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

Percentage Accurate: 90.3% → 96.8%

Time: 18.2s

Alternatives: 21

Speedup: 1.1×

• 46.4% 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

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = 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: 90.3% 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

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = 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: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot y + z \cdot t\right) - \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+272}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), \mathsf{fma}\left(z, t, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(i \cdot \mathsf{fma}\left(c, b, a\right)\right) \cdot \left(-c\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (- (+ (* x y) (* z t)) (* (* c (+ a (* b c))) i)) 1e+272)
   (* 2.0 (fma (fma b c a) (* c (- i)) (fma z t (* x y))))
   (* 2.0 (fma z t (fma y x (* (* i (fma c b a)) (- c)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((((x * y) + (z * t)) - ((c * (a + (b * c))) * i)) <= 1e+272) {
		tmp = 2.0 * fma(fma(b, c, a), (c * -i), fma(z, t, (x * y)));
	} else {
		tmp = 2.0 * fma(z, t, fma(y, x, ((i * fma(c, b, a)) * -c)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(c * Float64(a + Float64(b * c))) * i)) <= 1e+272)
		tmp = Float64(2.0 * fma(fma(b, c, a), Float64(c * Float64(-i)), fma(z, t, Float64(x * y))));
	else
		tmp = Float64(2.0 * fma(z, t, fma(y, x, Float64(Float64(i * fma(c, b, a)) * Float64(-c)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], 1e+272], N[(2.0 * N[(N[(b * c + a), $MachinePrecision] * N[(c * (-i)), $MachinePrecision] + N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(z * t + N[(y * x + N[(N[(i * N[(c * b + a), $MachinePrecision]), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)) < 1.0000000000000001e272

   1. Initial program 98.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 99.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   if 1.0000000000000001e272 < (-.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i))

   1. Initial program 77.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 90.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lift-*.f64 N/A

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

      14. associate-+l+ N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

   6. Applied rewrites 87.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-*l* N/A

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

      17. lower-neg.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 98.9

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

   8. Applied rewrites 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y + z \cdot t\right) - \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+272}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), \mathsf{fma}\left(z, t, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(i \cdot \mathsf{fma}\left(c, b, a\right)\right) \cdot \left(-c\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(-i\right)\\ t_2 := 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), t\_1, x \cdot y\right)\\ t_3 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{-179}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(b \cdot c, t\_1, \mathsf{fma}\left(z, t, x \cdot y\right)\right)\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* c (- i)))
        (t_2 (* 2.0 (fma (fma b c a) t_1 (* x y))))
        (t_3 (* (* c (+ a (* b c))) i)))
   (if (<= t_3 -5e+27)
     t_2
     (if (<= t_3 1e-179)
       (* 2.0 (fma (* b c) t_1 (fma z t (* x y))))
       (if (<= t_3 5e-18) (* 2.0 (- (* z t) (* c (* i (fma b c a))))) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = c * -i;
	double t_2 = 2.0 * fma(fma(b, c, a), t_1, (x * y));
	double t_3 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_3 <= -5e+27) {
		tmp = t_2;
	} else if (t_3 <= 1e-179) {
		tmp = 2.0 * fma((b * c), t_1, fma(z, t, (x * y)));
	} else if (t_3 <= 5e-18) {
		tmp = 2.0 * ((z * t) - (c * (i * fma(b, c, a))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(c * Float64(-i))
	t_2 = Float64(2.0 * fma(fma(b, c, a), t_1, Float64(x * y)))
	t_3 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_3 <= -5e+27)
		tmp = t_2;
	elseif (t_3 <= 1e-179)
		tmp = Float64(2.0 * fma(Float64(b * c), t_1, fma(z, t, Float64(x * y))));
	elseif (t_3 <= 5e-18)
		tmp = Float64(2.0 * Float64(Float64(z * t) - Float64(c * Float64(i * fma(b, c, a)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(c * (-i)), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(b * c + a), $MachinePrecision] * t$95$1 + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+27], t$95$2, If[LessEqual[t$95$3, 1e-179], N[(2.0 * N[(N[(b * c), $MachinePrecision] * t$95$1 + N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e-18], N[(2.0 * N[(N[(z * t), $MachinePrecision] - N[(c * N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.99999999999999979e27 or 5.00000000000000036e-18 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 86.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 95.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 90.9

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

   7. Applied rewrites 90.9%

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

   if -4.99999999999999979e27 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e-179

   1. Initial program 97.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 97.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if 1e-179 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.00000000000000036e-18

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), x \cdot y\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{-179}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(b \cdot c, c \cdot \left(-i\right), \mathsf{fma}\left(z, t, x \cdot y\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 5 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), x \cdot y\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-179}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(i \cdot \left(b \cdot c\right)\right) \cdot \left(-c\right)\right)\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (fma (fma b c a) (* c (- i)) (* x y))))
        (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -5e+27)
     t_1
     (if (<= t_2 1e-179)
       (* 2.0 (fma z t (fma y x (* (* i (* b c)) (- c)))))
       (if (<= t_2 5e-18) (* 2.0 (- (* z t) (* c (* i (fma b c a))))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * fma(fma(b, c, a), (c * -i), (x * y));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -5e+27) {
		tmp = t_1;
	} else if (t_2 <= 1e-179) {
		tmp = 2.0 * fma(z, t, fma(y, x, ((i * (b * c)) * -c)));
	} else if (t_2 <= 5e-18) {
		tmp = 2.0 * ((z * t) - (c * (i * fma(b, c, a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(2.0 * fma(fma(b, c, a), Float64(c * Float64(-i)), Float64(x * y)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -5e+27)
		tmp = t_1;
	elseif (t_2 <= 1e-179)
		tmp = Float64(2.0 * fma(z, t, fma(y, x, Float64(Float64(i * Float64(b * c)) * Float64(-c)))));
	elseif (t_2 <= 5e-18)
		tmp = Float64(2.0 * Float64(Float64(z * t) - Float64(c * Float64(i * fma(b, c, a)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[(N[(b * c + a), $MachinePrecision] * N[(c * (-i)), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+27], t$95$1, If[LessEqual[t$95$2, 1e-179], N[(2.0 * N[(z * t + N[(y * x + N[(N[(i * N[(b * c), $MachinePrecision]), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-18], N[(2.0 * N[(N[(z * t), $MachinePrecision] - N[(c * N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.99999999999999979e27 or 5.00000000000000036e-18 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 86.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 95.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 90.9

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

   7. Applied rewrites 90.9%

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

   if -4.99999999999999979e27 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e-179

   1. Initial program 97.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 97.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lift-*.f64 N/A

