Linear.V4:$cdot from linear-1.19.1.3, C

Percentage Accurate: 96.2% → 99.1%

Time: 11.3s

Alternatives: 14

Speedup: 1.3×

• 41.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((x * y) + (z * t)) + (a * b)) + (c * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (((x * y) + (z * t)) + (a * b)) + (c * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((x * y) + (z * t)) + (a * b)) + (c * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((x * y) + (z * t)) + (a * b)) + (c * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(x * y) + Float64(z * t)) + Float64(a * b)) + Float64(c * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((x * y) + (z * t)) + (a * b)) + (c * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision]

Initial Program: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((x * y) + (z * t)) + (a * b)) + (c * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (((x * y) + (z * t)) + (a * b)) + (c * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((x * y) + (z * t)) + (a * b)) + (c * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((x * y) + (z * t)) + (a * b)) + (c * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(x * y) + Float64(z * t)) + Float64(a * b)) + Float64(c * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((x * y) + (z * t)) + (a * b)) + (c * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(a, \frac{b}{c}, \mathsf{fma}\left(x, \frac{y}{c}, \mathsf{fma}\left(t, \frac{z}{c}, i\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i))))
   (if (<= t_1 INFINITY)
     t_1
     (* c (fma a (/ b c) (fma x (/ y c) (fma t (/ z c) i)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((x * y) + (z * t)) + (a * b)) + (c * i);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = c * fma(a, (b / c), fma(x, (y / c), fma(t, (z / c), i)));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(x * y) + Float64(z * t)) + Float64(a * b)) + Float64(c * i))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(c * fma(a, Float64(b / c), fma(x, Float64(y / c), fma(t, Float64(z / c), i))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(c * N[(a * N[(b / c), $MachinePrecision] + N[(x * N[(y / c), $MachinePrecision] + N[(t * N[(z / c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) (*.f64 c i)) < +inf.0

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   if +inf.0 < (+.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) (*.f64 c i))

   1. Initial program 0.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(i + \left(\frac{a \cdot b}{c} + \left(\frac{t \cdot z}{c} + \frac{x \cdot y}{c}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(\left(\frac{a \cdot b}{c} + \left(\frac{t \cdot z}{c} + \frac{x \cdot y}{c}\right)\right) + i\right)} \]

      3. associate-+l+ N/A

         \[\leadsto c \cdot \color{blue}{\left(\frac{a \cdot b}{c} + \left(\left(\frac{t \cdot z}{c} + \frac{x \cdot y}{c}\right) + i\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto c \cdot \left(\color{blue}{a \cdot \frac{b}{c}} + \left(\left(\frac{t \cdot z}{c} + \frac{x \cdot y}{c}\right) + i\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(a, \frac{b}{c}, \left(\frac{t \cdot z}{c} + \frac{x \cdot y}{c}\right) + i\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(a, \color{blue}{\frac{b}{c}}, \left(\frac{t \cdot z}{c} + \frac{x \cdot y}{c}\right) + i\right) \]

      7. +-commutative N/A

         \[\leadsto c \cdot \mathsf{fma}\left(a, \frac{b}{c}, \color{blue}{\left(\frac{x \cdot y}{c} + \frac{t \cdot z}{c}\right)} + i\right) \]

      8. associate-+l+ N/A

         \[\leadsto c \cdot \mathsf{fma}\left(a, \frac{b}{c}, \color{blue}{\frac{x \cdot y}{c} + \left(\frac{t \cdot z}{c} + i\right)}\right) \]

      9. associate-/l* N/A

         \[\leadsto c \cdot \mathsf{fma}\left(a, \frac{b}{c}, \color{blue}{x \cdot \frac{y}{c}} + \left(\frac{t \cdot z}{c} + i\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(a, \frac{b}{c}, \color{blue}{\mathsf{fma}\left(x, \frac{y}{c}, \frac{t \cdot z}{c} + i\right)}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(a, \frac{b}{c}, \mathsf{fma}\left(x, \color{blue}{\frac{y}{c}}, \frac{t \cdot z}{c} + i\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto c \cdot \mathsf{fma}\left(a, \frac{b}{c}, \mathsf{fma}\left(x, \frac{y}{c}, \color{blue}{t \cdot \frac{z}{c}} + i\right)\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(a, \frac{b}{c}, \mathsf{fma}\left(x, \frac{y}{c}, \color{blue}{\mathsf{fma}\left(t, \frac{z}{c}, i\right)}\right)\right) \]

