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

Percentage Accurate: 90.3% → 96.0%

Time: 10.1s

Alternatives: 13

Speedup: 0.5×

• 47.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = 2.0d0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = 2.0d0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+296}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, y \cdot x - i \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\mathsf{fma}\left(i, \frac{a}{c}, i \cdot b\right) \cdot c\right) \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 (- INFINITY))
     (* 2.0 (fma z t (* (- c) (* (fma b c a) i))))
     (if (<= t_1 1e+296)
       (* 2.0 (fma z t (- (* y x) (* i (* (fma c b a) c)))))
       (* 2.0 (- (+ (* x y) (* z t)) (* (* (fma i (/ a c) (* i b)) c) c)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 2.0 * fma(z, t, (-c * (fma(b, c, a) * i)));
	} else if (t_1 <= 1e+296) {
		tmp = 2.0 * fma(z, t, ((y * x) - (i * (fma(c, b, a) * c))));
	} else {
		tmp = 2.0 * (((x * y) + (z * t)) - ((fma(i, (a / c), (i * b)) * c) * c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(2.0 * fma(z, t, Float64(Float64(-c) * Float64(fma(b, c, a) * i))));
	elseif (t_1 <= 1e+296)
		tmp = Float64(2.0 * fma(z, t, Float64(Float64(y * x) - Float64(i * Float64(fma(c, b, a) * c)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(fma(i, Float64(a / c), Float64(i * b)) * c) * c)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(2.0 * N[(z * t + N[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+296], N[(2.0 * N[(z * t + N[(N[(y * x), $MachinePrecision] - N[(i * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(i * N[(a / c), $MachinePrecision] + N[(i * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 66.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 68.2

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 68.2

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 68.2

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

   4. Applied rewrites 68.2%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 94.1

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

   7. Applied rewrites 94.1%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.99999999999999981e295

   1. Initial program 98.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 98.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 98.6

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if 9.99999999999999981e295 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 71.8%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 90.3

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

   5. Applied rewrites 90.3%

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

Alternative 2: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 10^{+302}\right):\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, y \cdot x - i \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+302)))
     (* 2.0 (fma z t (* (- c) (* (fma b c a) i))))
     (* 2.0 (fma z t (- (* y x) (* i (* (fma c b a) c))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+302)) {
		tmp = 2.0 * fma(z, t, (-c * (fma(b, c, a) * i)));
	} else {
		tmp = 2.0 * fma(z, t, ((y * x) - (i * (fma(c, b, a) * c))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+302))
		tmp = Float64(2.0 * fma(z, t, Float64(Float64(-c) * Float64(fma(b, c, a) * i))));
	else
		tmp = Float64(2.0 * fma(z, t, Float64(Float64(y * x) - Float64(i * Float64(fma(c, b, a) * c)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+302]], $MachinePrecision]], N[(2.0 * N[(z * t + N[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(z * t + N[(N[(y * x), $MachinePrecision] - N[(i * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 1.0000000000000001e302 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 68.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 71.8

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 71.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 71.8

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 71.8

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 90.3

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

   7. Applied rewrites 90.3%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.0000000000000001e302

   1. Initial program 98.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 98.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 98.6

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 98.6

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

   4. Applied rewrites 98.6%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq -\infty \lor \neg \left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq 10^{+302}\right):\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, y \cdot x - i \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+242)))
     (* 2.0 (fma z t (* (- c) (* (fma b c a) i))))
     (* 2.0 (fma z t (- (* y x) (* i (* a c))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+242)) {
		tmp = 2.0 * fma(z, t, (-c * (fma(b, c, a) * i)));
	} else {
		tmp = 2.0 * fma(z, t, ((y * x) - (i * (a * c))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+242))
		tmp = Float64(2.0 * fma(z, t, Float64(Float64(-c) * Float64(fma(b, c, a) * i))));
	else
		tmp = Float64(2.0 * fma(z, t, Float64(Float64(y * x) - Float64(i * Float64(a * c)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+242]], $MachinePrecision]], N[(2.0 * N[(z * t + N[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(z * t + N[(N[(y * x), $MachinePrecision] - N[(i * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 1.00000000000000005e242 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 70.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 73.0

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 73.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.0