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

      14. associate-+l+ N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-*l* N/A

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

      17. lower-neg.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 98.8

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in c around inf

      \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \mathsf{neg}\left(c \cdot \color{blue}{\left(b \cdot \left(c \cdot i\right)\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 96.1

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

   11. Applied rewrites 96.1%

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

   if 1e-179 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.00000000000000036e-18

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), x \cdot y\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{-179}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(i \cdot \left(b \cdot c\right)\right) \cdot \left(-c\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 5 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), x \cdot y\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-179}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (fma (fma b c a) (* c (- i)) (* x y))))
        (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -5e+27)
     t_1
     (if (<= t_2 1e-179)
       (* 2.0 (fma t z (* x y)))
       (if (<= t_2 5e-18) (* 2.0 (- (* z t) (* c (* i (fma b c a))))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * fma(fma(b, c, a), (c * -i), (x * y));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -5e+27) {
		tmp = t_1;
	} else if (t_2 <= 1e-179) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else if (t_2 <= 5e-18) {
		tmp = 2.0 * ((z * t) - (c * (i * fma(b, c, a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(2.0 * fma(fma(b, c, a), Float64(c * Float64(-i)), Float64(x * y)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -5e+27)
		tmp = t_1;
	elseif (t_2 <= 1e-179)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	elseif (t_2 <= 5e-18)
		tmp = Float64(2.0 * Float64(Float64(z * t) - Float64(c * Float64(i * fma(b, c, a)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[(N[(b * c + a), $MachinePrecision] * N[(c * (-i)), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+27], t$95$1, If[LessEqual[t$95$2, 1e-179], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-18], N[(2.0 * N[(N[(z * t), $MachinePrecision] - N[(c * N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.99999999999999979e27 or 5.00000000000000036e-18 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 86.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 95.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 90.9

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

   7. Applied rewrites 90.9%

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

   if -4.99999999999999979e27 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e-179

   1. Initial program 97.7%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if 1e-179 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.00000000000000036e-18

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), x \cdot y\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{-179}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 5 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;t\_2 \leq 10^{-179}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \left(z \cdot t - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot y - t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* c (* i (fma b c a)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -2e+95)
     (* (fma c b a) (* i (* c -2.0)))
     (if (<= t_2 1e-179)
       (* 2.0 (fma t z (* x y)))
       (if (<= t_2 5e-18) (* 2.0 (- (* z t) t_1)) (* 2.0 (- (* x y) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = c * (i * fma(b, c, a));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -2e+95) {
		tmp = fma(c, b, a) * (i * (c * -2.0));
	} else if (t_2 <= 1e-179) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else if (t_2 <= 5e-18) {
		tmp = 2.0 * ((z * t) - t_1);
	} else {
		tmp = 2.0 * ((x * y) - t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(c * Float64(i * fma(b, c, a)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -2e+95)
		tmp = Float64(fma(c, b, a) * Float64(i * Float64(c * -2.0)));
	elseif (t_2 <= 1e-179)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	elseif (t_2 <= 5e-18)
		tmp = Float64(2.0 * Float64(Float64(z * t) - t_1));
	else
		tmp = Float64(2.0 * Float64(Float64(x * y) - t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(c * N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+95], N[(N[(c * b + a), $MachinePrecision] * N[(i * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-179], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-18], N[(2.0 * N[(N[(z * t), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.00000000000000004e95

   1. Initial program 83.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot \left({c}^{2} \cdot i\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(\color{blue}{\left({c}^{2} \cdot i\right) \cdot b}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(i \cdot {c}^{2}\right)} \cdot b\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(\color{blue}{i \cdot \left({c}^{2} \cdot b\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(i \cdot \color{blue}{\left(b \cdot {c}^{2}\right)}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto 2 \cdot \color{blue}{\left(i \cdot \left(\mathsf{neg}\left(b \cdot {c}^{2}\right)\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(-1 \cdot \left(b \cdot {c}^{2}\right)\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(i \cdot \left(-1 \cdot \left(b \cdot {c}^{2}\right)\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot {c}^{2}\right)\right)}\right) \]

      10. distribute-rgt-neg-out N/A

          \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(b \cdot \left(\mathsf{neg}\left({c}^{2}\right)\right)\right)}\right) \]

      11. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(-1 \cdot {c}^{2}\right)}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(b \cdot \left(-1 \cdot {c}^{2}\right)\right)}\right) \]

      13. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left({c}^{2}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \left(\mathsf{neg}\left(\color{blue}{c \cdot c}\right)\right)\right)\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)}\right)\right) \]

      16. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot c\right)}\right)\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(c \cdot \left(-1 \cdot c\right)\right)}\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \left(c \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)}\right)\right)\right) \]

      19. lower-neg.f64 50.0

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

   5. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(i \cdot \left(b \cdot \left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot \left(b \cdot \left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \cdot 2} \]

      3. lower-*.f64 50.0

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 80.2

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

   10. Applied rewrites 80.2%

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

   12. Applied rewrites 88.9%

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

   if -2.00000000000000004e95 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e-179

   1. Initial program 97.8%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if 1e-179 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.00000000000000036e-18

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 5.00000000000000036e-18 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 88.6%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 85.4

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

   5. Applied rewrites 85.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{-179}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 5 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(b \cdot \left(i \cdot \left(c \cdot -2\right)\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+286}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+284}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* c (* b (* i (* c -2.0))))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -2e+286)
     t_1
     (if (<= t_2 2e+161)
       (* 2.0 (fma t z (* x y)))
       (if (<= t_2 1e+284) (* a (* -2.0 (* c i))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = c * (b * (i * (c * -2.0)));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -2e+286) {
		tmp = t_1;
	} else if (t_2 <= 2e+161) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else if (t_2 <= 1e+284) {
		tmp = a * (-2.0 * (c * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(c * Float64(b * Float64(i * Float64(c * -2.0))))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -2e+286)
		tmp = t_1;
	elseif (t_2 <= 2e+161)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	elseif (t_2 <= 1e+284)
		tmp = Float64(a * Float64(-2.0 * Float64(c * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(c * N[(b * N[(i * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+286], t$95$1, If[LessEqual[t$95$2, 2e+161], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+284], N[(a * N[(-2.0 * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.00000000000000007e286 or 1.00000000000000008e284 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 79.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 68.4%

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

   9. Applied rewrites 72.9%

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

   if -2.00000000000000007e286 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 98.5%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000008e284