      14. /-lowering-/.f64 88.9

          \[\leadsto c \cdot \mathsf{fma}\left(a, \frac{b}{c}, \mathsf{fma}\left(x, \frac{y}{c}, \mathsf{fma}\left(t, \color{blue}{\frac{z}{c}}, i\right)\right)\right) \]

   5. Simplified 88.9%

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

Alternative 2: 66.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;c \cdot i \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(c, i, z \cdot t\right)\\ \mathbf{elif}\;c \cdot i \leq -2 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c \cdot i\right)\\ \mathbf{elif}\;c \cdot i \leq -1.2 \cdot 10^{-295}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5e+140)
   (fma i c (* a b))
   (if (<= (* c i) -1e+81)
     (fma c i (* z t))
     (if (<= (* c i) -2e-18)
       (fma y x (* c i))
       (if (<= (* c i) -1.2e-295)
         (fma z t (* a b))
         (if (<= (* c i) 2e+98) (fma z t (* x y)) (fma z t (* c i))))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -5e+140) {
		tmp = fma(i, c, (a * b));
	} else if ((c * i) <= -1e+81) {
		tmp = fma(c, i, (z * t));
	} else if ((c * i) <= -2e-18) {
		tmp = fma(y, x, (c * i));
	} else if ((c * i) <= -1.2e-295) {
		tmp = fma(z, t, (a * b));
	} else if ((c * i) <= 2e+98) {
		tmp = fma(z, t, (x * y));
	} else {
		tmp = fma(z, t, (c * i));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -5e+140)
		tmp = fma(i, c, Float64(a * b));
	elseif (Float64(c * i) <= -1e+81)
		tmp = fma(c, i, Float64(z * t));
	elseif (Float64(c * i) <= -2e-18)
		tmp = fma(y, x, Float64(c * i));
	elseif (Float64(c * i) <= -1.2e-295)
		tmp = fma(z, t, Float64(a * b));
	elseif (Float64(c * i) <= 2e+98)
		tmp = fma(z, t, Float64(x * y));
	else
		tmp = fma(z, t, Float64(c * i));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -5e+140], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -1e+81], N[(c * i + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -2e-18], N[(y * x + N[(c * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -1.2e-295], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e+98], N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if (*.f64 c i) < -5.00000000000000008e140

   1. Initial program 86.4%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 88.9

         \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]

   5. Simplified 88.9%

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + a \cdot b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + a \cdot b \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, a \cdot b\right)} \]

      4. *-lowering-*.f64 91.2

         \[\leadsto \mathsf{fma}\left(i, c, \color{blue}{a \cdot b}\right) \]

   7. Applied egg-rr 91.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, a \cdot b\right)} \]

   if -5.00000000000000008e140 < (*.f64 c i) < -9.99999999999999921e80

   1. Initial program 86.6%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + t \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot i + t \cdot z\right) + a \cdot b} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{c \cdot i + \left(t \cdot z + a \cdot b\right)} \]

      3. +-commutative N/A

         \[\leadsto c \cdot i + \color{blue}{\left(a \cdot b + t \cdot z\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, a \cdot b + t \cdot z\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, i, \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)}\right) \]

      6. *-lowering-*.f64 93.2

         \[\leadsto \mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, \color{blue}{t \cdot z}\right)\right) \]

   5. Simplified 93.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, t \cdot z\right)\right)} \]

   6. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 80.1

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

   8. Simplified 80.1%

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

   if -9.99999999999999921e80 < (*.f64 c i) < -2.0000000000000001e-18

   1. Initial program 93.3%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 86.8

         \[\leadsto \color{blue}{x \cdot y} + c \cdot i \]

   5. Simplified 86.8%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + c \cdot i \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 86.8

