Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A

Percentage Accurate: 90.0% → 95.7%

Time: 4.7s

Alternatives: 19

Speedup: 0.9×

• 46.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * 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 = 2.0d0 * (((x * y) + (z * t)) - (((a + (b * c)) * 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 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

Python

def code(x, y, z, t, a, b, c, i):
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(Float64(a + Float64(b * c)) * c) * i)))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * 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 = 2.0d0 * (((x * y) + (z * t)) - (((a + (b * c)) * 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 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

Python

def code(x, y, z, t, a, b, c, i):
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(Float64(a + Float64(b * c)) * c) * i)))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* 2.0 (fma y x (* (- z) (fma c (/ (* (fma c b a) i) z) (- t))))))
        (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 1e+307) (* 2.0 (- (+ (* x y) (* z t)) t_2)) 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 = 2.0 * fma(y, x, (-z * fma(c, ((fma(c, b, a) * i) / z), -t)));
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+307) {
		tmp = 2.0 * (((x * y) + (z * t)) - t_2);
	} 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 = Float64(2.0 * fma(y, x, Float64(Float64(-z) * fma(c, Float64(Float64(fma(c, b, a) * i) / z), Float64(-t)))))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1e+307)
		tmp = Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - t_2));
	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[(2.0 * N[(y * x + N[((-z) * N[(c * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] / z), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1e+307], N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 9.99999999999999986e306 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in z around -inf

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, -1 \cdot \left(z \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(\frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z} + -1 \cdot t\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(c \cdot \frac{i \cdot \left(a + b \cdot c\right)}{z} + -1 \cdot t\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(a + b \cdot c\right)}{z}, -1 \cdot t\right)\right) \]

      8. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(b \cdot c + a\right)}{z}, -1 \cdot t\right)\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(c \cdot b + a\right)}{z}, -1 \cdot t\right)\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      12. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, \mathsf{neg}\left(t\right)\right)\right) \]

      15. lower-neg.f64 85.7

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \]

   7. Applied rewrites 85.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \]

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.99999999999999986e306

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, a\right) \cdot i\\ t_2 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+258}:\\ \;\;\;\;\left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{t\_1}{z}, -t\right)\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+256}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - t\_2 \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x - t\_1 \cdot c\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma c b a) i)) (t_2 (* (+ a (* b c)) c)))
   (if (<= t_2 -1e+258)
     (* (* (- z) (fma c (/ t_1 z) (- t))) 2.0)
     (if (<= t_2 5e+256)
       (* 2.0 (- (+ (* x y) (* z t)) (* t_2 i)))
       (* (- (* y x) (* t_1 c)) 2.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(c, b, a) * i;
	double t_2 = (a + (b * c)) * c;
	double tmp;
	if (t_2 <= -1e+258) {
		tmp = (-z * fma(c, (t_1 / z), -t)) * 2.0;
	} else if (t_2 <= 5e+256) {
		tmp = 2.0 * (((x * y) + (z * t)) - (t_2 * i));
	} else {
		tmp = ((y * x) - (t_1 * c)) * 2.0;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(c, b, a) * i)
	t_2 = Float64(Float64(a + Float64(b * c)) * c)
	tmp = 0.0
	if (t_2 <= -1e+258)
		tmp = Float64(Float64(Float64(-z) * fma(c, Float64(t_1 / z), Float64(-t))) * 2.0);
	elseif (t_2 <= 5e+256)
		tmp = Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(t_2 * i)));
	else
		tmp = Float64(Float64(Float64(y * x) - Float64(t_1 * c)) * 2.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+258], N[(N[((-z) * N[(c * N[(t$95$1 / z), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 5e+256], N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * x), $MachinePrecision] - N[(t$95$1 * c), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -1.00000000000000006e258

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \cdot 2 \]

      3. +-commutative N/A

         \[\leadsto \left(t \cdot z - \left(i \cdot \left(b \cdot c + a\right)\right) \cdot c\right) \cdot 2 \]

      4. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(i \cdot \left(c \cdot b + a\right)\right) \cdot c\right) \cdot 2 \]

      5. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot \color{blue}{2} \]

   7. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right) \cdot 2} \]

   8. Taylor expanded in a around 0

      \[\leadsto \left(t \cdot z - \left(\left(b \cdot c\right) \cdot c\right) \cdot i\right) \cdot 2 \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(\left(c \cdot b\right) \cdot c\right) \cdot i\right) \cdot 2 \]

      2. lower-*.f64 54.4

         \[\leadsto \left(t \cdot z - \left(\left(c \cdot b\right) \cdot c\right) \cdot i\right) \cdot 2 \]

   10. Applied rewrites 54.4%

       \[\leadsto \left(t \cdot z - \left(\left(c \cdot b\right) \cdot c\right) \cdot i\right) \cdot 2 \]

   11. Taylor expanded in z around -inf

       \[\leadsto \left(-1 \cdot \left(z \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right)\right) \cdot 2 \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \cdot 2 \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \cdot 2 \]

       3. mul-1-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \cdot 2 \]

       4. lower-neg.f64 N/A

          \[\leadsto \left(\left(-z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \cdot 2 \]

       5. +-commutative N/A

          \[\leadsto \left(\left(-z\right) \cdot \left(\frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z} + -1 \cdot t\right)\right) \cdot 2 \]

       6. associate-/l* N/A

          \[\leadsto \left(\left(-z\right) \cdot \left(c \cdot \frac{i \cdot \left(a + b \cdot c\right)}{z} + -1 \cdot t\right)\right) \cdot 2 \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(a + b \cdot c\right)}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       8. +-commutative N/A

          \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(b \cdot c + a\right)}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       9. *-commutative N/A

          \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(c \cdot b + a\right)}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       10. *-commutative N/A

           \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       11. lower-/.f64 N/A

           \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       12. lift-fma.f64 N/A

           \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       14. mul-1-neg N/A

           \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, \mathsf{neg}\left(t\right)\right)\right) \cdot 2 \]

       15. lower-neg.f64 65.5

           \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \cdot 2 \]

   13. Applied rewrites 65.5%

       \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \cdot 2 \]

   if -1.00000000000000006e258 < (*.f64 (+.f64 a (*.f64 b c)) c) < 5.00000000000000015e256

   1. Initial program 90.0%

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

   if 5.00000000000000015e256 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in z around -inf

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, -1 \cdot \left(z \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(\frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z} + -1 \cdot t\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(c \cdot \frac{i \cdot \left(a + b \cdot c\right)}{z} + -1 \cdot t\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(a + b \cdot c\right)}{z}, -1 \cdot t\right)\right) \]

