1, If[LessEqual[x, 6.2e-10], N[(y * 5.0 + N[(x * N[(t + N[(z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -80 or 6.2000000000000003e-10 < 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 x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -80 < x < 6.2000000000000003e-10

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lower-+.f64 99.4

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

   4. Applied rewrites 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -80:\\ \;\;\;\;\mathsf{fma}\left(2, y + z, t\right) \cdot x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y + z, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, y + z, t\right) \cdot x\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 (+ y z) t) x)))
   (if (<= x -4.8e-67) t_1 (if (<= x 3.05e-21) (fma y 5.0 (* t x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, (y + z), t) * x;
	double tmp;
	if (x <= -4.8e-67) {
		tmp = t_1;
	} else if (x <= 3.05e-21) {
		tmp = fma(y, 5.0, (t * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, Float64(y + z), t) * x)
	tmp = 0.0
	if (x <= -4.8e-67)
		tmp = t_1;
	elseif (x <= 3.05e-21)
		tmp = fma(y, 5.0, Float64(t * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4.8e-67], t$95$1, If[LessEqual[x, 3.05e-21], N[(y * 5.0 + N[(t * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8e-67 or 3.05000000000000007e-21 < 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 x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if -4.8e-67 < x < 3.05000000000000007e-21

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lift-+.f64 N/A

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

      15. associate-+l+ N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. count-2 N/A

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

      19. lift-fma.f64 100.0

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

   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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.7

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

   9. Applied rewrites 86.7%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(2, y + z, t\right) \cdot x\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y + z, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -2.55 \cdot 10^{-68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -2.55e-68) t_1 (if (<= y 1.4e+88) (* x (fma 2.0 z t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * fma(x, 2.0, 5.0);
	double tmp;
	if (y <= -2.55e-68) {
		tmp = t_1;
	} else if (y <= 1.4e+88) {
		tmp = x * fma(2.0, z, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * fma(x, 2.0, 5.0))
	tmp = 0.0
	if (y <= -2.55e-68)
		tmp = t_1;
	elseif (y <= 1.4e+88)
		tmp = Float64(x * fma(2.0, z, t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x * 2.0 + 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.55e-68], t$95$1, If[LessEqual[y, 1.4e+88], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.54999999999999983e-68 or 1.39999999999999994e88 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. neg-sub0 N/A

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

      7. lower-*.f64 N/A

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

      8. neg-sub0 N/A

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

      9. associate--r- N/A

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

      10. neg-sub0 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 76.5

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

   5. Applied rewrites 76.5%

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

   if -2.54999999999999983e-68 < y < 1.39999999999999994e88

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 79.6

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

   5. Applied rewrites 79.6%

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

Alternative 12: 63.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{if}\;x \leq -2.55 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 10^{-12}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 y t))))
   (if (<= x -2.55e-24) t_1 (if (<= x 1e-12) (* y 5.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, y, t);
	double tmp;
	if (x <= -2.55e-24) {
		tmp = t_1;
	} else if (x <= 1e-12) {
		tmp = y * 5.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, y, t))
	tmp = 0.0
	if (x <= -2.55e-24)
		tmp = t_1;
	elseif (x <= 1e-12)
		tmp = Float64(y * 5.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * y + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.55e-24], t$95$1, If[LessEqual[x, 1e-12], N[(y * 5.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.55000000000000013e-24 or 9.9999999999999998e-13 < 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 x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 70.6%

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

   if -2.55000000000000013e-24 < x < 9.9999999999999998e-13

   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 64.8

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

   5. Applied rewrites 64.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-24}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{elif}\;x \leq 10^{-12}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-24}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 10^{-12}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.55e-24) (* t x) (if (<= x 1e-12) (* y 5.0) (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.55e-24) {
		tmp = t * x;
	} else if (x <= 1e-12) {
		tmp = y * 5.0;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-2.55d-24)) then
        tmp = t * x
    else if (x <= 1d-12) then
        tmp = y * 5.0d0
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.55e-24) {
		tmp = t * x;
	} else if (x <= 1e-12) {
		tmp = y * 5.0;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.55e-24:
		tmp = t * x
	elif x <= 1e-12:
		tmp = y * 5.0
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.55e-24)
		tmp = Float64(t * x);
	elseif (x <= 1e-12)
		tmp = Float64(y * 5.0);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.55e-24)
		tmp = t * x;
	elseif (x <= 1e-12)
		tmp = y * 5.0;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.55e-24], N[(t * x), $MachinePrecision], If[LessEqual[x, 1e-12], N[(y * 5.0), $MachinePrecision], N[(t * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.55000000000000013e-24 or 9.9999999999999998e-13 < 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. *-commutative N/A

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

      2. lower-*.f64 37.9

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

   5. Applied rewrites 37.9%

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

   if -2.55000000000000013e-24 < x < 9.9999999999999998e-13

   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 64.8

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

   5. Applied rewrites 64.8%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-24}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 10^{-12}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 31.2

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

5. Applied rewrites 31.2%

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

6. Final simplification 31.2%

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

Reproduce

herbie shell --seed 2024233 
(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)))