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

Percentage Accurate: 95.9% → 98.3%

Time: 3.5s

Alternatives: 13

Speedup: 1.1×

• 41.2% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    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: 95.9% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    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: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;i \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(b, a, t \cdot z\right)\right) + c \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

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
 (if (<= i 5e+141)
   (+ (fma y x (fma b a (* t z))) (* c i))
   (fma i c (fma b a (fma t z (* y x))))))

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 (i <= 5e+141) {
		tmp = fma(y, x, fma(b, a, (t * z))) + (c * i);
	} else {
		tmp = fma(i, c, fma(b, a, fma(t, z, (y * x))));
	}
	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 (i <= 5e+141)
		tmp = Float64(fma(y, x, fma(b, a, Float64(t * z))) + Float64(c * i));
	else
		tmp = fma(i, c, fma(b, a, fma(t, z, Float64(y * x))));
	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[i, 5e+141], N[(N[(y * x + N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(b * a + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 5.00000000000000025e141

   1. Initial program 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

   if 5.00000000000000025e141 < i

   1. Initial program 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.3

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

   3. Applied rewrites 98.3%

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

Alternative 2: 97.6% accurate, 1.1× speedup

Math

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

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 i c (fma b a (fma t z (* y x)))))

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(i, c, fma(b, a, fma(t, z, (y * x))));
}

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(i, c, fma(b, a, fma(t, z, Float64(y * x))))
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[(i * c + N[(b * a + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. +-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 98.3

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

3. Applied rewrites 98.3%

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

Alternative 3: 89.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, c, \mathsf{fma}\left(x, y, a \cdot b\right)\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 i c (fma x y (* a b)))))
   (if (<= (* a b) -1e+86)
     t_1
     (if (<= (* a b) 4e-11) (fma i c (fma t z (* y x))) 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, fma(x, y, (a * b)));
	double tmp;
	if ((a * b) <= -1e+86) {
		tmp = t_1;
	} else if ((a * b) <= 4e-11) {
		tmp = fma(i, c, fma(t, z, (y * x)));
	} 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, fma(x, y, Float64(a * b)))
	tmp = 0.0
	if (Float64(a * b) <= -1e+86)
		tmp = t_1;
	elseif (Float64(a * b) <= 4e-11)
		tmp = fma(i, c, fma(t, z, Float64(y * x)));
	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[(x * y + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+86], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 4e-11], N[(i * c + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1e86 or 3.99999999999999976e-11 < (*.f64 a b)

   1. Initial program 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.3

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 74.7

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

   6. Applied rewrites 74.7%

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

   if -1e86 < (*.f64 a b) < 3.99999999999999976e-11

   1. Initial program 95.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   4. Applied rewrites 75.9%

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

Alternative 4: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(y, x, c \cdot i\right)\right)\\ \end{array} \end{array} \]

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
 (if (<= (* x y) -2e+59)
   (fma i c (fma t z (* y x)))
   (if (<= (* x y) 5e+96)
     (fma b a (fma i c (* t z)))
     (fma b a (fma y x (* 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 ((x * y) <= -2e+59) {
		tmp = fma(i, c, fma(t, z, (y * x)));
	} else if ((x * y) <= 5e+96) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} else {
		tmp = fma(b, a, fma(y, x, (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(x * y) <= -2e+59)
		tmp = fma(i, c, fma(t, z, Float64(y * x)));
	elseif (Float64(x * y) <= 5e+96)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	else
		tmp = fma(b, a, fma(y, x, 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[(x * y), $MachinePrecision], -2e+59], N[(i * c + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+96], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(y * x + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.99999999999999994e59

   1. Initial program 95.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   4. Applied rewrites 75.9%

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

   if -1.99999999999999994e59 < (*.f64 x y) < 5.0000000000000004e96

   1. Initial program 95.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

   if 5.0000000000000004e96 < (*.f64 x y)

   1. Initial program 95.9%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 74.6

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

   6. Applied rewrites 74.6%

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

Alternative 5: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(y, x, c \cdot i\right)\right)\\ \end{array} \end{array} \]

