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

Percentage Accurate: 90.4% → 94.9%

Time: 10.7s

Alternatives: 14

Speedup: 1.1×

• 47.5% 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: 90.4% 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: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+192}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, \mathsf{fma}\left(t, z, x \cdot y\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t, z, \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* i (* (+ (* c b) a) c)) 1e+192)
   (* (fma (fma c b a) (* (- c) i) (fma t z (* x y))) 2.0)
   (* (fma y x (fma t z (* (* (fma c b a) i) (- c)))) 2.0)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((i * (((c * b) + a) * c)) <= 1e+192) {
		tmp = fma(fma(c, b, a), (-c * i), fma(t, z, (x * y))) * 2.0;
	} else {
		tmp = fma(y, x, fma(t, z, ((fma(c, b, a) * i) * -c))) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(i * Float64(Float64(Float64(c * b) + a) * c)) <= 1e+192)
		tmp = Float64(fma(fma(c, b, a), Float64(Float64(-c) * i), fma(t, z, Float64(x * y))) * 2.0);
	else
		tmp = Float64(fma(y, x, fma(t, z, Float64(Float64(fma(c, b, a) * i) * Float64(-c)))) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], 1e+192], N[(N[(N[(c * b + a), $MachinePrecision] * N[((-c) * i), $MachinePrecision] + N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(y * x + N[(t * z + N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.00000000000000004e192

   1. Initial program 94.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 98.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 99.9

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 1.00000000000000004e192 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 74.1%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

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

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 98.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+192}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, \mathsf{fma}\left(t, z, x \cdot y\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t, z, \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+248}:\\ \;\;\;\;\mathsf{fma}\left(-c, \mathsf{fma}\left(b, c, a\right) \cdot i, z \cdot t\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, z \cdot t\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -1e+248)
     (* (fma (- c) (* (fma b c a) i) (* z t)) 2.0)
     (if (<= t_1 1e-14)
       (* (fma t z (* x y)) 2.0)
       (if (<= t_1 1e+261)
         (* (fma (- i) (* (fma c b a) c) (* z t)) 2.0)
         (* -2.0 (* (* i c) (fma b c a))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -1e+248) {
		tmp = fma(-c, (fma(b, c, a) * i), (z * t)) * 2.0;
	} else if (t_1 <= 1e-14) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else if (t_1 <= 1e+261) {
		tmp = fma(-i, (fma(c, b, a) * c), (z * t)) * 2.0;
	} else {
		tmp = -2.0 * ((i * c) * fma(b, c, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= -1e+248)
		tmp = Float64(fma(Float64(-c), Float64(fma(b, c, a) * i), Float64(z * t)) * 2.0);
	elseif (t_1 <= 1e-14)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	elseif (t_1 <= 1e+261)
		tmp = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(z * t)) * 2.0);
	else
		tmp = Float64(-2.0 * Float64(Float64(i * c) * fma(b, c, a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+248], N[(N[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e-14], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+261], N[(N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(-2.0 * N[(N[(i * c), $MachinePrecision] * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.00000000000000005e248

   1. Initial program 85.1%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 43.5

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

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.8

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

   8. Applied rewrites 96.8%

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

   if -1.00000000000000005e248 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.99999999999999999e-15

   1. Initial program 98.1%

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

   3. Taylor expanded in 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 95.7

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

   5. Applied rewrites 95.7%

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

   if 9.99999999999999999e-15 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999993e260

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if 9.9999999999999993e260 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 71.6%