      8. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(b \cdot c + a\right)}{z}, -1 \cdot t\right)\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(c \cdot b + a\right)}{z}, -1 \cdot t\right)\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      12. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, \mathsf{neg}\left(t\right)\right)\right) \]

      15. lower-neg.f64 85.7

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \]

   7. Applied rewrites 85.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \]

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \cdot 2 \]

      7. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      8. +-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      9. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      10. lower-*.f64 N/A

          \[\leadsto \left(y \cdot x - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      11. lift-fma.f64 N/A

          \[\leadsto \left(y \cdot x - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      12. lift-*.f64 69.7

          \[\leadsto \left(y \cdot x - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

   10. Applied rewrites 69.7%

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

Alternative 3: 91.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;i \leq 1.1 \cdot 10^{+159}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= i 1.1e+159)
   (* 2.0 (fma y x (- (* t z) (* (* (fma c b a) i) c))))
   (* (- (* t z) (* (* (fma c b a) c) i)) 2.0)))

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 <= 1.1e+159) {
		tmp = 2.0 * fma(y, x, ((t * z) - ((fma(c, b, a) * i) * c)));
	} else {
		tmp = ((t * z) - ((fma(c, b, a) * c) * i)) * 2.0;
	}
	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 <= 1.1e+159)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(t * z) - Float64(Float64(fma(c, b, a) * i) * c))));
	else
		tmp = Float64(Float64(Float64(t * z) - Float64(Float64(fma(c, b, a) * c) * i)) * 2.0);
	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, 1.1e+159], N[(2.0 * N[(y * x + N[(N[(t * z), $MachinePrecision] - N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * z), $MachinePrecision] - N[(N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.1e159

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   if 1.1e159 < i

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \cdot 2 \]

      3. +-commutative N/A

         \[\leadsto \left(t \cdot z - \left(i \cdot \left(b \cdot c + a\right)\right) \cdot c\right) \cdot 2 \]

      4. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(i \cdot \left(c \cdot b + a\right)\right) \cdot c\right) \cdot 2 \]

      5. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot \color{blue}{2} \]

   7. Applied rewrites 68.3%

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

Alternative 4: 86.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, a\right) \cdot i\\ t_2 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+252}:\\ \;\;\;\;\left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{t\_1}{z}, -t\right)\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+24}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x - t\_1 \cdot c\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma c b a) i)) (t_2 (* (+ a (* b c)) c)))
   (if (<= t_2 -5e+252)
     (* (* (- z) (fma c (/ t_1 z) (- t))) 2.0)
     (if (<= t_2 2e+24)
       (* 2.0 (- (fma t z (* y x)) (* (* i c) a)))
       (* (- (* y x) (* t_1 c)) 2.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(c, b, a) * i;
	double t_2 = (a + (b * c)) * c;
	double tmp;
	if (t_2 <= -5e+252) {
		tmp = (-z * fma(c, (t_1 / z), -t)) * 2.0;
	} else if (t_2 <= 2e+24) {
		tmp = 2.0 * (fma(t, z, (y * x)) - ((i * c) * a));
	} else {
		tmp = ((y * x) - (t_1 * c)) * 2.0;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(c, b, a) * i)
	t_2 = Float64(Float64(a + Float64(b * c)) * c)
	tmp = 0.0
	if (t_2 <= -5e+252)
		tmp = Float64(Float64(Float64(-z) * fma(c, Float64(t_1 / z), Float64(-t))) * 2.0);
	elseif (t_2 <= 2e+24)
		tmp = Float64(2.0 * Float64(fma(t, z, Float64(y * x)) - Float64(Float64(i * c) * a)));
	else
		tmp = Float64(Float64(Float64(y * x) - Float64(t_1 * c)) * 2.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+252], N[(N[((-z) * N[(c * N[(t$95$1 / z), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+24], N[(2.0 * N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] - N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * x), $MachinePrecision] - N[(t$95$1 * c), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -4.9999999999999997e252

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \cdot 2 \]

      3. +-commutative N/A

         \[\leadsto \left(t \cdot z - \left(i \cdot \left(b \cdot c + a\right)\right) \cdot c\right) \cdot 2 \]

      4. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(i \cdot \left(c \cdot b + a\right)\right) \cdot c\right) \cdot 2 \]

      5. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot \color{blue}{2} \]

   7. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right) \cdot 2} \]

   8. Taylor expanded in a around 0

      \[\leadsto \left(t \cdot z - \left(\left(b \cdot c\right) \cdot c\right) \cdot i\right) \cdot 2 \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(\left(c \cdot b\right) \cdot c\right) \cdot i\right) \cdot 2 \]

      2. lower-*.f64 54.4

         \[\leadsto \left(t \cdot z - \left(\left(c \cdot b\right) \cdot c\right) \cdot i\right) \cdot 2 \]

   10. Applied rewrites 54.4%

       \[\leadsto \left(t \cdot z - \left(\left(c \cdot b\right) \cdot c\right) \cdot i\right) \cdot 2 \]

   11. Taylor expanded in z around -inf

       \[\leadsto \left(-1 \cdot \left(z \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right)\right) \cdot 2 \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \cdot 2 \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \cdot 2 \]

       3. mul-1-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \cdot 2 \]

       4. lower-neg.f64 N/A

          \[\leadsto \left(\left(-z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \cdot 2 \]

       5. +-commutative N/A

          \[\leadsto \left(\left(-z\right) \cdot \left(\frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z} + -1 \cdot t\right)\right) \cdot 2 \]

       6. associate-/l* N/A

          \[\leadsto \left(\left(-z\right) \cdot \left(c \cdot \frac{i \cdot \left(a + b \cdot c\right)}{z} + -1 \cdot t\right)\right) \cdot 2 \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(a + b \cdot c\right)}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       8. +-commutative N/A

          \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(b \cdot c + a\right)}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       9. *-commutative N/A

          \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(c \cdot b + a\right)}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       10. *-commutative N/A

           \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       11. lower-/.f64 N/A

           \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       12. lift-fma.f64 N/A

           \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \cdot 2 \]

       14. mul-1-neg N/A

           \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, \mathsf{neg}\left(t\right)\right)\right) \cdot 2 \]

       15. lower-neg.f64 65.5

           \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \cdot 2 \]

   13. Applied rewrites 65.5%

       \[\leadsto \left(\left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \cdot 2 \]

   if -4.9999999999999997e252 < (*.f64 (+.f64 a (*.f64 b c)) c) < 2e24

   1. Initial program 90.0%

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

   2. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - a \cdot \left(c \cdot i\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - a \cdot \left(c \cdot i\right)\right) \]

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(i \cdot c\right) \cdot a\right) \]

      8. lower-*.f64 74.4

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(i \cdot c\right) \cdot a\right) \]