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 10.5

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

   5. Applied rewrites 10.5%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 62.2

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

   8. Applied rewrites 62.2%

      \[\leadsto \color{blue}{a \cdot \left(\left(c \cdot i\right) \cdot -2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+286}:\\ \;\;\;\;c \cdot \left(b \cdot \left(i \cdot \left(c \cdot -2\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+284}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(b \cdot \left(i \cdot \left(c \cdot -2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+286}:\\ \;\;\;\;c \cdot \left(c \cdot \left(i \cdot \left(b \cdot -2\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+284}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot -2\right) \cdot \left(c \cdot \left(b \cdot c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 -2e+286)
     (* c (* c (* i (* b -2.0))))
     (if (<= t_1 2e+161)
       (* 2.0 (fma t z (* x y)))
       (if (<= t_1 1e+284)
         (* a (* -2.0 (* c i)))
         (* (* i -2.0) (* c (* b c))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_1 <= -2e+286) {
		tmp = c * (c * (i * (b * -2.0)));
	} else if (t_1 <= 2e+161) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else if (t_1 <= 1e+284) {
		tmp = a * (-2.0 * (c * i));
	} else {
		tmp = (i * -2.0) * (c * (b * c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_1 <= -2e+286)
		tmp = Float64(c * Float64(c * Float64(i * Float64(b * -2.0))));
	elseif (t_1 <= 2e+161)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	elseif (t_1 <= 1e+284)
		tmp = Float64(a * Float64(-2.0 * Float64(c * i)));
	else
		tmp = Float64(Float64(i * -2.0) * Float64(c * Float64(b * c)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+286], N[(c * N[(c * N[(i * N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+161], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+284], N[(a * N[(-2.0 * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * -2.0), $MachinePrecision] * N[(c * N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.00000000000000007e286

   1. Initial program 78.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 8.4

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

   5. Applied rewrites 8.4%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   if -2.00000000000000007e286 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 98.5%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000008e284

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 10.5

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

   5. Applied rewrites 10.5%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 62.2

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

   8. Applied rewrites 62.2%

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

   if 1.00000000000000008e284 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

   7. Applied rewrites 83.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+286}:\\ \;\;\;\;c \cdot \left(c \cdot \left(i \cdot \left(b \cdot -2\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+284}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot -2\right) \cdot \left(c \cdot \left(b \cdot c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+286}:\\ \;\;\;\;c \cdot \left(c \cdot \left(i \cdot \left(b \cdot -2\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+284}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 -2e+286)
     (* c (* c (* i (* b -2.0))))
     (if (<= t_1 2e+161)
       (* 2.0 (fma t z (* x y)))
       (if (<= t_1 1e+284)
         (* a (* -2.0 (* c i)))
         (* b (* i (* -2.0 (* c c)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_1 <= -2e+286) {
		tmp = c * (c * (i * (b * -2.0)));
	} else if (t_1 <= 2e+161) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else if (t_1 <= 1e+284) {
		tmp = a * (-2.0 * (c * i));
	} else {
		tmp = b * (i * (-2.0 * (c * c)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_1 <= -2e+286)
		tmp = Float64(c * Float64(c * Float64(i * Float64(b * -2.0))));
	elseif (t_1 <= 2e+161)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	elseif (t_1 <= 1e+284)
		tmp = Float64(a * Float64(-2.0 * Float64(c * i)));
	else
		tmp = Float64(b * Float64(i * Float64(-2.0 * Float64(c * c))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+286], N[(c * N[(c * N[(i * N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+161], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+284], N[(a * N[(-2.0 * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(i * N[(-2.0 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.00000000000000007e286

   1. Initial program 78.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 8.4

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

   5. Applied rewrites 8.4%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   if -2.00000000000000007e286 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 98.5%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000008e284

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 10.5

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

   5. Applied rewrites 10.5%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 62.2

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

   8. Applied rewrites 62.2%

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

   if 1.00000000000000008e284 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+286}:\\ \;\;\;\;c \cdot \left(c \cdot \left(i \cdot \left(b \cdot -2\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+284}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+286}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+284}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (* i (* -2.0 (* c c))))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -2e+286)
     t_1
     (if (<= t_2 2e+161)
       (* 2.0 (fma t z (* x y)))
       (if (<= t_2 1e+284) (* a (* -2.0 (* c i))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = b * (i * (-2.0 * (c * c)));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -2e+286) {
		tmp = t_1;
	} else if (t_2 <= 2e+161) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else if (t_2 <= 1e+284) {
		tmp = a * (-2.0 * (c * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(b * Float64(i * Float64(-2.0 * Float64(c * c))))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -2e+286)
		tmp = t_1;
	elseif (t_2 <= 2e+161)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	elseif (t_2 <= 1e+284)
		tmp = Float64(a * Float64(-2.0 * Float64(c * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * N[(i * N[(-2.0 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+286], t$95$1, If[LessEqual[t$95$2, 2e+161], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+284], N[(a * N[(-2.0 * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.00000000000000007e286 or 1.00000000000000008e284 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 79.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -2.00000000000000007e286 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 98.5%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000008e284

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 10.5

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

   5. Applied rewrites 10.5%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 62.2

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

   8. Applied rewrites 62.2%

      \[\leadsto \color{blue}{a \cdot \left(\left(c \cdot i\right) \cdot -2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+286}:\\ \;\;\;\;b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+284}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-179}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 -2e+95)
     (* (fma c b a) (* i (* c -2.0)))
     (if (<= t_1 1e-179)
       (* 2.0 (fma t z (* x y)))
       (* 2.0 (- (* z t) (* c (* i (fma b c a)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_1 <= -2e+95) {
		tmp = fma(c, b, a) * (i * (c * -2.0));
	} else if (t_1 <= 1e-179) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = 2.0 * ((z * t) - (c * (i * fma(b, c, a))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_1 <= -2e+95)
		tmp = Float64(fma(c, b, a) * Float64(i * Float64(c * -2.0)));
	elseif (t_1 <= 1e-179)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = Float64(2.0 * Float64(Float64(z * t) - Float64(c * Float64(i * fma(b, c, a)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+95], N[(N[(c * b + a), $MachinePrecision] * N[(i * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-179], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(z * t), $MachinePrecision] - N[(c * N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.00000000000000004e95

   1. Initial program 83.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot \left({c}^{2} \cdot i\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(\color{blue}{\left({c}^{2} \cdot i\right) \cdot b}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(i \cdot {c}^{2}\right)} \cdot b\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(\color{blue}{i \cdot \left({c}^{2} \cdot b\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(i \cdot \color{blue}{\left(b \cdot {c}^{2}\right)}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto 2 \cdot \color{blue}{\left(i \cdot \left(\mathsf{neg}\left(b \cdot {c}^{2}\right)\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(-1 \cdot \left(b \cdot {c}^{2}\right)\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(i \cdot \left(-1 \cdot \left(b \cdot {c}^{2}\right)\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot {c}^{2}\right)\right)}\right) \]