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

   7. Applied egg-rr 86.8%

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

   if -2.0000000000000001e-18 < (*.f64 c i) < -1.1999999999999999e-295

   1. Initial program 97.7%

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

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + z \cdot t\right) + \left(a \cdot b + c \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot t + x \cdot y\right)} + \left(a \cdot b + c \cdot i\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{z \cdot t + \left(x \cdot y + \left(a \cdot b + c \cdot i\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(a, b, c \cdot i\right)}\right)\right) \]

      7. *-lowering-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, \color{blue}{c \cdot i}\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\right)} \]

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 85.3

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

   7. Simplified 85.3%

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

   if -1.1999999999999999e-295 < (*.f64 c i) < 2e98

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + z \cdot t\right) + \left(a \cdot b + c \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot t + x \cdot y\right)} + \left(a \cdot b + c \cdot i\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{z \cdot t + \left(x \cdot y + \left(a \cdot b + c \cdot i\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(a, b, c \cdot i\right)}\right)\right) \]

      7. *-lowering-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, \color{blue}{c \cdot i}\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 73.9

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

   7. Simplified 73.9%

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

   if 2e98 < (*.f64 c i)

   1. Initial program 86.2%

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

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + z \cdot t\right) + \left(a \cdot b + c \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot t + x \cdot y\right)} + \left(a \cdot b + c \cdot i\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{z \cdot t + \left(x \cdot y + \left(a \cdot b + c \cdot i\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(a, b, c \cdot i\right)}\right)\right) \]

      7. *-lowering-*.f64 93.1

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, \color{blue}{c \cdot i}\right)\right)\right) \]

   4. Applied egg-rr 93.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\right)} \]

   5. Taylor expanded in c around inf

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

      1. *-lowering-*.f64 78.0

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

   7. Simplified 78.0%

      \[\leadsto \mathsf{fma}\left(z, t, \color{blue}{c \cdot i}\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;c \cdot i \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(c, i, z \cdot t\right)\\ \mathbf{elif}\;c \cdot i \leq -2 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c \cdot i\right)\\ \mathbf{elif}\;c \cdot i \leq -1.2 \cdot 10^{-295}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{if}\;c \cdot i \leq -2 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(a, b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma c i (fma a b (* z t)))))
   (if (<= (* c i) -2e+63)
     t_1
     (if (<= (* c i) 2e+98) (fma z t (fma a b (* x y))) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(c, i, fma(a, b, (z * t)));
	double tmp;
	if ((c * i) <= -2e+63) {
		tmp = t_1;
	} else if ((c * i) <= 2e+98) {
		tmp = fma(z, t, fma(a, b, (x * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = fma(c, i, fma(a, b, Float64(z * t)))
	tmp = 0.0
	if (Float64(c * i) <= -2e+63)
		tmp = t_1;
	elseif (Float64(c * i) <= 2e+98)
		tmp = fma(z, t, fma(a, b, Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(c * i + N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -2e+63], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], 2e+98], N[(z * t + N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -2.00000000000000012e63 or 2e98 < (*.f64 c i)

   1. Initial program 86.7%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + t \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot i + t \cdot z\right) + a \cdot b} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{c \cdot i + \left(t \cdot z + a \cdot b\right)} \]

      3. +-commutative N/A

         \[\leadsto c \cdot i + \color{blue}{\left(a \cdot b + t \cdot z\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, a \cdot b + t \cdot z\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, i, \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)}\right) \]

      6. *-lowering-*.f64 90.7

         \[\leadsto \mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, \color{blue}{t \cdot z}\right)\right) \]

   5. Simplified 90.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, t \cdot z\right)\right)} \]

   if -2.00000000000000012e63 < (*.f64 c i) < 2e98

   1. Initial program 98.5%

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

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + z \cdot t\right) + \left(a \cdot b + c \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot t + x \cdot y\right)} + \left(a \cdot b + c \cdot i\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{z \cdot t + \left(x \cdot y + \left(a \cdot b + c \cdot i\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(a, b, c \cdot i\right)}\right)\right) \]

      7. *-lowering-*.f64 99.2

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, \color{blue}{c \cdot i}\right)\right)\right) \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\right)} \]