   4. Applied rewrites 74.4%

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

   if 2e24 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in z around -inf

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, -1 \cdot \left(z \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(\frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z} + -1 \cdot t\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(c \cdot \frac{i \cdot \left(a + b \cdot c\right)}{z} + -1 \cdot t\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(a + b \cdot c\right)}{z}, -1 \cdot t\right)\right) \]

      8. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(b \cdot c + a\right)}{z}, -1 \cdot t\right)\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(c \cdot b + a\right)}{z}, -1 \cdot t\right)\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      12. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, \mathsf{neg}\left(t\right)\right)\right) \]

      15. lower-neg.f64 85.7

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \]

   7. Applied rewrites 85.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \]

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \cdot 2 \]

      7. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      8. +-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      9. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      10. lower-*.f64 N/A

          \[\leadsto \left(y \cdot x - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      11. lift-fma.f64 N/A

          \[\leadsto \left(y \cdot x - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      12. lift-*.f64 69.7

          \[\leadsto \left(y \cdot x - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

   10. Applied rewrites 69.7%

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

Alternative 5: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\\ t_2 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+252}:\\ \;\;\;\;-2 \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+24}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(i \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x - t\_1\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (fma c b a) i) c)) (t_2 (* (+ a (* b c)) c)))
   (if (<= t_2 -5e+252)
     (* -2.0 t_1)
     (if (<= t_2 2e+24)
       (* 2.0 (- (fma t z (* y x)) (* (* i c) a)))
       (* (- (* y x) t_1) 2.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (fma(c, b, a) * i) * c;
	double t_2 = (a + (b * c)) * c;
	double tmp;
	if (t_2 <= -5e+252) {
		tmp = -2.0 * t_1;
	} else if (t_2 <= 2e+24) {
		tmp = 2.0 * (fma(t, z, (y * x)) - ((i * c) * a));
	} else {
		tmp = ((y * x) - t_1) * 2.0;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(fma(c, b, a) * i) * c)
	t_2 = Float64(Float64(a + Float64(b * c)) * c)
	tmp = 0.0
	if (t_2 <= -5e+252)
		tmp = Float64(-2.0 * t_1);
	elseif (t_2 <= 2e+24)
		tmp = Float64(2.0 * Float64(fma(t, z, Float64(y * x)) - Float64(Float64(i * c) * a)));
	else
		tmp = Float64(Float64(Float64(y * x) - t_1) * 2.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+252], N[(-2.0 * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2e+24], N[(2.0 * N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] - N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * x), $MachinePrecision] - t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -4.9999999999999997e252

   1. Initial program 90.0%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      4. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      6. +-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      8. lower-fma.f64 47.3

         \[\leadsto -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 47.3%

      \[\leadsto \color{blue}{-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   if -4.9999999999999997e252 < (*.f64 (+.f64 a (*.f64 b c)) c) < 2e24

   1. Initial program 90.0%

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

   2. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - a \cdot \left(c \cdot i\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - a \cdot \left(c \cdot i\right)\right) \]

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(i \cdot c\right) \cdot a\right) \]

      8. lower-*.f64 74.4

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(i \cdot c\right) \cdot a\right) \]

   4. Applied rewrites 74.4%

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

   if 2e24 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in z around -inf

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, -1 \cdot \left(z \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(\frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z} + -1 \cdot t\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(c \cdot \frac{i \cdot \left(a + b \cdot c\right)}{z} + -1 \cdot t\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(a + b \cdot c\right)}{z}, -1 \cdot t\right)\right) \]

      8. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(b \cdot c + a\right)}{z}, -1 \cdot t\right)\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(c \cdot b + a\right)}{z}, -1 \cdot t\right)\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      12. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, \mathsf{neg}\left(t\right)\right)\right) \]

      15. lower-neg.f64 85.7

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \]

   7. Applied rewrites 85.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \]

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \cdot 2 \]

      7. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      8. +-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      9. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      10. lower-*.f64 N/A

          \[\leadsto \left(y \cdot x - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      11. lift-fma.f64 N/A

          \[\leadsto \left(y \cdot x - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      12. lift-*.f64 69.7

          \[\leadsto \left(y \cdot x - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

   10. Applied rewrites 69.7%

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

Alternative 6: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\\ t_2 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \left(t \cdot z - t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+24}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x - t\_1\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (fma c b a) i) c)) (t_2 (* (+ a (* b c)) c)))
   (if (<= t_2 -1e+134)
     (* 2.0 (- (* t z) t_1))
     (if (<= t_2 2e+24)
       (* 2.0 (fma y x (- (* t z) (* (* i a) c))))
       (* (- (* y x) t_1) 2.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (fma(c, b, a) * i) * c;
	double t_2 = (a + (b * c)) * c;
	double tmp;
	if (t_2 <= -1e+134) {
		tmp = 2.0 * ((t * z) - t_1);
	} else if (t_2 <= 2e+24) {
		tmp = 2.0 * fma(y, x, ((t * z) - ((i * a) * c)));
	} else {
		tmp = ((y * x) - t_1) * 2.0;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(fma(c, b, a) * i) * c)
	t_2 = Float64(Float64(a + Float64(b * c)) * c)
	tmp = 0.0
	if (t_2 <= -1e+134)
		tmp = Float64(2.0 * Float64(Float64(t * z) - t_1));
	elseif (t_2 <= 2e+24)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(t * z) - Float64(Float64(i * a) * c))));
	else
		tmp = Float64(Float64(Float64(y * x) - t_1) * 2.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+134], N[(2.0 * N[(N[(t * z), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+24], N[(2.0 * N[(y * x + N[(N[(t * z), $MachinePrecision] - N[(N[(i * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * x), $MachinePrecision] - t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -9.99999999999999921e133

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \color{blue}{c} \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      9. lower-fma.f64 69.0

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 69.0%

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   if -9.99999999999999921e133 < (*.f64 (+.f64 a (*.f64 b c)) c) < 2e24

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right) \]

      2. lower-*.f64 70.8

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right) \]

   7. Applied rewrites 70.8%

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right) \]

   if 2e24 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in z around -inf

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, -1 \cdot \left(z \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(\frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z} + -1 \cdot t\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(c \cdot \frac{i \cdot \left(a + b \cdot c\right)}{z} + -1 \cdot t\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(a + b \cdot c\right)}{z}, -1 \cdot t\right)\right) \]

      8. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(b \cdot c + a\right)}{z}, -1 \cdot t\right)\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(c \cdot b + a\right)}{z}, -1 \cdot t\right)\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      12. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, \mathsf{neg}\left(t\right)\right)\right) \]

      15. lower-neg.f64 85.7

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \]

   7. Applied rewrites 85.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \]

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \cdot 2 \]

