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

Percentage Accurate: 89.6% → 95.7%

Time: 17.2s

Alternatives: 14

Speedup: 0.7×

• 46.9% 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: 89.6% 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: 95.7% accurate, 0.5× speedup

Math

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

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 (fma b c a) (* c (- i)) (fma z t (* x y))))
   (* (- t (/ (* a (* c i)) z)) (* z 2.0))))

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(fma(b, c, a), (c * -i), fma(z, t, (x * y)));
	} else {
		tmp = (t - ((a * (c * i)) / z)) * (z * 2.0);
	}
	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(fma(b, c, a), Float64(c * Float64(-i)), fma(z, t, Float64(x * y))));
	else
		tmp = Float64(Float64(t - Float64(Float64(a * Float64(c * i)) / z)) * Float64(z * 2.0));
	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[(N[(b * c + a), $MachinePrecision] * N[(c * (-i)), $MachinePrecision] + N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t - N[(N[(a * N[(c * i), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * N[(z * 2.0), $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 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 97.5%

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

   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. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 25.0

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

   5. Applied rewrites 25.0%

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

   6. Taylor expanded in z around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

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

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-*.f64 75.0

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

   8. Applied rewrites 75.0%

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

   9. Taylor expanded in a around inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

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

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

       4. neg-mul-1 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. neg-mul-1 N/A

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

       8. lower-neg.f64 83.3

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

   11. Applied rewrites 83.3%

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

4. Final simplification 96.8%

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

Alternative 2: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+292}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(-2 \cdot \mathsf{fma}\left(c, b, a\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+305}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, z \cdot t - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(c \cdot c\right)\right) \cdot \left(i \cdot \left(b + \frac{a}{c}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 -5e+292)
     (* (* c i) (* -2.0 (fma c b a)))
     (if (<= t_1 1e+305)
       (* 2.0 (fma y x (- (* z t) (* i (* c (fma b c a))))))
       (* (* -2.0 (* c c)) (* i (+ b (/ a 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 <= -5e+292) {
		tmp = (c * i) * (-2.0 * fma(c, b, a));
	} else if (t_1 <= 1e+305) {
		tmp = 2.0 * fma(y, x, ((z * t) - (i * (c * fma(b, c, a)))));
	} else {
		tmp = (-2.0 * (c * c)) * (i * (b + (a / 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 <= -5e+292)
		tmp = Float64(Float64(c * i) * Float64(-2.0 * fma(c, b, a)));
	elseif (t_1 <= 1e+305)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(z * t) - Float64(i * Float64(c * fma(b, c, a))))));
	else
		tmp = Float64(Float64(-2.0 * Float64(c * c)) * Float64(i * Float64(b + Float64(a / 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, -5e+292], N[(N[(c * i), $MachinePrecision] * N[(-2.0 * N[(c * b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+305], N[(2.0 * N[(y * x + N[(N[(z * t), $MachinePrecision] - N[(i * N[(c * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(i * N[(b + N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.9999999999999996e292

   1. Initial program 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in i around inf

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

      1. distribute-lft-in N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. distribute-rgt-in N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

   7. Applied rewrites 89.7%

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

   if -4.9999999999999996e292 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999994e304

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 99.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.3

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   if 9.9999999999999994e304 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 68.7%

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

   3. Taylor expanded in c around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{+292}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(-2 \cdot \mathsf{fma}\left(c, b, a\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+305}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, z \cdot t - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(c \cdot c\right)\right) \cdot \left(i \cdot \left(b + \frac{a}{c}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+292}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(-2 \cdot \mathsf{fma}\left(c, b, a\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+305}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, z \cdot t - i \cdot \left(a \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(c \cdot c\right)\right) \cdot \left(i \cdot \left(b + \frac{a}{c}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 -5e+292)
     (* (* c i) (* -2.0 (fma c b a)))
     (if (<= t_1 1e+305)
       (* 2.0 (fma y x (- (* z t) (* i (* a c)))))
       (* (* -2.0 (* c c)) (* i (+ b (/ a 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 <= -5e+292) {
		tmp = (c * i) * (-2.0 * fma(c, b, a));
	} else if (t_1 <= 1e+305) {
		tmp = 2.0 * fma(y, x, ((z * t) - (i * (a * c))));
	} else {
		tmp = (-2.0 * (c * c)) * (i * (b + (a / 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 <= -5e+292)
		tmp = Float64(Float64(c * i) * Float64(-2.0 * fma(c, b, a)));
	elseif (t_1 <= 1e+305)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(z * t) - Float64(i * Float64(a * c)))));
	else
		tmp = Float64(Float64(-2.0 * Float64(c * c)) * Float64(i * Float64(b + Float64(a / 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, -5e+292], N[(N[(c * i), $MachinePrecision] * N[(-2.0 * N[(c * b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+305], N[(2.0 * N[(y * x + N[(N[(z * t), $MachinePrecision] - N[(i * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(i * N[(b + N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.9999999999999996e292