   5. Taylor expanded in c around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. *-lowering-*.f64 93.7

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

   7. Simplified 93.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(z, t, \mathsf{fma}\left(a, b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, z \cdot t\right)\\ t_2 := \mathsf{fma}\left(c, i, t\_1\right)\\ \mathbf{if}\;c \cdot i \leq -2 \cdot 10^{+63}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma a b (* z t))) (t_2 (fma c i t_1)))
   (if (<= (* c i) -2e+63) t_2 (if (<= (* c i) 2e+98) (fma x y t_1) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(a, b, (z * t));
	double t_2 = fma(c, i, t_1);
	double tmp;
	if ((c * i) <= -2e+63) {
		tmp = t_2;
	} else if ((c * i) <= 2e+98) {
		tmp = fma(x, y, t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = fma(a, b, Float64(z * t))
	t_2 = fma(c, i, t_1)
	tmp = 0.0
	if (Float64(c * i) <= -2e+63)
		tmp = t_2;
	elseif (Float64(c * i) <= 2e+98)
		tmp = fma(x, y, t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c * i + t$95$1), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -2e+63], t$95$2, If[LessEqual[N[(c * i), $MachinePrecision], 2e+98], N[(x * y + t$95$1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -2.00000000000000012e63 or 2e98 < (*.f64 c i)

   1. Initial program 86.7%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + t \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot i + t \cdot z\right) + a \cdot b} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{c \cdot i + \left(t \cdot z + a \cdot b\right)} \]

      3. +-commutative N/A

         \[\leadsto c \cdot i + \color{blue}{\left(a \cdot b + t \cdot z\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, a \cdot b + t \cdot z\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, i, \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)}\right) \]

      6. *-lowering-*.f64 90.7

         \[\leadsto \mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, \color{blue}{t \cdot z}\right)\right) \]

   5. Simplified 90.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, t \cdot z\right)\right)} \]

   if -2.00000000000000012e63 < (*.f64 c i) < 2e98

   1. Initial program 98.5%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 93.7

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

   5. Simplified 93.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma c i (fma a b (* z t)))))
   (if (<= (* z t) -2e-60)
     t_1
     (if (<= (* z t) 5e+104) (fma a b (fma c i (* x y))) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(c, i, fma(a, b, (z * t)));
	double tmp;
	if ((z * t) <= -2e-60) {
		tmp = t_1;
	} else if ((z * t) <= 5e+104) {
		tmp = fma(a, b, fma(c, i, (x * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = fma(c, i, fma(a, b, Float64(z * t)))
	tmp = 0.0
	if (Float64(z * t) <= -2e-60)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e+104)
		tmp = fma(a, b, fma(c, i, Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(c * i + N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e-60], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e+104], N[(a * b + N[(c * i + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.9999999999999999e-60 or 4.9999999999999997e104 < (*.f64 z t)

   1. Initial program 91.4%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + t \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot i + t \cdot z\right) + a \cdot b} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{c \cdot i + \left(t \cdot z + a \cdot b\right)} \]

      3. +-commutative N/A

         \[\leadsto c \cdot i + \color{blue}{\left(a \cdot b + t \cdot z\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, a \cdot b + t \cdot z\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, i, \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)}\right) \]

      6. *-lowering-*.f64 90.1

         \[\leadsto \mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, \color{blue}{t \cdot z}\right)\right) \]

   5. Simplified 90.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, t \cdot z\right)\right)} \]

   if -1.9999999999999999e-60 < (*.f64 z t) < 4.9999999999999997e104

   1. Initial program 94.8%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, c \cdot i + x \cdot y\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{\mathsf{fma}\left(c, i, x \cdot y\right)}\right) \]

      3. *-lowering-*.f64 91.3

         \[\leadsto \mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, \color{blue}{x \cdot y}\right)\right) \]