      7. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      8. +-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      9. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      10. lower-*.f64 N/A

          \[\leadsto \left(y \cdot x - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      11. lift-fma.f64 N/A

          \[\leadsto \left(y \cdot x - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      12. lift-*.f64 69.7

          \[\leadsto \left(y \cdot x - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

   10. Applied rewrites 69.7%

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

Alternative 7: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, a\right) \cdot c\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+267}:\\ \;\;\;\;\left(t\_1 \cdot -2\right) \cdot i\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+90}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\left(t \cdot z - t\_1 \cdot i\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma c b a) c)) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -5e+267)
     (* (* t_1 -2.0) i)
     (if (<= t_2 2e+90)
       (* 2.0 (fma y x (- (* t z) (* (* i a) c))))
       (if (<= t_2 5e+279)
         (* (- (* t z) (* t_1 i)) 2.0)
         (* -2.0 (* (* (fma c b a) 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) {
	double t_1 = fma(c, b, a) * c;
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -5e+267) {
		tmp = (t_1 * -2.0) * i;
	} else if (t_2 <= 2e+90) {
		tmp = 2.0 * fma(y, x, ((t * z) - ((i * a) * c)));
	} else if (t_2 <= 5e+279) {
		tmp = ((t * z) - (t_1 * i)) * 2.0;
	} else {
		tmp = -2.0 * ((fma(c, b, a) * i) * c);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(c, b, a) * c)
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -5e+267)
		tmp = Float64(Float64(t_1 * -2.0) * i);
	elseif (t_2 <= 2e+90)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(t * z) - Float64(Float64(i * a) * c))));
	elseif (t_2 <= 5e+279)
		tmp = Float64(Float64(Float64(t * z) - Float64(t_1 * i)) * 2.0);
	else
		tmp = Float64(-2.0 * Float64(Float64(fma(c, b, a) * i) * c));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+267], N[(N[(t$95$1 * -2.0), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$2, 2e+90], N[(2.0 * N[(y * x + N[(N[(t * z), $MachinePrecision] - N[(N[(i * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+279], N[(N[(N[(t * z), $MachinePrecision] - N[(t$95$1 * i), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.9999999999999999e267

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(i \cdot \left(b \cdot c + \color{blue}{a}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(i \cdot \left(c \cdot b + a\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(c \cdot b + a\right) \cdot \color{blue}{i}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(\left(c \cdot b + a\right) \cdot i\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\color{blue}{\left(c \cdot b + a\right)} \cdot i\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \]

      8. lift-*.f64 47.3

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot \color{blue}{i}\right) \]

   7. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\color{blue}{\mathsf{fma}\left(c, b, a\right)} \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot \color{blue}{i}\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(c \cdot b + a\right) \cdot i\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(-2 \cdot c\right) \cdot \left(c \cdot b + a\right)\right) \cdot \color{blue}{i} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot c\right) \cdot \left(b \cdot c + a\right)\right) \cdot i \]

      7. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot c\right) \cdot \left(a + b \cdot c\right)\right) \cdot i \]

      8. associate-*r* N/A

         \[\leadsto \left(-2 \cdot \left(c \cdot \left(a + b \cdot c\right)\right)\right) \cdot i \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \left(c \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{i} \]

      10. *-commutative N/A

          \[\leadsto \left(\left(c \cdot \left(a + b \cdot c\right)\right) \cdot -2\right) \cdot i \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(c \cdot \left(a + b \cdot c\right)\right) \cdot -2\right) \cdot i \]

      12. +-commutative N/A

          \[\leadsto \left(\left(c \cdot \left(b \cdot c + a\right)\right) \cdot -2\right) \cdot i \]

      13. *-commutative N/A

          \[\leadsto \left(\left(c \cdot \left(c \cdot b + a\right)\right) \cdot -2\right) \cdot i \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\left(c \cdot b + a\right) \cdot c\right) \cdot -2\right) \cdot i \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(c \cdot b + a\right) \cdot c\right) \cdot -2\right) \cdot i \]

      16. lift-fma.f64 46.9

          \[\leadsto \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot -2\right) \cdot i \]

   9. Applied rewrites 46.9%

      \[\leadsto \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot -2\right) \cdot \color{blue}{i} \]

   if -4.9999999999999999e267 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999993e90

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right) \]

      2. lower-*.f64 70.8

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right) \]

   7. Applied rewrites 70.8%

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right) \]

   if 1.99999999999999993e90 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000002e279

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \cdot 2 \]

      3. +-commutative N/A

         \[\leadsto \left(t \cdot z - \left(i \cdot \left(b \cdot c + a\right)\right) \cdot c\right) \cdot 2 \]

      4. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(i \cdot \left(c \cdot b + a\right)\right) \cdot c\right) \cdot 2 \]

      5. *-commutative N/A

         \[\leadsto \left(t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot \color{blue}{2} \]

   7. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right) \cdot 2} \]

   if 5.0000000000000002e279 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      4. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      6. +-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      8. lower-fma.f64 47.3

         \[\leadsto -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 47.3%

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

Alternative 8: 84.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\\ t_2 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+252}:\\ \;\;\;\;-2 \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+24}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x - t\_1\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (fma c b a) i) c)) (t_2 (* (+ a (* b c)) c)))
   (if (<= t_2 -5e+252)
     (* -2.0 t_1)
     (if (<= t_2 2e+24)
       (* 2.0 (fma y x (- (* t z) (* (* i a) c))))
       (* (- (* y x) t_1) 2.0)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (fma(c, b, a) * i) * c;
	double t_2 = (a + (b * c)) * c;
	double tmp;
	if (t_2 <= -5e+252) {
		tmp = -2.0 * t_1;
	} else if (t_2 <= 2e+24) {
		tmp = 2.0 * fma(y, x, ((t * z) - ((i * a) * c)));
	} else {
		tmp = ((y * x) - t_1) * 2.0;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(fma(c, b, a) * i) * c)
	t_2 = Float64(Float64(a + Float64(b * c)) * c)
	tmp = 0.0
	if (t_2 <= -5e+252)
		tmp = Float64(-2.0 * t_1);
	elseif (t_2 <= 2e+24)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(t * z) - Float64(Float64(i * a) * c))));
	else
		tmp = Float64(Float64(Float64(y * x) - t_1) * 2.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+252], N[(-2.0 * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2e+24], N[(2.0 * N[(y * x + N[(N[(t * z), $MachinePrecision] - N[(N[(i * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * x), $MachinePrecision] - t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -4.9999999999999997e252

   1. Initial program 90.0%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      4. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      6. +-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      8. lower-fma.f64 47.3

         \[\leadsto -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 47.3%

      \[\leadsto \color{blue}{-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   if -4.9999999999999997e252 < (*.f64 (+.f64 a (*.f64 b c)) c) < 2e24