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
 (if (<= (* x y) -1e+140)
   (fma b a (fma i c (* y x)))
   (if (<= (* x y) 5e+96)
     (fma b a (fma i c (* t z)))
     (fma b a (fma y x (* 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 ((x * y) <= -1e+140) {
		tmp = fma(b, a, fma(i, c, (y * x)));
	} else if ((x * y) <= 5e+96) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} else {
		tmp = fma(b, a, fma(y, x, (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(x * y) <= -1e+140)
		tmp = fma(b, a, fma(i, c, Float64(y * x)));
	elseif (Float64(x * y) <= 5e+96)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	else
		tmp = fma(b, a, fma(y, x, 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[(x * y), $MachinePrecision], -1e+140], N[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+96], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(y * x + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.00000000000000006e140

   1. Initial program 95.9%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

   if -1.00000000000000006e140 < (*.f64 x y) < 5.0000000000000004e96

   1. Initial program 95.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

   if 5.0000000000000004e96 < (*.f64 x y)

   1. Initial program 95.9%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 74.6

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

   6. Applied rewrites 74.6%

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

Alternative 6: 89.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 b a (fma i c (* y x)))))
   (if (<= (* x y) -1e+140)
     t_1
     (if (<= (* x y) 5e+96) (fma b a (fma i c (* t z))) 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(b, a, fma(i, c, (y * x)));
	double tmp;
	if ((x * y) <= -1e+140) {
		tmp = t_1;
	} else if ((x * y) <= 5e+96) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} 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(b, a, fma(i, c, Float64(y * x)))
	tmp = 0.0
	if (Float64(x * y) <= -1e+140)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e+96)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	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[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+140], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e+96], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000006e140 or 5.0000000000000004e96 < (*.f64 x y)

   1. Initial program 95.9%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

   if -1.00000000000000006e140 < (*.f64 x y) < 5.0000000000000004e96

   1. Initial program 95.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

Alternative 7: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+265}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 i c (* y x))))
   (if (<= (* x y) -2e+265)
     t_1
     (if (<= (* x y) 2e+98) (fma b a (fma i c (* t z))) 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, (y * x));
	double tmp;
	if ((x * y) <= -2e+265) {
		tmp = t_1;
	} else if ((x * y) <= 2e+98) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} 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(y * x))
	tmp = 0.0
	if (Float64(x * y) <= -2e+265)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e+98)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	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[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+265], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e+98], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.00000000000000013e265 or 2e98 < (*.f64 x y)

   1. Initial program 95.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto t \cdot z + c \cdot \color{blue}{i} \]

      2. *-commutative N/A

         \[\leadsto z \cdot t + c \cdot i \]

      3. lower-fma.f64 N/A

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

      4. lift-*.f64 51.7

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

   7. Applied rewrites 51.7%

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

   8. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{t \cdot z}{c} + i\right) \cdot c \]

      4. associate-/l* N/A

         \[\leadsto \left(t \cdot \frac{z}{c} + i\right) \cdot c \]

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 47.3

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

   10. Applied rewrites 47.3%

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

   11. Taylor expanded in z around 0

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

       1. *-commutative N/A

          \[\leadsto i \cdot c + x \cdot y \]

       2. lower-fma.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 51.6

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

   13. Applied rewrites 51.6%

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

   if -2.00000000000000013e265 < (*.f64 x y) < 2e98

   1. Initial program 95.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

Alternative 8: 66.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 10^{-242}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \end{array} \end{array} \]

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 (<= (* a b) -2e+48)
   (fma i c (* a b))
   (if (<= (* a b) 1e-242)
     (fma i c (* y x))
     (if (<= (* a b) 4e-11) (fma z t (* c i)) (fma b a (* 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 ((a * b) <= -2e+48) {
		tmp = fma(i, c, (a * b));
	} else if ((a * b) <= 1e-242) {
		tmp = fma(i, c, (y * x));
	} else if ((a * b) <= 4e-11) {
		tmp = fma(z, t, (c * i));
	} else {
		tmp = fma(b, a, (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(a * b) <= -2e+48)
		tmp = fma(i, c, Float64(a * b));
	elseif (Float64(a * b) <= 1e-242)
		tmp = fma(i, c, Float64(y * x));
	elseif (Float64(a * b) <= 4e-11)
		tmp = fma(z, t, Float64(c * i));
	else
		tmp = fma(b, a, 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[(a * b), $MachinePrecision], -2e+48], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e-242], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 4e-11], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 a b) < -2.00000000000000009e48