   5. Simplified 91.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, i, z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -1e+132)
   (fma z t (* a b))
   (if (<= (* z t) 1e+152) (fma a b (fma c i (* x y))) (fma c i (* z t)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -1e+132) {
		tmp = fma(z, t, (a * b));
	} else if ((z * t) <= 1e+152) {
		tmp = fma(a, b, fma(c, i, (x * y)));
	} else {
		tmp = fma(c, i, (z * t));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -1e+132)
		tmp = fma(z, t, Float64(a * b));
	elseif (Float64(z * t) <= 1e+152)
		tmp = fma(a, b, fma(c, i, Float64(x * y)));
	else
		tmp = fma(c, i, Float64(z * t));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+132], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+152], N[(a * b + N[(c * i + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * i + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.99999999999999991e131

   1. Initial program 90.6%

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

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + z \cdot t\right) + \left(a \cdot b + c \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot t + x \cdot y\right)} + \left(a \cdot b + c \cdot i\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{z \cdot t + \left(x \cdot y + \left(a \cdot b + c \cdot i\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(a, b, c \cdot i\right)}\right)\right) \]

      7. *-lowering-*.f64 96.2

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, \color{blue}{c \cdot i}\right)\right)\right) \]

   4. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\right)} \]

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 86.9

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

   7. Simplified 86.9%

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

   if -9.99999999999999991e131 < (*.f64 z t) < 1e152

   1. Initial program 96.4%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, c \cdot i + x \cdot y\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{\mathsf{fma}\left(c, i, x \cdot y\right)}\right) \]

      3. *-lowering-*.f64 86.3

         \[\leadsto \mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, \color{blue}{x \cdot y}\right)\right) \]

   5. Simplified 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)} \]

   if 1e152 < (*.f64 z t)

   1. Initial program 79.4%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + t \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot i + t \cdot z\right) + a \cdot b} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{c \cdot i + \left(t \cdot z + a \cdot b\right)} \]

      3. +-commutative N/A

         \[\leadsto c \cdot i + \color{blue}{\left(a \cdot b + t \cdot z\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, a \cdot b + t \cdot z\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, i, \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)}\right) \]

      6. *-lowering-*.f64 91.2

         \[\leadsto \mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, \color{blue}{t \cdot z}\right)\right) \]

   5. Simplified 91.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, t \cdot z\right)\right)} \]

   6. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 81.3

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

   8. Simplified 81.3%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, i, z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.4 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 1.65 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -2.4e+118)
   (fma z t (* a b))
   (if (<= (* a b) 1.65e+160) (fma z t (* c i)) (fma i c (* a b)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -2.4e+118) {
		tmp = fma(z, t, (a * b));
	} else if ((a * b) <= 1.65e+160) {
		tmp = fma(z, t, (c * i));
	} else {
		tmp = fma(i, c, (a * b));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -2.4e+118)
		tmp = fma(z, t, Float64(a * b));
	elseif (Float64(a * b) <= 1.65e+160)
		tmp = fma(z, t, Float64(c * i));
	else
		tmp = fma(i, c, Float64(a * b));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -2.4e+118], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.65e+160], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.4e118

   1. Initial program 81.8%

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

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + z \cdot t\right) + \left(a \cdot b + c \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot t + x \cdot y\right)} + \left(a \cdot b + c \cdot i\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{z \cdot t + \left(x \cdot y + \left(a \cdot b + c \cdot i\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(a, b, c \cdot i\right)}\right)\right) \]

      7. *-lowering-*.f64 86.3

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, \color{blue}{c \cdot i}\right)\right)\right) \]

   4. Applied egg-rr 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\right)} \]

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 77.3

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

   7. Simplified 77.3%

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

   if -2.4e118 < (*.f64 a b) < 1.6499999999999999e160

   1. Initial program 95.6%

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

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + z \cdot t\right) + \left(a \cdot b + c \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot t + x \cdot y\right)} + \left(a \cdot b + c \cdot i\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{z \cdot t + \left(x \cdot y + \left(a \cdot b + c \cdot i\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(a, b, c \cdot i\right)}\right)\right) \]

      7. *-lowering-*.f64 99.4

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, \color{blue}{c \cdot i}\right)\right)\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\right)} \]

   5. Taylor expanded in c around inf

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

      1. *-lowering-*.f64 71.9

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

   7. Simplified 71.9%

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

   if 1.6499999999999999e160 < (*.f64 a b)