   1. Initial program 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in i around inf

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

      1. distribute-lft-in N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. distribute-rgt-in N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

   7. Applied rewrites 89.7%

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

   if -4.9999999999999996e292 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999994e304

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 99.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.3

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 92.2

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

   7. Applied rewrites 92.2%

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

   if 9.9999999999999994e304 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 68.7%

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

   3. Taylor expanded in c around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{+292}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(-2 \cdot \mathsf{fma}\left(c, b, a\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+305}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, z \cdot t - i \cdot \left(a \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(c \cdot c\right)\right) \cdot \left(i \cdot \left(b + \frac{a}{c}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot i\right) \cdot \left(-2 \cdot \mathsf{fma}\left(c, b, a\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, z \cdot t - i \cdot \left(a \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c i) (* -2.0 (fma c b a)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -5e+292)
     t_1
     (if (<= t_2 2e+161) (* 2.0 (fma y x (- (* z t) (* i (* a c))))) 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) * (-2.0 * fma(c, b, a));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -5e+292) {
		tmp = t_1;
	} else if (t_2 <= 2e+161) {
		tmp = 2.0 * fma(y, x, ((z * t) - (i * (a * c))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * i) * Float64(-2.0 * fma(c, b, a)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -5e+292)
		tmp = t_1;
	elseif (t_2 <= 2e+161)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(z * t) - Float64(i * Float64(a * c)))));
	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 * i), $MachinePrecision] * N[(-2.0 * N[(c * b + 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, -5e+292], t$95$1, If[LessEqual[t$95$2, 2e+161], N[(2.0 * N[(y * x + N[(N[(z * t), $MachinePrecision] - N[(i * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.9999999999999996e292 or 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 76.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 71.8

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in i around inf

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

      1. distribute-lft-in N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. distribute-rgt-in N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

   7. Applied rewrites 87.8%

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

   if -4.9999999999999996e292 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 99.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.3

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 94.3

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

   7. Applied rewrites 94.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -5 \cdot 10^{+292}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(-2 \cdot \mathsf{fma}\left(c, b, a\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, z \cdot t - i \cdot \left(a \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(-2 \cdot \mathsf{fma}\left(c, b, a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot i\right) \cdot \left(-2 \cdot \mathsf{fma}\left(c, b, a\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+193}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - c \cdot \left(a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c i) (* -2.0 (fma c b a)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+193)
     t_1
     (if (<= t_2 2e+161) (* 2.0 (- (fma t z (* x y)) (* c (* a 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 * i) * (-2.0 * fma(c, b, a));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+193) {
		tmp = t_1;
	} else if (t_2 <= 2e+161) {
		tmp = 2.0 * (fma(t, z, (x * y)) - (c * (a * i)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * i) * Float64(-2.0 * fma(c, b, a)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -1e+193)
		tmp = t_1;
	elseif (t_2 <= 2e+161)
		tmp = Float64(2.0 * Float64(fma(t, z, Float64(x * y)) - Float64(c * Float64(a * i))));
	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 * i), $MachinePrecision] * N[(-2.0 * N[(c * b + 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, -1e+193], t$95$1, If[LessEqual[t$95$2, 2e+161], N[(2.0 * N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(c * N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.00000000000000007e193 or 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.0

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

   4. Applied rewrites 73.0%

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

   5. Taylor expanded in i around inf

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

      1. distribute-lft-in N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. distribute-rgt-in N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

   7. Applied rewrites 86.5%

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

   if -1.00000000000000007e193 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 99.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+193}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(-2 \cdot \mathsf{fma}\left(c, b, a\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - c \cdot \left(a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(-2 \cdot \mathsf{fma}\left(c, b, a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot i\right) \cdot \left(-2 \cdot \mathsf{fma}\left(c, b, a\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+182}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c i) (* -2.0 (fma c b a)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+182)
     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) * (-2.0 * fma(c, b, a));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+182) {
		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(Float64(c * i) * Float64(-2.0 * fma(c, b, a)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -1e+182)
		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 * i), $MachinePrecision] * N[(-2.0 * N[(c * b + 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, -1e+182], 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) < -1.0000000000000001e182 or 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 73.5