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

   3. Taylor expanded in 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 94.4

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

   5. Applied rewrites 94.4%

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

   7. Applied rewrites 94.5%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+248}:\\ \;\;\;\;\mathsf{fma}\left(-c, \mathsf{fma}\left(b, c, a\right) \cdot i, z \cdot t\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, z \cdot t\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-c, \mathsf{fma}\left(b, c, a\right) \cdot i, z \cdot t\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(\left(-b\right) \cdot c, i \cdot c, z \cdot t\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 (- INFINITY))
     (* (fma (- c) (* (fma b c a) i) (* z t)) 2.0)
     (if (<= t_1 1e+261)
       (* (fma y x (fma (* (- b) c) (* i c) (* z t))) 2.0)
       (* -2.0 (* (* i c) (fma b c a)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-c, (fma(b, c, a) * i), (z * t)) * 2.0;
	} else if (t_1 <= 1e+261) {
		tmp = fma(y, x, fma((-b * c), (i * c), (z * t))) * 2.0;
	} else {
		tmp = -2.0 * ((i * c) * fma(b, c, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(-c), Float64(fma(b, c, a) * i), Float64(z * t)) * 2.0);
	elseif (t_1 <= 1e+261)
		tmp = Float64(fma(y, x, fma(Float64(Float64(-b) * c), Float64(i * c), Float64(z * t))) * 2.0);
	else
		tmp = Float64(-2.0 * Float64(Float64(i * c) * fma(b, c, a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+261], N[(N[(y * x + N[(N[((-b) * c), $MachinePrecision] * N[(i * c), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(-2.0 * N[(N[(i * c), $MachinePrecision] * N[(b * c + a), $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 83.8%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 45.4

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

   5. Applied rewrites 45.4%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999993e260

   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. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

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

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. lower-fma.f64 N/A

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

   7. Applied rewrites 92.0%

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

   if 9.9999999999999993e260 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 71.6%

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

   3. Taylor expanded in 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 94.4

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

   5. Applied rewrites 94.4%

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

   7. Applied rewrites 94.5%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-c, \mathsf{fma}\left(b, c, a\right) \cdot i, z \cdot t\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(\left(-b\right) \cdot c, i \cdot c, z \cdot t\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+248}:\\ \;\;\;\;\mathsf{fma}\left(-c, \mathsf{fma}\left(b, c, a\right) \cdot i, z \cdot t\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -1e+248)
     (* (fma (- c) (* (fma b c a) i) (* z t)) 2.0)
     (if (<= t_1 2e+119)
       (* (fma t z (* x y)) 2.0)
       (* -2.0 (* (* i c) (fma b c a)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -1e+248) {
		tmp = fma(-c, (fma(b, c, a) * i), (z * t)) * 2.0;
	} else if (t_1 <= 2e+119) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else {
		tmp = -2.0 * ((i * c) * fma(b, c, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= -1e+248)
		tmp = Float64(fma(Float64(-c), Float64(fma(b, c, a) * i), Float64(z * t)) * 2.0);
	elseif (t_1 <= 2e+119)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	else
		tmp = Float64(-2.0 * Float64(Float64(i * c) * fma(b, c, a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+248], N[(N[((-c) * N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+119], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(-2.0 * N[(N[(i * c), $MachinePrecision] * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.00000000000000005e248

   1. Initial program 85.1%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 43.5

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

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.8

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

   8. Applied rewrites 96.8%

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

   if -1.00000000000000005e248 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999989e119

   1. Initial program 98.4%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   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 91.1

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

   5. Applied rewrites 91.1%

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

   if 1.99999999999999989e119 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 85.2

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

   5. Applied rewrites 85.2%

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

   7. Applied rewrites 86.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+248}:\\ \;\;\;\;\mathsf{fma}\left(-c, \mathsf{fma}\left(b, c, a\right) \cdot i, z \cdot t\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 \cdot \left(\left(i \cdot c\right) \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* -2.0 (* (* i c) (fma b c a)))) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 2e+119) (* (fma t z (* x y)) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -2.0 * ((i * c) * fma(b, c, a));
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+119) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(-2.0 * Float64(Float64(i * c) * fma(b, c, a)))
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+119)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(-2.0 * N[(N[(i * c), $MachinePrecision] * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 2e+119], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 1.99999999999999989e119 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.3%

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

   3. Taylor expanded in 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 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 92.2%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999989e119

   1. Initial program 98.4%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   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 89.6

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

   5. Applied rewrites 89.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -\infty:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(i \cdot c\right) \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* -2.0 (* (fma c b a) i)) c)) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 5e+205) (* (fma t z (* x y)) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (-2.0 * (fma(c, b, a) * i)) * c;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+205) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+205)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+205], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 5.0000000000000002e205 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 78.8%