   1. Initial program 93.1%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 86.2

         \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]

   5. Simplified 86.2%

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + a \cdot b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + a \cdot b \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, a \cdot b\right)} \]

      4. *-lowering-*.f64 89.7

         \[\leadsto \mathsf{fma}\left(i, c, \color{blue}{a \cdot b}\right) \]

   7. Applied egg-rr 89.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, a \cdot b\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.4 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(c, i, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -2.4e+124)
   (fma z t (* a b))
   (if (<= (* a b) 1e+160) (fma c i (* z t)) (fma i c (* a b)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -2.4e+124) {
		tmp = fma(z, t, (a * b));
	} else if ((a * b) <= 1e+160) {
		tmp = fma(c, i, (z * t));
	} else {
		tmp = fma(i, c, (a * b));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -2.4e+124)
		tmp = fma(z, t, Float64(a * b));
	elseif (Float64(a * b) <= 1e+160)
		tmp = fma(c, i, Float64(z * t));
	else
		tmp = fma(i, c, Float64(a * b));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -2.4e+124], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e+160], N[(c * i + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.40000000000000006e124

   1. Initial program 81.8%

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

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + z \cdot t\right) + \left(a \cdot b + c \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot t + x \cdot y\right)} + \left(a \cdot b + c \cdot i\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{z \cdot t + \left(x \cdot y + \left(a \cdot b + c \cdot i\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(a, b, c \cdot i\right)}\right)\right) \]

      7. *-lowering-*.f64 86.3

         \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, \color{blue}{c \cdot i}\right)\right)\right) \]

   4. Applied egg-rr 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\right)} \]

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 77.3

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

   7. Simplified 77.3%

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

   if -2.40000000000000006e124 < (*.f64 a b) < 1.00000000000000001e160

   1. Initial program 95.6%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + t \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot i + t \cdot z\right) + a \cdot b} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{c \cdot i + \left(t \cdot z + a \cdot b\right)} \]

      3. +-commutative N/A

         \[\leadsto c \cdot i + \color{blue}{\left(a \cdot b + t \cdot z\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, a \cdot b + t \cdot z\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, i, \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)}\right) \]

      6. *-lowering-*.f64 78.0

         \[\leadsto \mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, \color{blue}{t \cdot z}\right)\right) \]

   5. Simplified 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, t \cdot z\right)\right)} \]

   6. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 71.4

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

   8. Simplified 71.4%

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

   if 1.00000000000000001e160 < (*.f64 a b)

   1. Initial program 93.1%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 86.2

         \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]

   5. Simplified 86.2%

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + a \cdot b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + a \cdot b \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, a \cdot b\right)} \]

      4. *-lowering-*.f64 89.7

         \[\leadsto \mathsf{fma}\left(i, c, \color{blue}{a \cdot b}\right) \]

   7. Applied egg-rr 89.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, a \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.4 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(c, i, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 5.4 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(c, i, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i c (* a b))))
   (if (<= (* a b) -1e+151)
     t_1
     (if (<= (* a b) 5.4e+159) (fma c i (* z t)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(i, c, (a * b));
	double tmp;
	if ((a * b) <= -1e+151) {
		tmp = t_1;
	} else if ((a * b) <= 5.4e+159) {
		tmp = fma(c, i, (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, c, Float64(a * b))
	tmp = 0.0
	if (Float64(a * b) <= -1e+151)
		tmp = t_1;
	elseif (Float64(a * b) <= 5.4e+159)
		tmp = fma(c, i, Float64(z * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+151], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 5.4e+159], N[(c * i + N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.00000000000000002e151 or 5.40000000000000016e159 < (*.f64 a b)

   1. Initial program 87.1%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 79.7

         \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]

   5. Simplified 79.7%

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + a \cdot b} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + a \cdot b \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, a \cdot b\right)} \]

      4. *-lowering-*.f64 84.0

         \[\leadsto \mathsf{fma}\left(i, c, \color{blue}{a \cdot b}\right) \]