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 73.0

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

   4. Applied rewrites 73.0%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 88.3

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

   7. Applied rewrites 88.3%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000005e242

   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

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 99.2

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.2

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 92.7

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

   7. Applied rewrites 92.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq -\infty \lor \neg \left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq 10^{+242}\right):\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, y \cdot x - i \cdot \left(a \cdot c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+242)))
     (* 2.0 (fma z t (* (- c) (* (fma b c a) i))))
     (* 2.0 (fma (- i) (* c a) (fma t z (* y x)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+242)) {
		tmp = 2.0 * fma(z, t, (-c * (fma(b, c, a) * i)));
	} else {
		tmp = 2.0 * fma(-i, (c * a), fma(t, z, (y * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+242))
		tmp = Float64(2.0 * fma(z, t, Float64(Float64(-c) * Float64(fma(b, c, a) * i))));
	else
		tmp = Float64(2.0 * fma(Float64(-i), Float64(c * a), fma(t, z, Float64(y * x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+242]], $MachinePrecision]], N[(2.0 * N[(z * t + N[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[((-i) * N[(c * a), $MachinePrecision] + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 1.00000000000000005e242 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 70.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 73.0

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 73.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.0

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 73.0

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

   4. Applied rewrites 73.0%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 88.3

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

   7. Applied rewrites 88.3%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000005e242

   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

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. distribute-lft-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 92.6

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

   5. Applied rewrites 92.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq -\infty \lor \neg \left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq 10^{+242}\right):\\ \;\;\;\;2 \cdot \mathsf{fma}\left(z, t, \left(-c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, c \cdot a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+242)))
     (* (* -2.0 (* (fma c b a) i)) c)
     (* 2.0 (fma (- i) (* c a) (fma t z (* y x)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+242)) {
		tmp = (-2.0 * (fma(c, b, a) * i)) * c;
	} else {
		tmp = 2.0 * fma(-i, (c * a), fma(t, z, (y * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+242))
		tmp = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c);
	else
		tmp = Float64(2.0 * fma(Float64(-i), Float64(c * a), fma(t, z, Float64(y * x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+242]], $MachinePrecision]], N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(2.0 * N[((-i) * N[(c * a), $MachinePrecision] + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 1.00000000000000005e242 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 70.2%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 83.6

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

   5. Applied rewrites 83.6%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000005e242

   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

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. distribute-lft-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 92.6

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

   5. Applied rewrites 92.6%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq -\infty \lor \neg \left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq 10^{+242}\right):\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-i, c \cdot a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+161)))
     (* (* -2.0 (* (fma c b a) i)) c)
     (* 2.0 (fma t z (* y x))))))

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+161]], $MachinePrecision]], N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 72.7%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if -inf.0 < (*.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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

4. Final simplification 77.1%

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

Alternative 7: 73.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+161)))
     (* (* (* c i) (* b c)) -2.0)
     (* 2.0 (fma t z (* y x))))))

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+161]], $MachinePrecision]], N[(N[(N[(c * i), $MachinePrecision] * N[(b * c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 72.7%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 60.3%

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

   if -inf.0 < (*.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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

4. Final simplification 67.7%

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

Alternative 8: 73.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+161)))
     (* (* (* (* c i) -2.0) b) c)
     (* 2.0 (fma t z (* y x))))))

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+161]], $MachinePrecision]], N[(N[(N[(N[(c * i), $MachinePrecision] * -2.0), $MachinePrecision] * b), $MachinePrecision] * c), $MachinePrecision], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 2.0000000000000001e161 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 72.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 75.2

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 75.2

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 75.2

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

   4. Applied rewrites 75.2%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 59.1

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

   7. Applied rewrites 59.1%

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

   if -inf.0 < (*.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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

4. Final simplification 67.1%

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

Alternative 9: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(c \cdot i\right) \cdot \left(b \cdot c\right)\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+266}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot i\right) \cdot c\right) \cdot b\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 (- INFINITY))
     (* (* (* c i) (* b c)) -2.0)
     (if (<= t_1 4e+266)
       (* 2.0 (fma t z (* y x)))
       (* (* (* (* c i) c) b) -2.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((c * i) * (b * c)) * -2.0;
	} else if (t_1 <= 4e+266) {
		tmp = 2.0 * fma(t, z, (y * x));
	} else {
		tmp = (((c * i) * c) * b) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(c * i) * Float64(b * c)) * -2.0);
	elseif (t_1 <= 4e+266)
		tmp = Float64(2.0 * fma(t, z, Float64(y * x)));
	else
		tmp = Float64(Float64(Float64(Float64(c * i) * c) * b) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(c * i), $MachinePrecision] * N[(b * c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 4e+266], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * i), $MachinePrecision] * c), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 66.1%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

   7. Applied rewrites 65.8%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.0000000000000001e266

   1. Initial program 98.5%

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

   3. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   if 4.0000000000000001e266 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 73.3%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 62.5