   1. Initial program 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.3

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 74.7

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

   6. Applied rewrites 74.7%

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

   7. Taylor expanded in x around 0

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

      1. lift-*.f64 50.5

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

   9. Applied rewrites 50.5%

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

   if -2.00000000000000009e48 < (*.f64 a b) < 1e-242

   1. Initial program 95.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto t \cdot z + c \cdot \color{blue}{i} \]

      2. *-commutative N/A

         \[\leadsto z \cdot t + c \cdot i \]

      3. lower-fma.f64 N/A

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

      4. lift-*.f64 51.7

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

   7. Applied rewrites 51.7%

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

   8. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{t \cdot z}{c} + i\right) \cdot c \]

      4. associate-/l* N/A

         \[\leadsto \left(t \cdot \frac{z}{c} + i\right) \cdot c \]

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 47.3

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

   10. Applied rewrites 47.3%

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

   11. Taylor expanded in z around 0

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

       1. *-commutative N/A

          \[\leadsto i \cdot c + x \cdot y \]

       2. lower-fma.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 51.6

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

   13. Applied rewrites 51.6%

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

   if 1e-242 < (*.f64 a b) < 3.99999999999999976e-11

   1. Initial program 95.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto t \cdot z + c \cdot \color{blue}{i} \]

      2. *-commutative N/A

         \[\leadsto z \cdot t + c \cdot i \]

      3. lower-fma.f64 N/A

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

      4. lift-*.f64 51.7

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

   7. Applied rewrites 51.7%

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

   if 3.99999999999999976e-11 < (*.f64 a b)

   1. Initial program 95.9%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 50.5

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

   7. Applied rewrites 50.5%

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

Alternative 9: 65.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 10^{-242}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 b a (* c i))))
   (if (<= (* a b) -2e+48)
     t_1
     (if (<= (* a b) 1e-242)
       (fma i c (* y x))
       (if (<= (* a b) 4e-11) (fma z t (* c i)) 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(b, a, (c * i));
	double tmp;
	if ((a * b) <= -2e+48) {
		tmp = t_1;
	} else if ((a * b) <= 1e-242) {
		tmp = fma(i, c, (y * x));
	} else if ((a * b) <= 4e-11) {
		tmp = fma(z, t, (c * i));
	} 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(b, a, Float64(c * i))
	tmp = 0.0
	if (Float64(a * b) <= -2e+48)
		tmp = t_1;
	elseif (Float64(a * b) <= 1e-242)
		tmp = fma(i, c, Float64(y * x));
	elseif (Float64(a * b) <= 4e-11)
		tmp = fma(z, t, Float64(c * i));
	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[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+48], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 1e-242], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 4e-11], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.00000000000000009e48 or 3.99999999999999976e-11 < (*.f64 a b)

   1. Initial program 95.9%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 50.5

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

   7. Applied rewrites 50.5%

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

   if -2.00000000000000009e48 < (*.f64 a b) < 1e-242

   1. Initial program 95.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto t \cdot z + c \cdot \color{blue}{i} \]

      2. *-commutative N/A

         \[\leadsto z \cdot t + c \cdot i \]

      3. lower-fma.f64 N/A

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

      4. lift-*.f64 51.7

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

   7. Applied rewrites 51.7%

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

   8. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{t \cdot z}{c} + i\right) \cdot c \]

      4. associate-/l* N/A

         \[\leadsto \left(t \cdot \frac{z}{c} + i\right) \cdot c \]