   7. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, a \cdot b\right)} \]

   if -1.00000000000000002e151 < (*.f64 a b) < 5.40000000000000016e159

   1. Initial program 95.1%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + t \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot i + t \cdot z\right) + a \cdot b} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{c \cdot i + \left(t \cdot z + a \cdot b\right)} \]

      3. +-commutative N/A

         \[\leadsto c \cdot i + \color{blue}{\left(a \cdot b + t \cdot z\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, a \cdot b + t \cdot z\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, i, \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)}\right) \]

      6. *-lowering-*.f64 77.3

         \[\leadsto \mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, \color{blue}{t \cdot z}\right)\right) \]

   5. Simplified 77.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, t \cdot z\right)\right)} \]

   6. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 70.8

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

   8. Simplified 70.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 5.4 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(c, i, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4 \cdot 10^{+134}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 7 \cdot 10^{+198}:\\ \;\;\;\;\mathsf{fma}\left(c, i, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -4e+134)
   (* a b)
   (if (<= (* a b) 7e+198) (fma c i (* z t)) (* a b))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -4e+134) {
		tmp = a * b;
	} else if ((a * b) <= 7e+198) {
		tmp = fma(c, i, (z * t));
	} else {
		tmp = a * b;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -4e+134)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 7e+198)
		tmp = fma(c, i, Float64(z * t));
	else
		tmp = Float64(a * b);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -4e+134], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 7e+198], N[(c * i + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -3.99999999999999969e134 or 7.00000000000000026e198 < (*.f64 a b)

   1. Initial program 85.3%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 73.8

         \[\leadsto \color{blue}{a \cdot b} \]

   5. Simplified 73.8%

      \[\leadsto \color{blue}{a \cdot b} \]

   if -3.99999999999999969e134 < (*.f64 a b) < 7.00000000000000026e198

   1. Initial program 95.7%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + t \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot i + t \cdot z\right) + a \cdot b} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{c \cdot i + \left(t \cdot z + a \cdot b\right)} \]

      3. +-commutative N/A

         \[\leadsto c \cdot i + \color{blue}{\left(a \cdot b + t \cdot z\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, a \cdot b + t \cdot z\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, i, \color{blue}{\mathsf{fma}\left(a, b, t \cdot z\right)}\right) \]

      6. *-lowering-*.f64 78.0

         \[\leadsto \mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, \color{blue}{t \cdot z}\right)\right) \]

   5. Simplified 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, i, \mathsf{fma}\left(a, b, t \cdot z\right)\right)} \]

   6. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 71.6

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

   8. Simplified 71.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4 \cdot 10^{+134}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 7 \cdot 10^{+198}:\\ \;\;\;\;\mathsf{fma}\left(c, i, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.25 \cdot 10^{+141}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 7.8 \cdot 10^{+91}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1.25e+141)
   (* c i)
   (if (<= (* c i) 7.8e+91) (* z t) (* c i))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -1.25e+141) {
		tmp = c * i;
	} else if ((c * i) <= 7.8e+91) {
		tmp = z * t;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((c * i) <= (-1.25d+141)) then
        tmp = c * i
    else if ((c * i) <= 7.8d+91) then
        tmp = z * t
    else
        tmp = c * i
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c && c < i;
public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -1.25e+141) {
		tmp = c * i;
	} else if ((c * i) <= 7.8e+91) {
		tmp = z * t;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c, i] = sort([x, y, z, t, a, b, c, i])
def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -1.25e+141:
		tmp = c * i
	elif (c * i) <= 7.8e+91:
		tmp = z * t
	else:
		tmp = c * i
	return tmp

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -1.25e+141)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= 7.8e+91)
		tmp = Float64(z * t);
	else
		tmp = Float64(c * i);
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c, i = num2cell(sort([x, y, z, t, a, b, c, i])){:}
function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((c * i) <= -1.25e+141)
		tmp = c * i;
	elseif ((c * i) <= 7.8e+91)
		tmp = z * t;
	else
		tmp = c * i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -1.25e+141], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 7.8e+91], N[(z * t), $MachinePrecision], N[(c * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -1.25000000000000006e141 or 7.79999999999999935e91 < (*.f64 c i)