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right) \]

      2. lower-*.f64 70.8

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right) \]

   7. Applied rewrites 70.8%

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right) \]

   if 2e24 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in z around -inf

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, -1 \cdot \left(z \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-1 \cdot z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(-1 \cdot t + \frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(\frac{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}{z} + -1 \cdot t\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \left(c \cdot \frac{i \cdot \left(a + b \cdot c\right)}{z} + -1 \cdot t\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(a + b \cdot c\right)}{z}, -1 \cdot t\right)\right) \]

      8. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(b \cdot c + a\right)}{z}, -1 \cdot t\right)\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{i \cdot \left(c \cdot b + a\right)}{z}, -1 \cdot t\right)\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot b + a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      12. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -1 \cdot t\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, \mathsf{neg}\left(t\right)\right)\right) \]

      15. lower-neg.f64 85.7

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \]

   7. Applied rewrites 85.7%

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-z\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(c, b, a\right) \cdot i}{z}, -t\right)\right) \]

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{2} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2 \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \cdot 2 \]

      7. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      8. +-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      9. *-commutative N/A

         \[\leadsto \left(y \cdot x - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      10. lower-*.f64 N/A

          \[\leadsto \left(y \cdot x - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      11. lift-fma.f64 N/A

          \[\leadsto \left(y \cdot x - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

      12. lift-*.f64 69.7

          \[\leadsto \left(y \cdot x - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \cdot 2 \]

   10. Applied rewrites 69.7%

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

Alternative 9: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+267}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot -2\right) \cdot i\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+279}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -5e+267)
     (* (* (* (fma c b a) c) -2.0) i)
     (if (<= t_1 5e+279)
       (* 2.0 (fma y x (- (* t z) (* (* i a) c))))
       (* -2.0 (* (* (fma c b a) 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) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -5e+267) {
		tmp = ((fma(c, b, a) * c) * -2.0) * i;
	} else if (t_1 <= 5e+279) {
		tmp = 2.0 * fma(y, x, ((t * z) - ((i * a) * c)));
	} else {
		tmp = -2.0 * ((fma(c, b, a) * i) * c);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_1 <= -5e+267)
		tmp = Float64(Float64(Float64(fma(c, b, a) * c) * -2.0) * i);
	elseif (t_1 <= 5e+279)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(t * z) - Float64(Float64(i * a) * c))));
	else
		tmp = Float64(-2.0 * Float64(Float64(fma(c, b, a) * i) * c));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+267], N[(N[(N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$1, 5e+279], N[(2.0 * N[(y * x + N[(N[(t * z), $MachinePrecision] - N[(N[(i * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.9999999999999999e267

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(i \cdot \left(b \cdot c + \color{blue}{a}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(i \cdot \left(c \cdot b + a\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(c \cdot b + a\right) \cdot \color{blue}{i}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(\left(c \cdot b + a\right) \cdot i\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\color{blue}{\left(c \cdot b + a\right)} \cdot i\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \]

      8. lift-*.f64 47.3

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot \color{blue}{i}\right) \]

   7. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\color{blue}{\mathsf{fma}\left(c, b, a\right)} \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot \color{blue}{i}\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(c \cdot b + a\right) \cdot i\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(-2 \cdot c\right) \cdot \left(c \cdot b + a\right)\right) \cdot \color{blue}{i} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot c\right) \cdot \left(b \cdot c + a\right)\right) \cdot i \]

      7. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot c\right) \cdot \left(a + b \cdot c\right)\right) \cdot i \]

      8. associate-*r* N/A

         \[\leadsto \left(-2 \cdot \left(c \cdot \left(a + b \cdot c\right)\right)\right) \cdot i \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \left(c \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{i} \]

      10. *-commutative N/A

          \[\leadsto \left(\left(c \cdot \left(a + b \cdot c\right)\right) \cdot -2\right) \cdot i \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(c \cdot \left(a + b \cdot c\right)\right) \cdot -2\right) \cdot i \]

      12. +-commutative N/A

          \[\leadsto \left(\left(c \cdot \left(b \cdot c + a\right)\right) \cdot -2\right) \cdot i \]

      13. *-commutative N/A

          \[\leadsto \left(\left(c \cdot \left(c \cdot b + a\right)\right) \cdot -2\right) \cdot i \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\left(c \cdot b + a\right) \cdot c\right) \cdot -2\right) \cdot i \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(c \cdot b + a\right) \cdot c\right) \cdot -2\right) \cdot i \]

      16. lift-fma.f64 46.9

          \[\leadsto \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot -2\right) \cdot i \]

   9. Applied rewrites 46.9%

      \[\leadsto \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot -2\right) \cdot \color{blue}{i} \]

   if -4.9999999999999999e267 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000002e279

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right) \]

      2. lower-*.f64 70.8

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right) \]

   7. Applied rewrites 70.8%

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot a\right) \cdot c\right) \]

   if 5.0000000000000002e279 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      4. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      6. +-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      8. lower-fma.f64 47.3

         \[\leadsto -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 47.3%

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

Alternative 10: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot -2\right) \cdot i\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -1e+138)
     (* (* (* (fma c b a) c) -2.0) i)
     (if (<= t_1 5e+279)
       (* (fma t z (* y x)) 2.0)
       (* -2.0 (* (* (fma c b a) 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) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -1e+138) {
		tmp = ((fma(c, b, a) * c) * -2.0) * i;
	} else if (t_1 <= 5e+279) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = -2.0 * ((fma(c, b, a) * i) * c);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_1 <= -1e+138)
		tmp = Float64(Float64(Float64(fma(c, b, a) * c) * -2.0) * i);
	elseif (t_1 <= 5e+279)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = Float64(-2.0 * Float64(Float64(fma(c, b, a) * i) * c));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+138], N[(N[(N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$1, 5e+279], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1e138

   1. Initial program 90.0%

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

   2. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \left(\color{blue}{t \cdot z} - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)\right) \]

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      13. lower-fma.f64 92.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, x, t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(i \cdot \left(b \cdot c + \color{blue}{a}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(i \cdot \left(c \cdot b + a\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(c \cdot b + a\right) \cdot \color{blue}{i}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(\left(c \cdot b + a\right) \cdot i\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\color{blue}{\left(c \cdot b + a\right)} \cdot i\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \]

      8. lift-*.f64 47.3

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot \color{blue}{i}\right) \]

   7. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\color{blue}{\mathsf{fma}\left(c, b, a\right)} \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot \color{blue}{i}\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(c \cdot b + a\right) \cdot i\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(-2 \cdot c\right) \cdot \left(c \cdot b + a\right)\right) \cdot \color{blue}{i} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot c\right) \cdot \left(b \cdot c + a\right)\right) \cdot i \]