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

   5. Applied rewrites 62.5%

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

   7. Applied rewrites 64.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq -\infty:\\ \;\;\;\;\left(\left(c \cdot i\right) \cdot \left(b \cdot c\right)\right) \cdot -2\\ \mathbf{elif}\;\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq 4 \cdot 10^{+266}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot i\right) \cdot c\right) \cdot b\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+104} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+119}\right):\\ \;\;\;\;2 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* x y) -4e+104) (not (<= (* x y) 2e+119)))
   (* 2.0 (* y x))
   (* (* (* i c) a) -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) <= -4e+104) || !((x * y) <= 2e+119)) {
		tmp = 2.0 * (y * x);
	} else {
		tmp = ((i * c) * a) * -2.0;
	}
	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) :: tmp
    if (((x * y) <= (-4d+104)) .or. (.not. ((x * y) <= 2d+119))) then
        tmp = 2.0d0 * (y * x)
    else
        tmp = ((i * c) * a) * (-2.0d0)
    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 tmp;
	if (((x * y) <= -4e+104) || !((x * y) <= 2e+119)) {
		tmp = 2.0 * (y * x);
	} else {
		tmp = ((i * c) * a) * -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((x * y) <= -4e+104) or not ((x * y) <= 2e+119):
		tmp = 2.0 * (y * x)
	else:
		tmp = ((i * c) * a) * -2.0
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(x * y) <= -4e+104) || !(Float64(x * y) <= 2e+119))
		tmp = Float64(2.0 * Float64(y * x));
	else
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((x * y) <= -4e+104) || ~(((x * y) <= 2e+119)))
		tmp = 2.0 * (y * x);
	else
		tmp = ((i * c) * a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -4e+104], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e+119]], $MachinePrecision]], N[(2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4e104 or 1.99999999999999989e119 < (*.f64 x y)

   1. Initial program 81.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if -4e104 < (*.f64 x y) < 1.99999999999999989e119

   1. Initial program 90.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 36.3

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

   5. Applied rewrites 36.3%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+104} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+119}\right):\\ \;\;\;\;2 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -50000 \lor \neg \left(z \cdot t \leq 10^{+34}\right):\\ \;\;\;\;t \cdot \left(z + z\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* z t) -50000.0) (not (<= (* z t) 1e+34)))
   (* t (+ z z))
   (* 2.0 (* y x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((z * t) <= -50000.0) || !((z * t) <= 1e+34)) {
		tmp = t * (z + z);
	} else {
		tmp = 2.0 * (y * x);
	}
	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) :: tmp
    if (((z * t) <= (-50000.0d0)) .or. (.not. ((z * t) <= 1d+34))) then
        tmp = t * (z + z)
    else
        tmp = 2.0d0 * (y * x)
    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 tmp;
	if (((z * t) <= -50000.0) || !((z * t) <= 1e+34)) {
		tmp = t * (z + z);
	} else {
		tmp = 2.0 * (y * x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((z * t) <= -50000.0) or not ((z * t) <= 1e+34):
		tmp = t * (z + z)
	else:
		tmp = 2.0 * (y * x)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(z * t) <= -50000.0) || !(Float64(z * t) <= 1e+34))
		tmp = Float64(t * Float64(z + z));
	else
		tmp = Float64(2.0 * Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((z * t) <= -50000.0) || ~(((z * t) <= 1e+34)))
		tmp = t * (z + z);
	else
		tmp = 2.0 * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -50000.0], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+34]], $MachinePrecision]], N[(t * N[(z + z), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5e4 or 9.99999999999999946e33 < (*.f64 z t)

   1. Initial program 84.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

   7. Applied rewrites 52.2%

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

   if -5e4 < (*.f64 z t) < 9.99999999999999946e33

   1. Initial program 89.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 39.0

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

   5. Applied rewrites 39.0%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -50000 \lor \neg \left(z \cdot t \leq 10^{+34}\right):\\ \;\;\;\;t \cdot \left(z + z\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a -4.5e+153) (* (* (* i c) a) -2.0) (* 2.0 (fma t z (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= -4.5e+153) {
		tmp = ((i * c) * a) * -2.0;
	} else {
		tmp = 2.0 * fma(t, z, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= -4.5e+153)
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	else
		tmp = Float64(2.0 * fma(t, z, Float64(y * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, -4.5e+153], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5000000000000001e153

   1. Initial program 81.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   if -4.5000000000000001e153 < a

   1. Initial program 88.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.3% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

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 + z)
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 + z);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 24.8

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

5. Applied rewrites 24.8%

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

7. Applied rewrites 24.8%

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

8. Final simplification 24.8%

   \[\leadsto t \cdot \left(z + z\right) \]
9. Add Preprocessing

Developer Target 1: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = 2.0d0 * (((x * y) + (z * t)) - ((a + (b * c)) * (c * i)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024342 
(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))))