      10. distribute-rgt-neg-out N/A

          \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(b \cdot \left(\mathsf{neg}\left({c}^{2}\right)\right)\right)}\right) \]

      11. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(-1 \cdot {c}^{2}\right)}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(b \cdot \left(-1 \cdot {c}^{2}\right)\right)}\right) \]

      13. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left({c}^{2}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \left(\mathsf{neg}\left(\color{blue}{c \cdot c}\right)\right)\right)\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)}\right)\right) \]

      16. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot c\right)}\right)\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(c \cdot \left(-1 \cdot c\right)\right)}\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \left(c \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)}\right)\right)\right) \]

      19. lower-neg.f64 50.0

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

   5. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(i \cdot \left(b \cdot \left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot \left(b \cdot \left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \cdot 2} \]

      3. lower-*.f64 50.0

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 80.2

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

   10. Applied rewrites 80.2%

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

   12. Applied rewrites 88.9%

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

   if -2.00000000000000004e95 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e-179

   1. Initial program 97.8%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if 1e-179 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 91.1%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 80.5

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

   5. Applied rewrites 80.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{-179}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot y + z \cdot t\right) - \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+272}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \left(-i\right) \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(i \cdot \mathsf{fma}\left(c, b, a\right)\right) \cdot \left(-c\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (- (+ (* x y) (* z t)) (* (* c (+ a (* b c))) i)) 1e+272)
   (* 2.0 (fma z t (fma x y (* (- i) (* c (fma b c a))))))
   (* 2.0 (fma z t (fma y x (* (* i (fma c b a)) (- c)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((((x * y) + (z * t)) - ((c * (a + (b * c))) * i)) <= 1e+272) {
		tmp = 2.0 * fma(z, t, fma(x, y, (-i * (c * fma(b, c, a)))));
	} else {
		tmp = 2.0 * fma(z, t, fma(y, x, ((i * fma(c, b, a)) * -c)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(c * Float64(a + Float64(b * c))) * i)) <= 1e+272)
		tmp = Float64(2.0 * fma(z, t, fma(x, y, Float64(Float64(-i) * Float64(c * fma(b, c, a))))));
	else
		tmp = Float64(2.0 * fma(z, t, fma(y, x, Float64(Float64(i * fma(c, b, a)) * Float64(-c)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], 1e+272], N[(2.0 * N[(z * t + N[(x * y + N[((-i) * N[(c * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(z * t + N[(y * x + N[(N[(i * N[(c * b + a), $MachinePrecision]), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)) < 1.0000000000000001e272

   1. Initial program 98.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 99.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lift-*.f64 N/A

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

      14. associate-+l+ N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

   6. Applied rewrites 98.7%

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

   if 1.0000000000000001e272 < (-.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i))

   1. Initial program 77.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 90.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lift-*.f64 N/A

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

      14. associate-+l+ N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

   6. Applied rewrites 87.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-*l* N/A

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

      17. lower-neg.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 98.9

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

   8. Applied rewrites 98.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y + z \cdot t\right) - \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+272}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \left(-i\right) \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \left(i \cdot \mathsf{fma}\left(c, b, a\right)\right) \cdot \left(-c\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 94.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (- (+ (* x y) (* z t)) (* (* c (+ a (* b c))) i)) INFINITY)
   (* 2.0 (fma y x (- (* z t) (* c (* i (fma b c a))))))
   (* 2.0 (fma z t (fma y x (* (* i (* b c)) (- c)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((((x * y) + (z * t)) - ((c * (a + (b * c))) * i)) <= ((double) INFINITY)) {
		tmp = 2.0 * fma(y, x, ((z * t) - (c * (i * fma(b, c, a)))));
	} else {
		tmp = 2.0 * fma(z, t, fma(y, x, ((i * (b * c)) * -c)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(c * Float64(a + Float64(b * c))) * i)) <= Inf)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(z * t) - Float64(c * Float64(i * fma(b, c, a))))));
	else
		tmp = Float64(2.0 * fma(z, t, fma(y, x, Float64(Float64(i * Float64(b * c)) * Float64(-c)))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 99.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. lift-*.f64 N/A

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

      18. +-commutative N/A

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

   6. Applied rewrites 95.6%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 18.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 45.5

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

   4. Applied rewrites 45.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lift-*.f64 N/A

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

      14. associate-+l+ N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

   6. Applied rewrites 81.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-*l* N/A

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

      17. lower-neg.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 90.9

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

   8. Applied rewrites 90.9%

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

   9. Taylor expanded in c around inf

      \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \mathsf{neg}\left(c \cdot \color{blue}{\left(b \cdot \left(c \cdot i\right)\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 90.9

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

   11. Applied rewrites 90.9%

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

4. Final simplification 95.4%

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

Alternative 13: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma c b a) (* i (* c -2.0)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -2e+95)
     t_1
     (if (<= t_2 2e+161) (* 2.0 (fma t z (* x y))) t_1))))

C

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 * -2.0));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -2e+95) {
		tmp = t_1;
	} else if (t_2 <= 2e+161) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(c, b, a) * Float64(i * Float64(c * -2.0)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -2e+95)
		tmp = t_1;
	elseif (t_2 <= 2e+161)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * b + a), $MachinePrecision] * N[(i * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+95], t$95$1, If[LessEqual[t$95$2, 2e+161], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.00000000000000004e95 or 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 84.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot \left({c}^{2} \cdot i\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(\color{blue}{\left({c}^{2} \cdot i\right) \cdot b}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(i \cdot {c}^{2}\right)} \cdot b\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(\color{blue}{i \cdot \left({c}^{2} \cdot b\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(i \cdot \color{blue}{\left(b \cdot {c}^{2}\right)}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto 2 \cdot \color{blue}{\left(i \cdot \left(\mathsf{neg}\left(b \cdot {c}^{2}\right)\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(-1 \cdot \left(b \cdot {c}^{2}\right)\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(i \cdot \left(-1 \cdot \left(b \cdot {c}^{2}\right)\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot {c}^{2}\right)\right)}\right) \]

      10. distribute-rgt-neg-out N/A

          \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(b \cdot \left(\mathsf{neg}\left({c}^{2}\right)\right)\right)}\right) \]

      11. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(-1 \cdot {c}^{2}\right)}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(b \cdot \left(-1 \cdot {c}^{2}\right)\right)}\right) \]

      13. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left({c}^{2}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \left(\mathsf{neg}\left(\color{blue}{c \cdot c}\right)\right)\right)\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)}\right)\right) \]