      7. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot c\right) \cdot \left(a + b \cdot c\right)\right) \cdot i \]

      8. associate-*r* N/A

         \[\leadsto \left(-2 \cdot \left(c \cdot \left(a + b \cdot c\right)\right)\right) \cdot i \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \left(c \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{i} \]

      10. *-commutative N/A

          \[\leadsto \left(\left(c \cdot \left(a + b \cdot c\right)\right) \cdot -2\right) \cdot i \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(c \cdot \left(a + b \cdot c\right)\right) \cdot -2\right) \cdot i \]

      12. +-commutative N/A

          \[\leadsto \left(\left(c \cdot \left(b \cdot c + a\right)\right) \cdot -2\right) \cdot i \]

      13. *-commutative N/A

          \[\leadsto \left(\left(c \cdot \left(c \cdot b + a\right)\right) \cdot -2\right) \cdot i \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\left(c \cdot b + a\right) \cdot c\right) \cdot -2\right) \cdot i \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(c \cdot b + a\right) \cdot c\right) \cdot -2\right) \cdot i \]

      16. lift-fma.f64 46.9

          \[\leadsto \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot -2\right) \cdot i \]

   9. Applied rewrites 46.9%

      \[\leadsto \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot -2\right) \cdot \color{blue}{i} \]

   if -1e138 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000002e279

   1. Initial program 90.0%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.1

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

   4. Applied rewrites 55.1%

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

   if 5.0000000000000002e279 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      4. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      6. +-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      8. lower-fma.f64 47.3

         \[\leadsto -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 47.3%

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

Alternative 11: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* -2.0 (* (* (fma c b a) i) c))) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -1e+138)
     t_1
     (if (<= t_2 5e+279) (* (fma t z (* y x)) 2.0) 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 = -2.0 * ((fma(c, b, a) * i) * c);
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -1e+138) {
		tmp = t_1;
	} else if (t_2 <= 5e+279) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} 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 = Float64(-2.0 * Float64(Float64(fma(c, b, a) * i) * c))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -1e+138)
		tmp = t_1;
	elseif (t_2 <= 5e+279)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	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[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+138], t$95$1, If[LessEqual[t$95$2, 5e+279], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1e138 or 5.0000000000000002e279 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      4. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      6. +-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      8. lower-fma.f64 47.3

         \[\leadsto -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   4. Applied rewrites 47.3%

      \[\leadsto \color{blue}{-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   if -1e138 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000002e279

   1. Initial program 90.0%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.1

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

   4. Applied rewrites 55.1%

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

Alternative 12: 73.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2\\ t_2 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+252}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (* (* c (* i c)) b) -2.0)) (t_2 (* (+ a (* b c)) c)))
   (if (<= t_2 -5e+252)
     t_1
     (if (<= t_2 5e+305) (* (fma t z (* y x)) 2.0) 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 = ((c * (i * c)) * b) * -2.0;
	double t_2 = (a + (b * c)) * c;
	double tmp;
	if (t_2 <= -5e+252) {
		tmp = t_1;
	} else if (t_2 <= 5e+305) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} 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 = Float64(Float64(Float64(c * Float64(i * c)) * b) * -2.0)
	t_2 = Float64(Float64(a + Float64(b * c)) * c)
	tmp = 0.0
	if (t_2 <= -5e+252)
		tmp = t_1;
	elseif (t_2 <= 5e+305)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	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[(N[(N[(c * N[(i * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+252], t$95$1, If[LessEqual[t$95$2, 5e+305], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -4.9999999999999997e252 or 5.00000000000000009e305 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.0%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      7. lower-*.f64 32.8

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

   4. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(c \cdot \left(c \cdot i\right)\right) \cdot b\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot \left(c \cdot i\right)\right) \cdot b\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

      6. lift-*.f64 33.8

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

   6. Applied rewrites 33.8%

      \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

   if -4.9999999999999997e252 < (*.f64 (+.f64 a (*.f64 b c)) c) < 5.00000000000000009e305

   1. Initial program 90.0%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.1

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

   4. Applied rewrites 55.1%

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

Alternative 13: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+302}:\\ \;\;\;\;\left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -5e+302)
     (* (* (* c c) (* i b)) -2.0)
     (if (<= t_1 5e+279)
       (* (fma t z (* y x)) 2.0)
       (* (* c (* c (* i b))) -2.0)))))

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 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -5e+302) {
		tmp = ((c * c) * (i * b)) * -2.0;
	} else if (t_1 <= 5e+279) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = (c * (c * (i * b))) * -2.0;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_1 <= -5e+302)
		tmp = Float64(Float64(Float64(c * c) * Float64(i * b)) * -2.0);
	elseif (t_1 <= 5e+279)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(c * Float64(c * Float64(i * b))) * -2.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+302], N[(N[(N[(c * c), $MachinePrecision] * N[(i * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 5e+279], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(c * N[(c * N[(i * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5e302

   1. Initial program 90.0%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      7. lower-*.f64 32.8

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

   4. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      4. pow2 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. associate-*l* N/A

         \[\leadsto \left({c}^{2} \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      6. *-commutative N/A

         \[\leadsto \left({c}^{2} \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \left({c}^{2} \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      8. pow2 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      10. *-commutative N/A

          \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      11. lower-*.f64 31.7

          \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

   6. Applied rewrites 31.7%

      \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

   if -5e302 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000002e279

   1. Initial program 90.0%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.1

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

   4. Applied rewrites 55.1%

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

   if 5.0000000000000002e279 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      7. lower-*.f64 32.8

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

   4. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      4. pow2 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. associate-*l* N/A

         \[\leadsto \left({c}^{2} \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      6. *-commutative N/A

         \[\leadsto \left({c}^{2} \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \left({c}^{2} \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      8. pow2 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      10. *-commutative N/A

          \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      11. lower-*.f64 31.7

          \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

   6. Applied rewrites 31.7%

      \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      4. associate-*l* N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(b \cdot i\right)\right)\right) \cdot -2 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(b \cdot i\right)\right)\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(b \cdot i\right)\right)\right) \cdot -2 \]

      8. *-commutative N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

      9. lift-*.f64 33.2

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

   8. Applied rewrites 33.2%

      \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 73.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+302}:\\ \;\;\;\;\left(c \cdot \left(\left(c \cdot b\right) \cdot i\right)\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -5e+302)
     (* (* c (* (* c b) i)) -2.0)
     (if (<= t_1 5e+279)
       (* (fma t z (* y x)) 2.0)
       (* (* c (* c (* i b))) -2.0)))))