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

   4. Applied rewrites 73.5%

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

   5. Taylor expanded in i around inf

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

      1. distribute-lft-in N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. distribute-rgt-in N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

   7. Applied rewrites 86.0%

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

   if -1.0000000000000001e182 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+182}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(-2 \cdot \mathsf{fma}\left(c, b, a\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot i\right) \cdot \left(-2 \cdot \mathsf{fma}\left(c, b, a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+182}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -1e+182)
     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 <= -1e+182) {
		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 <= -1e+182)
		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, -1e+182], 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) < -1.0000000000000001e182 or 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.5%

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

   3. Taylor expanded in 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 81.9

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

   5. Applied rewrites 81.9%

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

   if -1.0000000000000001e182 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+182}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(i \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+193}:\\ \;\;\;\;b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(c \cdot \left(c \cdot \left(i \cdot -2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 -1e+193)
     (* b (* i (* -2.0 (* c c))))
     (if (<= t_1 2e+161)
       (* 2.0 (fma t z (* x y)))
       (* b (* c (* c (* i -2.0))))))))

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 <= -1e+193) {
		tmp = b * (i * (-2.0 * (c * c)));
	} else if (t_1 <= 2e+161) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = b * (c * (c * (i * -2.0)));
	}
	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 <= -1e+193)
		tmp = Float64(b * Float64(i * Float64(-2.0 * Float64(c * c))));
	elseif (t_1 <= 2e+161)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = Float64(b * Float64(c * Float64(c * Float64(i * -2.0))));
	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, -1e+193], N[(b * N[(i * N[(-2.0 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+161], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(c * N[(c * N[(i * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.00000000000000007e193

   1. Initial program 79.3%

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

   3. Taylor expanded in 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 69.8

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

   5. Applied rewrites 69.8%

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

   if -1.00000000000000007e193 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

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

   1. Initial program 74.3%

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

   3. Taylor expanded in 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 69.1

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

   5. Applied rewrites 69.1%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 69.1

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

   7. Applied rewrites 69.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+193}:\\ \;\;\;\;b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(c \cdot \left(c \cdot \left(i \cdot -2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+193}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* b (* i (* -2.0 (* c c))))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+193)
     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 = b * (i * (-2.0 * (c * c)));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+193) {
		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(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 <= -1e+193)
		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[(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, -1e+193], 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) < -1.00000000000000007e193 or 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.1%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 69.5

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

   5. Applied rewrites 69.5%

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

   if -1.00000000000000007e193 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000001e161

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+193}:\\ \;\;\;\;b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(i \cdot \left(-2 \cdot \left(c \cdot c\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(c \cdot \left(i \cdot -2\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+255}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+233}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* a (* c (* i -2.0)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+255)
     t_1
     (if (<= t_2 2e+233) (* 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 * (c * (i * -2.0));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+255) {
		tmp = t_1;
	} else if (t_2 <= 2e+233) {
		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(c * Float64(i * -2.0)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -1e+255)
		tmp = t_1;
	elseif (t_2 <= 2e+233)
		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[(c * N[(i * -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, -1e+255], t$95$1, If[LessEqual[t$95$2, 2e+233], 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) < -9.99999999999999988e254 or 1.99999999999999995e233 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   6. 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. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 49.0

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

   7. Applied rewrites 49.0%

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

   if -9.99999999999999988e254 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999995e233

   1. Initial program 99.3%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 85.8

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

   5. Applied rewrites 85.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+255}:\\ \;\;\;\;a \cdot \left(c \cdot \left(i \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+233}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(c \cdot \left(i \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(c \cdot \left(i \cdot -2\right)\right)\\ t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+255}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+233}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* a (* c (* i -2.0)))) (t_2 (* (* c (+ a (* b c))) i)))
   (if (<= t_2 -1e+255)
     t_1
     (if (<= t_2 2e+233) (* 2.0 (fma y x (* 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 * (c * (i * -2.0));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_2 <= -1e+255) {
		tmp = t_1;
	} else if (t_2 <= 2e+233) {
		tmp = 2.0 * fma(y, x, (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a * Float64(c * Float64(i * -2.0)))
	t_2 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_2 <= -1e+255)
		tmp = t_1;
	elseif (t_2 <= 2e+233)
		tmp = Float64(2.0 * fma(y, x, Float64(z * t)));
	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[(c * N[(i * -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, -1e+255], t$95$1, If[LessEqual[t$95$2, 2e+233], N[(2.0 * N[(y * x + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.99999999999999988e254 or 1.99999999999999995e233 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   6. 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. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 49.0