      16. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot c\right)}\right)\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(c \cdot \left(-1 \cdot c\right)\right)}\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \left(c \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)}\right)\right)\right) \]

      19. lower-neg.f64 55.4

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

   5. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(i \cdot \left(b \cdot \left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot \left(b \cdot \left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \cdot 2} \]

      3. lower-*.f64 55.4

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 82.1

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

   10. Applied rewrites 82.1%

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

   12. Applied rewrites 88.4%

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

   if -2.00000000000000004e95 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 98.3%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot \left(c \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+95}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(-2 \cdot \left(c \cdot \mathsf{fma}\left(c, b, a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 -2e+95)
     (* c (* i (* (fma b c a) -2.0)))
     (if (<= t_1 2e+161)
       (* 2.0 (fma t z (* x y)))
       (* i (* -2.0 (* c (fma c b a))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_1 <= -2e+95) {
		tmp = c * (i * (fma(b, c, a) * -2.0));
	} else if (t_1 <= 2e+161) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = i * (-2.0 * (c * fma(c, b, a)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_1 <= -2e+95)
		tmp = Float64(c * Float64(i * Float64(fma(b, c, a) * -2.0)));
	elseif (t_1 <= 2e+161)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = Float64(i * Float64(-2.0 * Float64(c * fma(c, b, a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+95], N[(c * N[(i * N[(N[(b * c + a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+161], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(-2.0 * N[(c * N[(c * b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.00000000000000004e95

   1. Initial program 83.2%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 84.5

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

   5. Applied rewrites 84.5%

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

   if -2.00000000000000004e95 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 98.3%

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

   3. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   if 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 85.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot \left({c}^{2} \cdot i\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(\color{blue}{\left({c}^{2} \cdot i\right) \cdot b}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(i \cdot {c}^{2}\right)} \cdot b\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(\color{blue}{i \cdot \left({c}^{2} \cdot b\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{neg}\left(i \cdot \color{blue}{\left(b \cdot {c}^{2}\right)}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto 2 \cdot \color{blue}{\left(i \cdot \left(\mathsf{neg}\left(b \cdot {c}^{2}\right)\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(-1 \cdot \left(b \cdot {c}^{2}\right)\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(i \cdot \left(-1 \cdot \left(b \cdot {c}^{2}\right)\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot {c}^{2}\right)\right)}\right) \]

      10. distribute-rgt-neg-out N/A

          \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(b \cdot \left(\mathsf{neg}\left({c}^{2}\right)\right)\right)}\right) \]

      11. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(-1 \cdot {c}^{2}\right)}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(i \cdot \color{blue}{\left(b \cdot \left(-1 \cdot {c}^{2}\right)\right)}\right) \]

      13. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left({c}^{2}\right)\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \left(\mathsf{neg}\left(\color{blue}{c \cdot c}\right)\right)\right)\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)}\right)\right) \]

      16. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot c\right)}\right)\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \color{blue}{\left(c \cdot \left(-1 \cdot c\right)\right)}\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto 2 \cdot \left(i \cdot \left(b \cdot \left(c \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)}\right)\right)\right) \]

      19. lower-neg.f64 65.0

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

   5. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(i \cdot \left(b \cdot \left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot \left(b \cdot \left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right)\right) \cdot 2} \]

      3. lower-*.f64 65.0

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

   7. Applied rewrites 65.0%

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

   8. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot \left(-2 \cdot \left(c \cdot \left(a + b \cdot c\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-2 \cdot \left(c \cdot \left(a + b \cdot c\right)\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto i \cdot \color{blue}{\left(-2 \cdot \left(c \cdot \left(a + b \cdot c\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto i \cdot \left(-2 \cdot \color{blue}{\left(c \cdot \left(a + b \cdot c\right)\right)}\right) \]

      8. +-commutative N/A

         \[\leadsto i \cdot \left(-2 \cdot \left(c \cdot \color{blue}{\left(b \cdot c + a\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto i \cdot \left(-2 \cdot \left(c \cdot \left(\color{blue}{c \cdot b} + a\right)\right)\right) \]

      10. lower-fma.f64 85.5

          \[\leadsto i \cdot \left(-2 \cdot \left(c \cdot \color{blue}{\mathsf{fma}\left(c, b, a\right)}\right)\right) \]

   10. Applied rewrites 85.5%

       \[\leadsto \color{blue}{i \cdot \left(-2 \cdot \left(c \cdot \mathsf{fma}\left(c, b, a\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+95}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(-2 \cdot \left(c \cdot \mathsf{fma}\left(c, b, a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* c (* i (* (fma b c a) -2.0)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -2e+95)
     t_1
     (if (<= t_2 2e+161) (* 2.0 (fma t z (* x y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = c * (i * (fma(b, c, a) * -2.0));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -2e+95) {
		tmp = t_1;
	} else if (t_2 <= 2e+161) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(c * Float64(i * Float64(fma(b, c, a) * -2.0)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -2e+95)
		tmp = t_1;
	elseif (t_2 <= 2e+161)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(c * N[(i * N[(N[(b * c + a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+95], t$95$1, If[LessEqual[t$95$2, 2e+161], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.00000000000000004e95 or 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 84.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot c} \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\left(a \cdot i + \left(b \cdot c\right) \cdot i\right)}\right) \cdot c \]

      4. associate-*r* N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot i + \color{blue}{b \cdot \left(c \cdot i\right)}\right)\right) \cdot c \]

      5. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \left(a \cdot i\right) + -2 \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)} \]

      8. distribute-lft-out N/A

         \[\leadsto c \cdot \color{blue}{\left(-2 \cdot \left(a \cdot i + b \cdot \left(c \cdot i\right)\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \left(-2 \cdot \left(a \cdot i + \color{blue}{\left(b \cdot c\right) \cdot i}\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto c \cdot \left(-2 \cdot \color{blue}{\left(\left(a + b \cdot c\right) \cdot i\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-2 \cdot \left(a + b \cdot c\right)\right) \cdot i\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto c \cdot \left(\color{blue}{\left(-2 \cdot \left(a + b \cdot c\right)\right)} \cdot i\right) \]

      15. +-commutative N/A

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\left(b \cdot c + a\right)}\right) \cdot i\right) \]

      16. lower-fma.f64 83.5

          \[\leadsto c \cdot \left(\left(-2 \cdot \color{blue}{\mathsf{fma}\left(b, c, a\right)}\right) \cdot i\right) \]

   5. Applied rewrites 83.5%

      \[\leadsto \color{blue}{c \cdot \left(\left(-2 \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot i\right)} \]

   if -2.00000000000000004e95 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 98.3%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \]

      2. lower-*.f64 86.4

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right) \]