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 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -5e+302) {
		tmp = (c * ((c * b) * i)) * -2.0;
	} else if (t_1 <= 5e+279) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} else {
		tmp = (c * (c * (i * b))) * -2.0;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_1 <= -5e+302)
		tmp = Float64(Float64(c * Float64(Float64(c * b) * i)) * -2.0);
	elseif (t_1 <= 5e+279)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(c * Float64(c * Float64(i * b))) * -2.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+302], N[(N[(c * N[(N[(c * b), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 5e+279], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(c * N[(c * N[(i * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5e302

   1. Initial program 90.0%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      7. lower-*.f64 32.8

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

   4. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      4. pow2 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. associate-*l* N/A

         \[\leadsto \left({c}^{2} \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      6. *-commutative N/A

         \[\leadsto \left({c}^{2} \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \left({c}^{2} \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      8. pow2 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      10. *-commutative N/A

          \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      11. lower-*.f64 31.7

          \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

   6. Applied rewrites 31.7%

      \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      4. associate-*l* N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(b \cdot i\right)\right)\right) \cdot -2 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(b \cdot i\right)\right)\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(b \cdot i\right)\right)\right) \cdot -2 \]

      8. *-commutative N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

      9. lift-*.f64 33.2

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

   8. Applied rewrites 33.2%

      \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

      3. associate-*r* N/A

         \[\leadsto \left(c \cdot \left(\left(c \cdot i\right) \cdot b\right)\right) \cdot -2 \]

      4. *-commutative N/A

         \[\leadsto \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \cdot -2 \]

      5. associate-*r* N/A

         \[\leadsto \left(c \cdot \left(\left(b \cdot c\right) \cdot i\right)\right) \cdot -2 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(\left(b \cdot c\right) \cdot i\right)\right) \cdot -2 \]

      7. *-commutative N/A

         \[\leadsto \left(c \cdot \left(\left(c \cdot b\right) \cdot i\right)\right) \cdot -2 \]

      8. lift-*.f64 33.2

         \[\leadsto \left(c \cdot \left(\left(c \cdot b\right) \cdot i\right)\right) \cdot -2 \]

   10. Applied rewrites 33.2%

       \[\leadsto \left(c \cdot \left(\left(c \cdot b\right) \cdot i\right)\right) \cdot -2 \]

   if -5e302 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000002e279

   1. Initial program 90.0%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.1

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

   4. Applied rewrites 55.1%

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

   if 5.0000000000000002e279 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      7. lower-*.f64 32.8

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

   4. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      4. pow2 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. associate-*l* N/A

         \[\leadsto \left({c}^{2} \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      6. *-commutative N/A

         \[\leadsto \left({c}^{2} \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \left({c}^{2} \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      8. pow2 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      10. *-commutative N/A

          \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      11. lower-*.f64 31.7

          \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

   6. Applied rewrites 31.7%

      \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      4. associate-*l* N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(b \cdot i\right)\right)\right) \cdot -2 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(b \cdot i\right)\right)\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(b \cdot i\right)\right)\right) \cdot -2 \]

      8. *-commutative N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

      9. lift-*.f64 33.2

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

   8. Applied rewrites 33.2%

      \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 71.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+302}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (* c (* c (* i b))) -2.0)) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -5e+302)
     t_1
     (if (<= t_2 5e+279) (* (fma t z (* y x)) 2.0) 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 = (c * (c * (i * b))) * -2.0;
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -5e+302) {
		tmp = t_1;
	} else if (t_2 <= 5e+279) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} 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 = Float64(Float64(c * Float64(c * Float64(i * b))) * -2.0)
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -5e+302)
		tmp = t_1;
	elseif (t_2 <= 5e+279)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	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[(N[(c * N[(c * N[(i * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+302], t$95$1, If[LessEqual[t$95$2, 5e+279], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5e302 or 5.0000000000000002e279 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      7. lower-*.f64 32.8

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

   4. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      4. pow2 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. associate-*l* N/A

         \[\leadsto \left({c}^{2} \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      6. *-commutative N/A

         \[\leadsto \left({c}^{2} \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \left({c}^{2} \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      8. pow2 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(b \cdot i\right)\right) \cdot -2 \]

      10. *-commutative N/A

          \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      11. lower-*.f64 31.7

          \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

   6. Applied rewrites 31.7%

      \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot c\right) \cdot \left(i \cdot b\right)\right) \cdot -2 \]

      4. associate-*l* N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(b \cdot i\right)\right)\right) \cdot -2 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(b \cdot i\right)\right)\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(b \cdot i\right)\right)\right) \cdot -2 \]

      8. *-commutative N/A

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

      9. lift-*.f64 33.2

         \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

   8. Applied rewrites 33.2%

      \[\leadsto \left(c \cdot \left(c \cdot \left(i \cdot b\right)\right)\right) \cdot -2 \]

   if -5e302 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000002e279

   1. Initial program 90.0%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.1

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

   4. Applied rewrites 55.1%

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

Alternative 16: 63.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+182}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (* (* i c) a) -2.0)) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -5e+153)
     t_1
     (if (<= t_2 4e+182) (* (fma t z (* y x)) 2.0) 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 = ((i * c) * a) * -2.0;
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -5e+153) {
		tmp = t_1;
	} else if (t_2 <= 4e+182) {
		tmp = fma(t, z, (y * x)) * 2.0;
	} 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 = Float64(Float64(Float64(i * c) * a) * -2.0)
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -5e+153)
		tmp = t_1;
	elseif (t_2 <= 4e+182)
		tmp = Float64(fma(t, z, Float64(y * x)) * 2.0);
	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[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+153], t$95$1, If[LessEqual[t$95$2, 4e+182], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.00000000000000018e153 or 4.0000000000000003e182 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

      6. lower-*.f64 25.9

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

   4. Applied rewrites 25.9%

      \[\leadsto \color{blue}{\left(\left(i \cdot c\right) \cdot a\right) \cdot -2} \]

   if -5.00000000000000018e153 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.0000000000000003e182