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

   3. Taylor expanded in 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 94.9

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

   5. Applied rewrites 94.9%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.0000000000000002e205

   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 86.9

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

   5. Applied rewrites 86.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -\infty:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(-2 \cdot b\right) \cdot \left(\left(i \cdot c\right) \cdot c\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 (- INFINITY))
     (* (* -2.0 b) (* (* i c) c))
     (if (<= t_1 2e+306)
       (* (fma t z (* x y)) 2.0)
       (* (* (* (* c c) i) 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 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-2.0 * b) * ((i * c) * c);
	} else if (t_1 <= 2e+306) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else {
		tmp = (((c * c) * i) * b) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-2.0 * b) * Float64(Float64(i * c) * c));
	elseif (t_1 <= 2e+306)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(c * c) * i) * b) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-2.0 * b), $MachinePrecision] * N[(N[(i * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * i), $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 83.8%

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

   3. Taylor expanded in 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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 73.9%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000003e306

   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. 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 85.4

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

   5. Applied rewrites 85.4%

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

   if 2.00000000000000003e306 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 71.1%

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

   3. Taylor expanded in 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 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 84.8%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -\infty:\\ \;\;\;\;\left(-2 \cdot b\right) \cdot \left(\left(i \cdot c\right) \cdot c\right)\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot c\right) \cdot \left(i \cdot c\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 (- INFINITY))
     (* (* (* -2.0 b) c) (* i c))
     (if (<= t_1 2e+306)
       (* (fma t z (* x y)) 2.0)
       (* (* (* (* c c) i) 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 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((-2.0 * b) * c) * (i * c);
	} else if (t_1 <= 2e+306) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else {
		tmp = (((c * c) * i) * b) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-2.0 * b) * c) * Float64(i * c));
	elseif (t_1 <= 2e+306)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(c * c) * i) * b) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-2.0 * b), $MachinePrecision] * c), $MachinePrecision] * N[(i * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * i), $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 83.8%

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

   3. Taylor expanded in 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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 67.5%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000003e306

   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. 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 85.4

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

   5. Applied rewrites 85.4%

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

   if 2.00000000000000003e306 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 71.1%

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

   3. Taylor expanded in 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 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 84.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -\infty:\\ \;\;\;\;\left(\left(-2 \cdot b\right) \cdot c\right) \cdot \left(i \cdot c\right)\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* (* c c) i) b) -2.0)) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 2e+306) (* (fma t z (* x y)) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((c * c) * i) * b) * -2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+306) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(c * c) * i) * b) * -2.0)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+306)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(c * c), $MachinePrecision] * i), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 2e+306], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 2.00000000000000003e306 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 77.8%

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

   3. Taylor expanded in 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 96.4

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 74.8%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000003e306

   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. 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 85.4

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

   5. Applied rewrites 85.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -\infty:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* i c) a) -2.0)) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 1e+261) (* (fma t z (* x y)) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((i * c) * a) * -2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+261) {
		tmp = fma(t, z, (x * y)) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(i * c) * a) * -2.0)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1e+261)
		tmp = Float64(fma(t, z, Float64(x * y)) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1e+261], N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 9.9999999999999993e260 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 78.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 48.6

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

   5. Applied rewrites 48.6%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999993e260

   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. 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 85.9