   5. Applied rewrites 86.4%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+95}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* a (* -2.0 (* c i)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -5e+111)
     t_1
     (if (<= t_2 2e+161) (* 2.0 (fma t z (* x y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = a * (-2.0 * (c * i));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -5e+111) {
		tmp = t_1;
	} else if (t_2 <= 2e+161) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a * Float64(-2.0 * Float64(c * i)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -5e+111)
		tmp = t_1;
	elseif (t_2 <= 2e+161)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a * N[(-2.0 * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+111], t$95$1, If[LessEqual[t$95$2, 2e+161], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.9999999999999997e111 or 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 83.4%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 9.3

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

   5. Applied rewrites 9.3%

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot i\right)\right) \cdot -2} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{a \cdot \left(\left(c \cdot i\right) \cdot -2\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(-2 \cdot \left(c \cdot i\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\left(c \cdot i\right) \cdot -2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(\left(c \cdot i\right) \cdot -2\right)} \]

      7. lower-*.f64 50.7

         \[\leadsto a \cdot \left(\color{blue}{\left(c \cdot i\right)} \cdot -2\right) \]

   8. Applied rewrites 50.7%

      \[\leadsto \color{blue}{a \cdot \left(\left(c \cdot i\right) \cdot -2\right)} \]

   if -4.9999999999999997e111 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 98.4%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \]

      2. lower-*.f64 84.6

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right) \]

   5. Applied rewrites 84.6%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{+111}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+122}:\\ \;\;\;\;2 \cdot \left(z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* a (* -2.0 (* c i)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -5e+27) t_1 (if (<= t_2 1e+122) (* 2.0 (* z t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = a * (-2.0 * (c * i));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -5e+27) {
		tmp = t_1;
	} else if (t_2 <= 1e+122) {
		tmp = 2.0 * (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a * ((-2.0d0) * (c * i))
    t_2 = (c * (a + (b * c))) * i
    if (t_2 <= (-5d+27)) then
        tmp = t_1
    else if (t_2 <= 1d+122) then
        tmp = 2.0d0 * (z * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = a * (-2.0 * (c * i));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -5e+27) {
		tmp = t_1;
	} else if (t_2 <= 1e+122) {
		tmp = 2.0 * (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = a * (-2.0 * (c * i))
	t_2 = (c * (a + (b * c))) * i
	tmp = 0
	if t_2 <= -5e+27:
		tmp = t_1
	elif t_2 <= 1e+122:
		tmp = 2.0 * (z * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a * Float64(-2.0 * Float64(c * i)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -5e+27)
		tmp = t_1;
	elseif (t_2 <= 1e+122)
		tmp = Float64(2.0 * Float64(z * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = a * (-2.0 * (c * i));
	t_2 = (c * (a + (b * c))) * i;
	tmp = 0.0;
	if (t_2 <= -5e+27)
		tmp = t_1;
	elseif (t_2 <= 1e+122)
		tmp = 2.0 * (z * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a * N[(-2.0 * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+27], t$95$1, If[LessEqual[t$95$2, 1e+122], N[(2.0 * N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.99999999999999979e27 or 1.00000000000000001e122 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 84.9%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 10.0

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

   5. Applied rewrites 10.0%

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot i\right)\right) \cdot -2} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{a \cdot \left(\left(c \cdot i\right) \cdot -2\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(-2 \cdot \left(c \cdot i\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\left(c \cdot i\right) \cdot -2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(\left(c \cdot i\right) \cdot -2\right)} \]

      7. lower-*.f64 49.0

         \[\leadsto a \cdot \left(\color{blue}{\left(c \cdot i\right)} \cdot -2\right) \]

   8. Applied rewrites 49.0%

      \[\leadsto \color{blue}{a \cdot \left(\left(c \cdot i\right) \cdot -2\right)} \]

   if -4.99999999999999979e27 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000001e122

   1. Initial program 98.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 52.6

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

   5. Applied rewrites 52.6%

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{+27}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+122}:\\ \;\;\;\;2 \cdot \left(z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \left(c \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+306}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \left(-i\right) \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* (* c (+ a (* b c))) i) -2e+306)
   (* 2.0 (fma y x (- (* z t) (* c (* i (fma b c a))))))
   (* 2.0 (fma z t (fma x y (* (- i) (* c (fma b c a))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * (a + (b * c))) * i) <= -2e+306) {
		tmp = 2.0 * fma(y, x, ((z * t) - (c * (i * fma(b, c, a)))));
	} else {
		tmp = 2.0 * fma(z, t, fma(x, y, (-i * (c * fma(b, c, a)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(c * Float64(a + Float64(b * c))) * i) <= -2e+306)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(z * t) - Float64(c * Float64(i * fma(b, c, a))))));
	else
		tmp = Float64(2.0 * fma(z, t, fma(x, y, Float64(Float64(-i) * Float64(c * fma(b, c, a))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], -2e+306], N[(2.0 * N[(y * x + N[(N[(z * t), $MachinePrecision] - N[(c * N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(z * t + N[(x * y + N[((-i) * N[(c * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.00000000000000003e306

   1. Initial program 78.1%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)} \]

      2. sub-neg N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(x \cdot y + z \cdot t\right) + \left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right) + \left(x \cdot y + z \cdot t\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i}\right)\right) + \left(x \cdot y + z \cdot t\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right)} \cdot i\right)\right) + \left(x \cdot y + z \cdot t\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(a + b \cdot c\right) \cdot \left(c \cdot i\right)}\right)\right) + \left(x \cdot y + z \cdot t\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto 2 \cdot \left(\color{blue}{\left(a + b \cdot c\right) \cdot \left(\mathsf{neg}\left(c \cdot i\right)\right)} + \left(x \cdot y + z \cdot t\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(a + b \cdot c, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{a + b \cdot c}, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right) \]

      10. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{b \cdot c + a}, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right) \]

      11. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{b \cdot c} + a, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, c, a\right)}, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), \color{blue}{c \cdot \left(\mathsf{neg}\left(i\right)\right)}, x \cdot y + z \cdot t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), \color{blue}{c \cdot \left(\mathsf{neg}\left(i\right)\right)}, x \cdot y + z \cdot t\right) \]

      15. lower-neg.f64 93.8

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \color{blue}{\left(-i\right)}, x \cdot y + z \cdot t\right) \]

      16. lift-+.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(\mathsf{neg}\left(i\right)\right), \color{blue}{x \cdot y + z \cdot t}\right) \]

      17. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(\mathsf{neg}\left(i\right)\right), \color{blue}{z \cdot t + x \cdot y}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(\mathsf{neg}\left(i\right)\right), \color{blue}{z \cdot t} + x \cdot y\right) \]