   1. Initial program 86.2%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. *-lowering-*.f64 67.5

         \[\leadsto \color{blue}{c \cdot i} \]

   5. Simplified 67.5%

      \[\leadsto \color{blue}{c \cdot i} \]

   if -1.25000000000000006e141 < (*.f64 c i) < 7.79999999999999935e91

   1. Initial program 97.4%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 40.4

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

   5. Simplified 40.4%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.25 \cdot 10^{+141}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 7.8 \cdot 10^{+91}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.2 \cdot 10^{+93}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.55 \cdot 10^{+196}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -5.2e+93)
   (* a b)
   (if (<= (* a b) 1.55e+196) (* c i) (* a b))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -5.2e+93) {
		tmp = a * b;
	} else if ((a * b) <= 1.55e+196) {
		tmp = c * i;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((a * b) <= (-5.2d+93)) then
        tmp = a * b
    else if ((a * b) <= 1.55d+196) then
        tmp = c * i
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c && c < i;
public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -5.2e+93) {
		tmp = a * b;
	} else if ((a * b) <= 1.55e+196) {
		tmp = c * i;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c, i] = sort([x, y, z, t, a, b, c, i])
def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -5.2e+93:
		tmp = a * b
	elif (a * b) <= 1.55e+196:
		tmp = c * i
	else:
		tmp = a * b
	return tmp

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -5.2e+93)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 1.55e+196)
		tmp = Float64(c * i);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c, i = num2cell(sort([x, y, z, t, a, b, c, i])){:}
function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((a * b) <= -5.2e+93)
		tmp = a * b;
	elseif ((a * b) <= 1.55e+196)
		tmp = c * i;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -5.2e+93], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.55e+196], N[(c * i), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5.19999999999999999e93 or 1.55000000000000005e196 < (*.f64 a b)

   1. Initial program 86.1%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 69.9

         \[\leadsto \color{blue}{a \cdot b} \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{a \cdot b} \]

   if -5.19999999999999999e93 < (*.f64 a b) < 1.55000000000000005e196

   1. Initial program 95.6%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. *-lowering-*.f64 38.5

         \[\leadsto \color{blue}{c \cdot i} \]

   5. Simplified 38.5%

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

Alternative 13: 98.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\right) \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (fma z t (fma x y (fma a b (* c i)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(z, t, fma(x, y, fma(a, b, (c * i))));
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	return fma(z, t, fma(x, y, fma(a, b, Float64(c * i))))
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(z * t + N[(x * y + N[(a * b + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.9%

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

   1. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(x \cdot y + z \cdot t\right) + \left(a \cdot b + c \cdot i\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(z \cdot t + x \cdot y\right)} + \left(a \cdot b + c \cdot i\right) \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{z \cdot t + \left(x \cdot y + \left(a \cdot b + c \cdot i\right)\right)} \]

   4. accelerator-lowering-fma.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(a, b, c \cdot i\right)}\right)\right) \]

   7. *-lowering-*.f64 96.5

      \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, \color{blue}{c \cdot i}\right)\right)\right) \]

4. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\right)} \]
5. Add Preprocessing

Alternative 14: 27.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ a \cdot b \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i) :precision binary64 (* a b))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return a * b;
}

Fortran

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = a * b
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c && c < i;
public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return a * b;
}

Python

[x, y, z, t, a, b, c, i] = sort([x, y, z, t, a, b, c, i])
def code(x, y, z, t, a, b, c, i):
	return a * b

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	return Float64(a * b)
end

MATLAB

x, y, z, t, a, b, c, i = num2cell(sort([x, y, z, t, a, b, c, i])){:}
function tmp = code(x, y, z, t, a, b, c, i)
	tmp = a * b;
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(a * b), $MachinePrecision]

Derivation

1. Initial program 92.9%

   \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
2. Add Preprocessing

3. Taylor expanded in a around inf

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

   1. *-lowering-*.f64 25.4

      \[\leadsto \color{blue}{a \cdot b} \]

5. Simplified 25.4%

   \[\leadsto \color{blue}{a \cdot b} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3, C"
  :precision binary64
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))