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 47.3

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

   10. Applied rewrites 47.3%

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

   11. Taylor expanded in z around 0

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

       1. *-commutative N/A

          \[\leadsto i \cdot c + x \cdot y \]

       2. lower-fma.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 51.6

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

   13. Applied rewrites 51.6%

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

   if 1e-242 < (*.f64 a b) < 3.99999999999999976e-11

   1. Initial program 95.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto t \cdot z + c \cdot \color{blue}{i} \]

      2. *-commutative N/A

         \[\leadsto z \cdot t + c \cdot i \]

      3. lower-fma.f64 N/A

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

      4. lift-*.f64 51.7

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

   7. Applied rewrites 51.7%

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

Alternative 10: 65.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* y x))))
   (if (<= (* x y) -1e+140)
     t_1
     (if (<= (* x y) 2e+98) (fma z t (* c i)) 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, (y * x));
	double tmp;
	if ((x * y) <= -1e+140) {
		tmp = t_1;
	} else if ((x * y) <= 2e+98) {
		tmp = fma(z, t, (c * i));
	} 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(y * x))
	tmp = 0.0
	if (Float64(x * y) <= -1e+140)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e+98)
		tmp = fma(z, t, Float64(c * i));
	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[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+140], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e+98], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000006e140 or 2e98 < (*.f64 x y)

   1. Initial program 95.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto t \cdot z + c \cdot \color{blue}{i} \]

      2. *-commutative N/A

         \[\leadsto z \cdot t + c \cdot i \]

      3. lower-fma.f64 N/A

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

      4. lift-*.f64 51.7

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

   7. Applied rewrites 51.7%

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

   8. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{t \cdot z}{c} + i\right) \cdot c \]

      4. associate-/l* N/A

         \[\leadsto \left(t \cdot \frac{z}{c} + i\right) \cdot c \]

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 47.3

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

   10. Applied rewrites 47.3%

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

   11. Taylor expanded in z around 0

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

       1. *-commutative N/A

          \[\leadsto i \cdot c + x \cdot y \]

       2. lower-fma.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 51.6

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

   13. Applied rewrites 51.6%

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

   if -1.00000000000000006e140 < (*.f64 x y) < 2e98

   1. Initial program 95.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto t \cdot z + c \cdot \color{blue}{i} \]

      2. *-commutative N/A

         \[\leadsto z \cdot t + c \cdot i \]

      3. lower-fma.f64 N/A

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

      4. lift-*.f64 51.7

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

   7. Applied rewrites 51.7%

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

Alternative 11: 51.6% accurate, 2.4× speedup

Math

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

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 i c (* y x)))

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(i, c, (y * x));
}

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(i, c, Float64(y * x))
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[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 75.9

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

4. Applied rewrites 75.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto t \cdot z + c \cdot \color{blue}{i} \]

   2. *-commutative N/A

      \[\leadsto z \cdot t + c \cdot i \]

   3. lower-fma.f64 N/A

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

   4. lift-*.f64 51.7

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

7. Applied rewrites 51.7%

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

8. Taylor expanded in c around inf

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

   1. *-commutative N/A

      \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

   2. lower-*.f64 N/A

      \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

   3. +-commutative N/A

      \[\leadsto \left(\frac{t \cdot z}{c} + i\right) \cdot c \]

   4. associate-/l* N/A

      \[\leadsto \left(t \cdot \frac{z}{c} + i\right) \cdot c \]

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 47.3

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

10. Applied rewrites 47.3%

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

11. Taylor expanded in z around 0

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

    1. *-commutative N/A

       \[\leadsto i \cdot c + x \cdot y \]

    2. lower-fma.f64 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 51.6

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

13. Applied rewrites 51.6%

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

Alternative 12: 43.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.45 \cdot 10^{+44}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 3.5 \cdot 10^{+100}:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

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 (<= (* x y) -1.45e+44)
   (* x y)
   (if (<= (* x y) 3.5e+100) (* i c) (* x y))))