      19. lower-fma.f64 95.3

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), \color{blue}{\mathsf{fma}\left(z, t, x \cdot y\right)}\right) \]

   4. Applied rewrites 95.3%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), \mathsf{fma}\left(z, t, x \cdot y\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{fma}\left(b, c, a\right) \cdot \left(c \cdot \left(\mathsf{neg}\left(i\right)\right)\right) + \mathsf{fma}\left(z, t, x \cdot y\right)\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot \left(c \cdot \left(\mathsf{neg}\left(i\right)\right)\right) + \color{blue}{\left(z \cdot t + x \cdot y\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\mathsf{fma}\left(b, c, a\right) \cdot \left(c \cdot \left(\mathsf{neg}\left(i\right)\right)\right) + z \cdot t\right) + x \cdot y\right)} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{fma}\left(b, c, a\right) \cdot \left(c \cdot \left(\mathsf{neg}\left(i\right)\right)\right) + z \cdot t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\color{blue}{x \cdot y} + \left(\mathsf{fma}\left(b, c, a\right) \cdot \left(c \cdot \left(\mathsf{neg}\left(i\right)\right)\right) + z \cdot t\right)\right) \]

      6. *-commutative N/A

         \[\leadsto 2 \cdot \left(\color{blue}{y \cdot x} + \left(\mathsf{fma}\left(b, c, a\right) \cdot \left(c \cdot \left(\mathsf{neg}\left(i\right)\right)\right) + z \cdot t\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(b, c, a\right) \cdot \left(c \cdot \left(\mathsf{neg}\left(i\right)\right)\right) + z \cdot t\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, c, a\right) \cdot \color{blue}{\left(c \cdot \left(\mathsf{neg}\left(i\right)\right)\right)} + z \cdot t\right) \]

      9. associate-*r* N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot \left(\mathsf{neg}\left(i\right)\right)} + z \cdot t\right) \]

      10. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\color{blue}{\left(b \cdot c + a\right)} \cdot c\right) \cdot \left(\mathsf{neg}\left(i\right)\right) + z \cdot t\right) \]

      11. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\left(\color{blue}{b \cdot c} + a\right) \cdot c\right) \cdot \left(\mathsf{neg}\left(i\right)\right) + z \cdot t\right) \]

      12. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\color{blue}{\left(a + b \cdot c\right)} \cdot c\right) \cdot \left(\mathsf{neg}\left(i\right)\right) + z \cdot t\right) \]

      13. lift-+.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\color{blue}{\left(a + b \cdot c\right)} \cdot c\right) \cdot \left(\mathsf{neg}\left(i\right)\right) + z \cdot t\right) \]

      14. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right)} \cdot \left(\mathsf{neg}\left(i\right)\right) + z \cdot t\right) \]

      15. lift-neg.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\left(a + b \cdot c\right) \cdot c\right) \cdot \color{blue}{\left(\mathsf{neg}\left(i\right)\right)} + z \cdot t\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right)} + z \cdot t\right) \]

      17. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(\color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i}\right)\right) + z \cdot t\right) \]

      18. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \color{blue}{z \cdot t + \left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right)}\right) \]

   6. Applied rewrites 98.4%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(y, x, z \cdot t - c \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)} \]

   if -2.00000000000000003e306 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 95.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)} \]

      2. sub-neg N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(x \cdot y + z \cdot t\right) + \left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right) + \left(x \cdot y + z \cdot t\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i}\right)\right) + \left(x \cdot y + z \cdot t\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right)} \cdot i\right)\right) + \left(x \cdot y + z \cdot t\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(a + b \cdot c\right) \cdot \left(c \cdot i\right)}\right)\right) + \left(x \cdot y + z \cdot t\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto 2 \cdot \left(\color{blue}{\left(a + b \cdot c\right) \cdot \left(\mathsf{neg}\left(c \cdot i\right)\right)} + \left(x \cdot y + z \cdot t\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(a + b \cdot c, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{a + b \cdot c}, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right) \]

      10. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{b \cdot c + a}, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right) \]

      11. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{b \cdot c} + a, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, c, a\right)}, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), \color{blue}{c \cdot \left(\mathsf{neg}\left(i\right)\right)}, x \cdot y + z \cdot t\right) \]

      14. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), \color{blue}{c \cdot \left(\mathsf{neg}\left(i\right)\right)}, x \cdot y + z \cdot t\right) \]

      15. lower-neg.f64 96.8

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \color{blue}{\left(-i\right)}, x \cdot y + z \cdot t\right) \]

      16. lift-+.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(\mathsf{neg}\left(i\right)\right), \color{blue}{x \cdot y + z \cdot t}\right) \]

      17. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(\mathsf{neg}\left(i\right)\right), \color{blue}{z \cdot t + x \cdot y}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(\mathsf{neg}\left(i\right)\right), \color{blue}{z \cdot t} + x \cdot y\right) \]

      19. lower-fma.f64 97.8

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), \color{blue}{\mathsf{fma}\left(z, t, x \cdot y\right)}\right) \]

   4. Applied rewrites 97.8%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), \mathsf{fma}\left(z, t, x \cdot y\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{fma}\left(b, c, a\right) \cdot \left(c \cdot \left(\mathsf{neg}\left(i\right)\right)\right) + \mathsf{fma}\left(z, t, x \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{fma}\left(z, t, x \cdot y\right) + \mathsf{fma}\left(b, c, a\right) \cdot \left(c \cdot \left(\mathsf{neg}\left(i\right)\right)\right)\right)} \]

      3. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \left(\color{blue}{\left(z \cdot t + x \cdot y\right)} + \mathsf{fma}\left(b, c, a\right) \cdot \left(c \cdot \left(\mathsf{neg}\left(i\right)\right)\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(z \cdot t + x \cdot y\right) + \mathsf{fma}\left(b, c, a\right) \cdot \color{blue}{\left(c \cdot \left(\mathsf{neg}\left(i\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto 2 \cdot \left(\left(z \cdot t + x \cdot y\right) + \color{blue}{\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot \left(\mathsf{neg}\left(i\right)\right)}\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \left(\left(z \cdot t + x \cdot y\right) + \left(\color{blue}{\left(b \cdot c + a\right)} \cdot c\right) \cdot \left(\mathsf{neg}\left(i\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(z \cdot t + x \cdot y\right) + \left(\left(\color{blue}{b \cdot c} + a\right) \cdot c\right) \cdot \left(\mathsf{neg}\left(i\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto 2 \cdot \left(\left(z \cdot t + x \cdot y\right) + \left(\color{blue}{\left(a + b \cdot c\right)} \cdot c\right) \cdot \left(\mathsf{neg}\left(i\right)\right)\right) \]