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

   7. Applied rewrites 49.0%

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

   if -9.99999999999999988e254 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999995e233

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 99.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.3

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -1 \cdot 10^{+255}:\\ \;\;\;\;a \cdot \left(c \cdot \left(i \cdot -2\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 2 \cdot 10^{+233}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(c \cdot \left(i \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 94.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+305}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), \mathsf{fma}\left(z, t, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(c \cdot c\right)\right) \cdot \left(i \cdot \left(b + \frac{a}{c}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* (* c (+ a (* b c))) i) 1e+305)
   (* 2.0 (fma (fma b c a) (* c (- i)) (fma z t (* x y))))
   (* (* -2.0 (* c c)) (* i (+ 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 (((c * (a + (b * c))) * i) <= 1e+305) {
		tmp = 2.0 * fma(fma(b, c, a), (c * -i), fma(z, t, (x * y)));
	} else {
		tmp = (-2.0 * (c * c)) * (i * (b + (a / c)));
	}
	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) <= 1e+305)
		tmp = Float64(2.0 * fma(fma(b, c, a), Float64(c * Float64(-i)), fma(z, t, Float64(x * y))));
	else
		tmp = Float64(Float64(-2.0 * Float64(c * c)) * Float64(i * Float64(b + Float64(a / c))));
	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], 1e+305], N[(2.0 * N[(N[(b * c + a), $MachinePrecision] * N[(c * (-i)), $MachinePrecision] + N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(i * N[(b + N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999994e304

   1. Initial program 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 96.6%

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

   if 9.9999999999999994e304 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 68.7%

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

   3. Taylor expanded in c around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 10^{+305}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), c \cdot \left(-i\right), \mathsf{fma}\left(z, t, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(c \cdot c\right)\right) \cdot \left(i \cdot \left(b + \frac{a}{c}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot 2\right)\\ \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+58}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* t (* z 2.0))))
   (if (<= (* z t) -4e+129) t_1 (if (<= (* z t) 1e+58) (* (* x y) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = t * (z * 2.0);
	double tmp;
	if ((z * t) <= -4e+129) {
		tmp = t_1;
	} else if ((z * t) <= 1e+58) {
		tmp = (x * y) * 2.0;
	} 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 = t * (z * 2.0d0)
    if ((z * t) <= (-4d+129)) then
        tmp = t_1
    else if ((z * t) <= 1d+58) then
        tmp = (x * y) * 2.0d0
    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 = t * (z * 2.0);
	double tmp;
	if ((z * t) <= -4e+129) {
		tmp = t_1;
	} else if ((z * t) <= 1e+58) {
		tmp = (x * y) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = t * (z * 2.0)
	tmp = 0
	if (z * t) <= -4e+129:
		tmp = t_1
	elif (z * t) <= 1e+58:
		tmp = (x * y) * 2.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(t * Float64(z * 2.0))
	tmp = 0.0
	if (Float64(z * t) <= -4e+129)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+58)
		tmp = Float64(Float64(x * y) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = t * (z * 2.0);
	tmp = 0.0;
	if ((z * t) <= -4e+129)
		tmp = t_1;
	elseif ((z * t) <= 1e+58)
		tmp = (x * y) * 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t * N[(z * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -4e+129], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+58], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4e129 or 9.99999999999999944e57 < (*.f64 z t)

   1. Initial program 84.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if -4e129 < (*.f64 z t) < 9.99999999999999944e57

   1. Initial program 92.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 43.8

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

   5. Applied rewrites 43.8%

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

Alternative 14: 29.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ t \cdot \left(z \cdot 2\right) \end{array} \]

FPCore

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

C

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

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 = t * (z * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return t * (z * 2.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(t * N[(z * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 28.9

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

5. Applied rewrites 28.9%

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

Developer Target 1: 94.1% 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 2024216 
(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))))