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

   5. Applied rewrites 85.9%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -\infty:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot t\right) \cdot 2\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+220}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{-172}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;z \cdot t \leq 10^{+132}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* z t) 2.0)))
   (if (<= (* z t) -5e+220)
     t_1
     (if (<= (* z t) -5e-172)
       (* (* (* i c) a) -2.0)
       (if (<= (* z t) 1e+132) (* (* x y) 2.0) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (z * t) * 2.0;
	double tmp;
	if ((z * t) <= -5e+220) {
		tmp = t_1;
	} else if ((z * t) <= -5e-172) {
		tmp = ((i * c) * a) * -2.0;
	} else if ((z * t) <= 1e+132) {
		tmp = (x * y) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * t) * 2.0d0
    if ((z * t) <= (-5d+220)) then
        tmp = t_1
    else if ((z * t) <= (-5d-172)) then
        tmp = ((i * c) * a) * (-2.0d0)
    else if ((z * t) <= 1d+132) then
        tmp = (x * y) * 2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (z * t) * 2.0;
	double tmp;
	if ((z * t) <= -5e+220) {
		tmp = t_1;
	} else if ((z * t) <= -5e-172) {
		tmp = ((i * c) * a) * -2.0;
	} else if ((z * t) <= 1e+132) {
		tmp = (x * y) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (z * t) * 2.0
	tmp = 0
	if (z * t) <= -5e+220:
		tmp = t_1
	elif (z * t) <= -5e-172:
		tmp = ((i * c) * a) * -2.0
	elif (z * t) <= 1e+132:
		tmp = (x * y) * 2.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(z * t) * 2.0)
	tmp = 0.0
	if (Float64(z * t) <= -5e+220)
		tmp = t_1;
	elseif (Float64(z * t) <= -5e-172)
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	elseif (Float64(z * t) <= 1e+132)
		tmp = Float64(Float64(x * y) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (z * t) * 2.0;
	tmp = 0.0;
	if ((z * t) <= -5e+220)
		tmp = t_1;
	elseif ((z * t) <= -5e-172)
		tmp = ((i * c) * a) * -2.0;
	elseif ((z * t) <= 1e+132)
		tmp = (x * y) * 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+220], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -5e-172], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+132], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000002e220 or 9.99999999999999991e131 < (*.f64 z t)

   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 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 66.1

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

   5. Applied rewrites 66.1%

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

   if -5.0000000000000002e220 < (*.f64 z t) < -4.9999999999999999e-172

   1. Initial program 89.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 40.3

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

   5. Applied rewrites 40.3%

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

   if -4.9999999999999999e-172 < (*.f64 z t) < 9.99999999999999991e131

   1. Initial program 91.0%

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

   3. Taylor expanded in 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 45.9

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

   5. Applied rewrites 45.9%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+220}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 2\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{-172}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;z \cdot t \leq 10^{+132}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

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

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

   16. lower-*.f64 N/A

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

4. Applied rewrites 97.3%

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

5. Final simplification 97.3%

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

Alternative 13: 43.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot t\right) \cdot 2\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+132}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* z t) 2.0)))
   (if (<= (* z t) -2e+58) t_1 (if (<= (* z t) 1e+132) (* (* x y) 2.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (z * t) * 2.0;
	double tmp;
	if ((z * t) <= -2e+58) {
		tmp = t_1;
	} else if ((z * t) <= 1e+132) {
		tmp = (x * y) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * t) * 2.0d0
    if ((z * t) <= (-2d+58)) then
        tmp = t_1
    else if ((z * t) <= 1d+132) then
        tmp = (x * y) * 2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (z * t) * 2.0;
	double tmp;
	if ((z * t) <= -2e+58) {
		tmp = t_1;
	} else if ((z * t) <= 1e+132) {
		tmp = (x * y) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (z * t) * 2.0
	tmp = 0
	if (z * t) <= -2e+58:
		tmp = t_1
	elif (z * t) <= 1e+132:
		tmp = (x * y) * 2.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(z * t) * 2.0)
	tmp = 0.0
	if (Float64(z * t) <= -2e+58)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+132)
		tmp = Float64(Float64(x * y) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (z * t) * 2.0;
	tmp = 0.0;
	if ((z * t) <= -2e+58)
		tmp = t_1;
	elseif ((z * t) <= 1e+132)
		tmp = (x * y) * 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+58], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+132], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.99999999999999989e58 or 9.99999999999999991e131 < (*.f64 z t)

   1. Initial program 89.3%

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

   3. Taylor expanded in 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 59.0

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

   5. Applied rewrites 59.0%

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

   if -1.99999999999999989e58 < (*.f64 z t) < 9.99999999999999991e131

   1. Initial program 90.2%

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

   3. Taylor expanded in 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 41.8

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

   5. Applied rewrites 41.8%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+58}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 2\\ \mathbf{elif}\;z \cdot t \leq 10^{+132}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot t\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

3. Taylor expanded in 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 31.4

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

5. Applied rewrites 31.4%

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

6. Final simplification 31.4%

   \[\leadsto \left(z \cdot t\right) \cdot 2 \]
7. Add Preprocessing

Developer Target 1: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024304 
(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))))