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 ((x * y) <= -1.45e+44) {
		tmp = x * y;
	} else if ((x * y) <= 3.5e+100) {
		tmp = i * c;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    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 ((x * y) <= (-1.45d+44)) then
        tmp = x * y
    else if ((x * y) <= 3.5d+100) then
        tmp = i * c
    else
        tmp = x * y
    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 ((x * y) <= -1.45e+44) {
		tmp = x * y;
	} else if ((x * y) <= 3.5e+100) {
		tmp = i * c;
	} else {
		tmp = x * y;
	}
	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 (x * y) <= -1.45e+44:
		tmp = x * y
	elif (x * y) <= 3.5e+100:
		tmp = i * c
	else:
		tmp = x * y
	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(x * y) <= -1.45e+44)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 3.5e+100)
		tmp = Float64(i * c);
	else
		tmp = Float64(x * y);
	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 ((x * y) <= -1.45e+44)
		tmp = x * y;
	elseif ((x * y) <= 3.5e+100)
		tmp = i * c;
	else
		tmp = x * y;
	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[(x * y), $MachinePrecision], -1.45e+44], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 3.5e+100], N[(i * c), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.4500000000000001e44 or 3.49999999999999976e100 < (*.f64 x y)

   1. Initial program 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.3

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 74.7

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

   6. Applied rewrites 74.7%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto x \cdot y \]

      2. associate-+r+ N/A

         \[\leadsto x \cdot y \]

      3. *-commutative N/A

         \[\leadsto x \cdot y \]

      4. *-commutative N/A

         \[\leadsto x \cdot y \]

      5. associate-+r+ N/A

         \[\leadsto x \cdot y \]

      6. +-commutative N/A

         \[\leadsto x \cdot y \]

      7. *-commutative N/A

         \[\leadsto x \cdot y \]

      8. +-commutative N/A

         \[\leadsto x \cdot y \]

      9. +-commutative N/A

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

      10. lower-*.f64 28.1

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

   9. Applied rewrites 28.1%

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

   if -1.4500000000000001e44 < (*.f64 x y) < 3.49999999999999976e100

   1. Initial program 95.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto t \cdot z + c \cdot \color{blue}{i} \]

      2. *-commutative N/A

         \[\leadsto z \cdot t + c \cdot i \]

      3. lower-fma.f64 N/A

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

      4. lift-*.f64 51.7

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

   7. Applied rewrites 51.7%

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

   8. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{t \cdot z}{c} + i\right) \cdot c \]

      4. associate-/l* N/A

         \[\leadsto \left(t \cdot \frac{z}{c} + i\right) \cdot c \]

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 47.3

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

   10. Applied rewrites 47.3%

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

   11. Taylor expanded in z around 0

       \[\leadsto i \cdot c \]
   12. Step-by-step derivation

   13. Applied rewrites 26.6%

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

Alternative 13: 26.6% accurate, 5.3× speedup

Math

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

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 (* i c))

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 i * c;
}

Fortran

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    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 = i * c
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 i * c;
}

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 i * c

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(i * c)
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 = i * c;
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[(i * c), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 75.9

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

4. Applied rewrites 75.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto t \cdot z + c \cdot \color{blue}{i} \]

   2. *-commutative N/A

      \[\leadsto z \cdot t + c \cdot i \]

   3. lower-fma.f64 N/A

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

   4. lift-*.f64 51.7

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

7. Applied rewrites 51.7%

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

8. Taylor expanded in c around inf

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

   1. *-commutative N/A

      \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

   2. lower-*.f64 N/A

      \[\leadsto \left(i + \frac{t \cdot z}{c}\right) \cdot c \]

   3. +-commutative N/A

      \[\leadsto \left(\frac{t \cdot z}{c} + i\right) \cdot c \]

   4. associate-/l* N/A

      \[\leadsto \left(t \cdot \frac{z}{c} + i\right) \cdot c \]

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 47.3

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

10. Applied rewrites 47.3%

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

11. Taylor expanded in z around 0

    \[\leadsto i \cdot c \]
12. Step-by-step derivation

13. Applied rewrites 26.6%

    \[\leadsto i \cdot c \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(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)))