      9. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\left(z \cdot t + x \cdot y\right) + \left(\color{blue}{\left(a + b \cdot c\right)} \cdot c\right) \cdot \left(\mathsf{neg}\left(i\right)\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(z \cdot t + x \cdot y\right) + \color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right)} \cdot \left(\mathsf{neg}\left(i\right)\right)\right) \]

      11. lift-neg.f64 N/A

          \[\leadsto 2 \cdot \left(\left(z \cdot t + x \cdot y\right) + \left(\left(a + b \cdot c\right) \cdot c\right) \cdot \color{blue}{\left(\mathsf{neg}\left(i\right)\right)}\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto 2 \cdot \left(\left(z \cdot t + x \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right)}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(z \cdot t + x \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i}\right)\right)\right) \]

      14. associate-+l+ N/A

          \[\leadsto 2 \cdot \color{blue}{\left(z \cdot t + \left(x \cdot y + \left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right)\right)\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(z, t, x \cdot y + \left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \color{blue}{x \cdot y} + \left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right)\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \color{blue}{\mathsf{fma}\left(x, y, \mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right)}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{neg}\left(\color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i}\right)\right)\right) \]

   6. Applied rewrites 97.8%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \left(-i\right) \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -2 \cdot 10^{+306}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, z \cdot t - c \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \left(-i\right) \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 93.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(c, -i \cdot \mathsf{fma}\left(b, c, a\right), x \cdot y\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (* 2.0 (fma z t (fma c (- (* i (fma b c a))) (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * fma(z, t, fma(c, -(i * fma(b, c, a)), (x * y)));
}

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * fma(z, t, fma(c, Float64(-Float64(i * fma(b, c, a))), Float64(x * y))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(z * t + N[(c * (-N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision]) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.9%

   \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto 2 \cdot \left(\color{blue}{\left(x \cdot y + z \cdot t\right)} - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   3. associate--l+ N/A

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y + \left(z \cdot t - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\left(z \cdot t - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) + x \cdot y\right)} \]

   5. sub-neg N/A

      \[\leadsto 2 \cdot \left(\color{blue}{\left(z \cdot t + \left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right)\right)} + x \cdot y\right) \]

   6. associate-+l+ N/A

      \[\leadsto 2 \cdot \color{blue}{\left(z \cdot t + \left(\left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right) + x \cdot y\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\color{blue}{z \cdot t} + \left(\left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right) + x \cdot y\right)\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(z, t, \left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right) + x \cdot y\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \left(\mathsf{neg}\left(\color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i}\right)\right) + x \cdot y\right) \]

   10. lift-*.f64 N/A

       \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \left(\mathsf{neg}\left(\color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right)} \cdot i\right)\right) + x \cdot y\right) \]

   11. *-commutative N/A

       \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot \left(a + b \cdot c\right)\right)} \cdot i\right)\right) + x \cdot y\right) \]

   12. associate-*l* N/A

       \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \left(\mathsf{neg}\left(\color{blue}{c \cdot \left(\left(a + b \cdot c\right) \cdot i\right)}\right)\right) + x \cdot y\right) \]

   13. distribute-rgt-neg-in N/A

       \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \color{blue}{c \cdot \left(\mathsf{neg}\left(\left(a + b \cdot c\right) \cdot i\right)\right)} + x \cdot y\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(\left(a + b \cdot c\right) \cdot i\right), x \cdot y\right)}\right) \]

4. Applied rewrites 93.9%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(c, -\mathsf{fma}\left(b, c, a\right) \cdot i, x \cdot y\right)\right)} \]

5. Final simplification 93.9%

   \[\leadsto 2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(c, -i \cdot \mathsf{fma}\left(b, c, a\right), x \cdot y\right)\right) \]
6. Add Preprocessing

Alternative 20: 44.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -2.6 \cdot 10^{+106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2.1 \cdot 10^{+111}:\\ \;\;\;\;2 \cdot \left(z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (* x y))))
   (if (<= (* x y) -2.6e+106)
     t_1
     (if (<= (* x y) 2.1e+111) (* 2.0 (* z t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * (x * y);
	double tmp;
	if ((x * y) <= -2.6e+106) {
		tmp = t_1;
	} else if ((x * y) <= 2.1e+111) {
		tmp = 2.0 * (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (x * y)
    if ((x * y) <= (-2.6d+106)) then
        tmp = t_1
    else if ((x * y) <= 2.1d+111) then
        tmp = 2.0d0 * (z * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * (x * y);
	double tmp;
	if ((x * y) <= -2.6e+106) {
		tmp = t_1;
	} else if ((x * y) <= 2.1e+111) {
		tmp = 2.0 * (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = 2.0 * (x * y)
	tmp = 0
	if (x * y) <= -2.6e+106:
		tmp = t_1
	elif (x * y) <= 2.1e+111:
		tmp = 2.0 * (z * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(2.0 * Float64(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -2.6e+106)
		tmp = t_1;
	elseif (Float64(x * y) <= 2.1e+111)
		tmp = Float64(2.0 * Float64(z * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = 2.0 * (x * y);
	tmp = 0.0;
	if ((x * y) <= -2.6e+106)
		tmp = t_1;
	elseif ((x * y) <= 2.1e+111)
		tmp = 2.0 * (z * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2.6e+106], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2.1e+111], N[(2.0 * N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.6000000000000002e106 or 2.09999999999999995e111 < (*.f64 x y)

   1. Initial program 86.9%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 65.7

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y\right)} \]

   5. Applied rewrites 65.7%

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y\right)} \]

   if -2.6000000000000002e106 < (*.f64 x y) < 2.09999999999999995e111

   1. Initial program 92.5%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 34.4

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

   5. Applied rewrites 34.4%

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.6 \cdot 10^{+106}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 2.1 \cdot 10^{+111}:\\ \;\;\;\;2 \cdot \left(z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 29.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(z \cdot t\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (* 2.0 (* z t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (z * t);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = 2.0d0 * (z * t)
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 * (z * t);
}

Python

def code(x, y, z, t, a, b, c, i):
	return 2.0 * (z * t)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * Float64(z * t))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = 2.0 * (z * t);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(z * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.9%

   \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 29.1

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

5. Applied rewrites 29.1%

   \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

6. Final simplification 29.1%

   \[\leadsto 2 \cdot \left(z \cdot t\right) \]
7. Add Preprocessing

Developer Target 1: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right) \end{array} \]

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

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = 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(a + Float64(b * c)) * Float64(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[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z t a b c i)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* 2 (- (+ (* x y) (* z t)) (* (+ a (* b c)) (* c i)))))

  (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))