   1. Initial program 90.0%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.1

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

   4. Applied rewrites 55.1%

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

Alternative 17: 42.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-303}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 0.0005:\\ \;\;\;\;\left(x + x\right) \cdot y\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+182}:\\ \;\;\;\;\left(t + t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (* (* i c) a) -2.0)) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -5e+153)
     t_1
     (if (<= t_2 5e-303)
       (* (* t z) 2.0)
       (if (<= t_2 0.0005)
         (* (+ x x) y)
         (if (<= t_2 4e+182) (* (+ t 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 = ((i * c) * a) * -2.0;
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -5e+153) {
		tmp = t_1;
	} else if (t_2 <= 5e-303) {
		tmp = (t * z) * 2.0;
	} else if (t_2 <= 0.0005) {
		tmp = (x + x) * y;
	} else if (t_2 <= 4e+182) {
		tmp = (t + t) * z;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((i * c) * a) * (-2.0d0)
    t_2 = ((a + (b * c)) * c) * i
    if (t_2 <= (-5d+153)) then
        tmp = t_1
    else if (t_2 <= 5d-303) then
        tmp = (t * z) * 2.0d0
    else if (t_2 <= 0.0005d0) then
        tmp = (x + x) * y
    else if (t_2 <= 4d+182) then
        tmp = (t + t) * z
    else
        tmp = t_1
    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 t_1 = ((i * c) * a) * -2.0;
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -5e+153) {
		tmp = t_1;
	} else if (t_2 <= 5e-303) {
		tmp = (t * z) * 2.0;
	} else if (t_2 <= 0.0005) {
		tmp = (x + x) * y;
	} else if (t_2 <= 4e+182) {
		tmp = (t + t) * z;
	} else {
		tmp = t_1;
	}
	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):
	t_1 = ((i * c) * a) * -2.0
	t_2 = ((a + (b * c)) * c) * i
	tmp = 0
	if t_2 <= -5e+153:
		tmp = t_1
	elif t_2 <= 5e-303:
		tmp = (t * z) * 2.0
	elif t_2 <= 0.0005:
		tmp = (x + x) * y
	elif t_2 <= 4e+182:
		tmp = (t + 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 = Float64(Float64(Float64(i * c) * a) * -2.0)
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -5e+153)
		tmp = t_1;
	elseif (t_2 <= 5e-303)
		tmp = Float64(Float64(t * z) * 2.0);
	elseif (t_2 <= 0.0005)
		tmp = Float64(Float64(x + x) * y);
	elseif (t_2 <= 4e+182)
		tmp = Float64(Float64(t + t) * z);
	else
		tmp = t_1;
	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)
	t_1 = ((i * c) * a) * -2.0;
	t_2 = ((a + (b * c)) * c) * i;
	tmp = 0.0;
	if (t_2 <= -5e+153)
		tmp = t_1;
	elseif (t_2 <= 5e-303)
		tmp = (t * z) * 2.0;
	elseif (t_2 <= 0.0005)
		tmp = (x + x) * y;
	elseif (t_2 <= 4e+182)
		tmp = (t + t) * z;
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+153], t$95$1, If[LessEqual[t$95$2, 5e-303], N[(N[(t * z), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 0.0005], N[(N[(x + x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$2, 4e+182], N[(N[(t + t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.00000000000000018e153 or 4.0000000000000003e182 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

      6. lower-*.f64 25.9

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

   4. Applied rewrites 25.9%

      \[\leadsto \color{blue}{\left(\left(i \cdot c\right) \cdot a\right) \cdot -2} \]

   if -5.00000000000000018e153 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.9999999999999998e-303

   1. Initial program 90.0%

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

   2. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

         \[\leadsto \left(t + t\right) \cdot z \]

      4. lower-+.f64 29.1

         \[\leadsto \left(t + t\right) \cdot z \]

   4. Applied rewrites 29.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

         \[\leadsto \left(t + t\right) \cdot z \]

      3. count-2-rev N/A

         \[\leadsto \left(2 \cdot t\right) \cdot z \]

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 29.1

         \[\leadsto \left(t \cdot z\right) \cdot 2 \]

   6. Applied rewrites 29.1%

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

   if 4.9999999999999998e-303 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000001e-4

   1. Initial program 90.0%

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

   2. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 29.3

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

   4. Applied rewrites 29.3%

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

   if 5.0000000000000001e-4 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.0000000000000003e182

   1. Initial program 90.0%

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

   2. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

         \[\leadsto \left(t + t\right) \cdot z \]

      4. lower-+.f64 29.1

         \[\leadsto \left(t + t\right) \cdot z \]

   4. Applied rewrites 29.1%

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

Alternative 18: 41.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \left(t + t\right) \cdot z\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+280}:\\ \;\;\;\;\left(x + x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (+ t t) z)))
   (if (<= (* z t) -1e-25) t_1 (if (<= (* z t) 2e+280) (* (+ x x) y) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (t + t) * z;
	double tmp;
	if ((z * t) <= -1e-25) {
		tmp = t_1;
	} else if ((z * t) <= 2e+280) {
		tmp = (x + x) * y;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (t + t) * z
    if ((z * t) <= (-1d-25)) then
        tmp = t_1
    else if ((z * t) <= 2d+280) then
        tmp = (x + x) * y
    else
        tmp = t_1
    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 t_1 = (t + t) * z;
	double tmp;
	if ((z * t) <= -1e-25) {
		tmp = t_1;
	} else if ((z * t) <= 2e+280) {
		tmp = (x + x) * y;
	} else {
		tmp = t_1;
	}
	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):
	t_1 = (t + t) * z
	tmp = 0
	if (z * t) <= -1e-25:
		tmp = t_1
	elif (z * t) <= 2e+280:
		tmp = (x + x) * y
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(t + t) * z)
	tmp = 0.0
	if (Float64(z * t) <= -1e-25)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+280)
		tmp = Float64(Float64(x + x) * y);
	else
		tmp = t_1;
	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)
	t_1 = (t + t) * z;
	tmp = 0.0;
	if ((z * t) <= -1e-25)
		tmp = t_1;
	elseif ((z * t) <= 2e+280)
		tmp = (x + x) * y;
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(N[(t + t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e-25], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+280], N[(N[(x + x), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.00000000000000004e-25 or 2.0000000000000001e280 < (*.f64 z t)

   1. Initial program 90.0%

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

   2. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

         \[\leadsto \left(t + t\right) \cdot z \]

      4. lower-+.f64 29.1

         \[\leadsto \left(t + t\right) \cdot z \]

   4. Applied rewrites 29.1%

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

   if -1.00000000000000004e-25 < (*.f64 z t) < 2.0000000000000001e280

   1. Initial program 90.0%

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

   2. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 29.3

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

   4. Applied rewrites 29.3%

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

Alternative 19: 29.1% accurate, 4.0× speedup

Math

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

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 (* (+ t t) z))

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 (t + t) * z;
}

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 = (t + t) * z
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 (t + t) * z;
}

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 (t + t) * z

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(Float64(t + t) * z)
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 = (t + t) * z;
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[(N[(t + t), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 90.0%

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

2. Taylor expanded in z around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. count-2-rev N/A

      \[\leadsto \left(t + t\right) \cdot z \]

   4. lower-+.f64 29.1

      \[\leadsto \left(t + t\right) \cdot z \]

4. Applied rewrites 29.1%

   \[\leadsto \color{blue}{\left(t + t\right) \cdot z} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025135 
(FPCore (x y z t a b c i)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))