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

Percentage Accurate: 90.4% → 97.0%

Time: 11.7s

Alternatives: 14

Speedup: 1.1×

• 47.2% 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: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x + t \cdot z\right) - i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, \mathsf{fma}\left(t, z, y \cdot x\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t, z, \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (- (+ (* y x) (* t z)) (* i (* (+ (* c b) a) c))) 1e+294)
   (* (fma (fma c b a) (* (- c) i) (fma t z (* y x))) 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 ((((y * x) + (t * z)) - (i * (((c * b) + a) * c))) <= 1e+294) {
		tmp = fma(fma(c, b, a), (-c * i), fma(t, z, (y * x))) * 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(Float64(Float64(y * x) + Float64(t * z)) - Float64(i * Float64(Float64(Float64(c * b) + a) * c))) <= 1e+294)
		tmp = Float64(fma(fma(c, b, a), Float64(Float64(-c) * i), fma(t, z, Float64(y * x))) * 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[(N[(N[(y * x), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision] - N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+294], N[(N[(N[(c * b + a), $MachinePrecision] * N[((-c) * i), $MachinePrecision] + N[(t * z + N[(y * x), $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 x y) (*.f64 z t)) (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)) < 1.00000000000000007e294

   1. Initial program 96.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-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 99.3

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

      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 99.3

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

      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 99.3

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

   4. Applied rewrites 99.3%

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

   if 1.00000000000000007e294 < (-.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i))

   1. Initial program 75.3%

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

      1. lift--.f64 N/A

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

      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 93.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x + t \cdot z\right) - i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, \mathsf{fma}\left(t, z, y \cdot x\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t, z, \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t, z, \left(i \cdot a\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -1e+197)
     (* (fma (fma c b a) (* (- c) i) (* t z)) 2.0)
     (if (<= t_1 2e+41)
       (* (fma y x (fma t z (* (* i a) (- c)))) 2.0)
       (if (<= t_1 2e+285)
         (* (fma (- i) (* (fma c b a) c) (* y x)) 2.0)
         (* (* -2.0 (* (fma c b a) i)) c))))))

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+197) {
		tmp = fma(fma(c, b, a), (-c * i), (t * z)) * 2.0;
	} else if (t_1 <= 2e+41) {
		tmp = fma(y, x, fma(t, z, ((i * a) * -c))) * 2.0;
	} else if (t_1 <= 2e+285) {
		tmp = fma(-i, (fma(c, b, a) * c), (y * x)) * 2.0;
	} else {
		tmp = (-2.0 * (fma(c, b, a) * i)) * c;
	}
	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+197)
		tmp = Float64(fma(fma(c, b, a), Float64(Float64(-c) * i), Float64(t * z)) * 2.0);
	elseif (t_1 <= 2e+41)
		tmp = Float64(fma(y, x, fma(t, z, Float64(Float64(i * a) * Float64(-c)))) * 2.0);
	elseif (t_1 <= 2e+285)
		tmp = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c);
	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+197], N[(N[(N[(c * b + a), $MachinePrecision] * N[((-c) * i), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+41], N[(N[(y * x + N[(t * z + N[(N[(i * a), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+285], N[(N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.9999999999999995e196

   1. Initial program 82.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-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 90.3

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

      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 90.3

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

      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 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 89.2

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

   7. Applied rewrites 89.2%

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

   if -9.9999999999999995e196 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000001e41

   1. Initial program 97.5%

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

      1. lift--.f64 N/A

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

      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 97.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.2

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

   7. Applied rewrites 96.2%

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

   if 2.00000000000000001e41 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e285

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if 2e285 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 72.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t, z, \left(i \cdot a\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, a\right) \cdot c\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(-i, t\_1, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t, z, \left(i \cdot a\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(-i, t\_1, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma c b a) c)) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -1e+197)
     (* (fma (- i) t_1 (* t z)) 2.0)
     (if (<= t_2 2e+41)
       (* (fma y x (fma t z (* (* i a) (- c)))) 2.0)
       (if (<= t_2 2e+285)
         (* (fma (- i) t_1 (* y x)) 2.0)
         (* (* -2.0 (* (fma c b a) i)) c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(c, b, a) * c;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -1e+197) {
		tmp = fma(-i, t_1, (t * z)) * 2.0;
	} else if (t_2 <= 2e+41) {
		tmp = fma(y, x, fma(t, z, ((i * a) * -c))) * 2.0;
	} else if (t_2 <= 2e+285) {
		tmp = fma(-i, t_1, (y * x)) * 2.0;
	} else {
		tmp = (-2.0 * (fma(c, b, a) * i)) * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(c, b, a) * c)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -1e+197)
		tmp = Float64(fma(Float64(-i), t_1, Float64(t * z)) * 2.0);
	elseif (t_2 <= 2e+41)
		tmp = Float64(fma(y, x, fma(t, z, Float64(Float64(i * a) * Float64(-c)))) * 2.0);
	elseif (t_2 <= 2e+285)
		tmp = Float64(fma(Float64(-i), t_1, Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+197], N[(N[((-i) * t$95$1 + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+41], N[(N[(y * x + N[(t * z + N[(N[(i * a), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+285], N[(N[((-i) * t$95$1 + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.9999999999999995e196

   1. Initial program 82.1%

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

   3. Taylor expanded in x around 0

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

      1. 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 85.2

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

   5. Applied rewrites 85.2%

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

   if -9.9999999999999995e196 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000001e41

   1. Initial program 97.5%

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

      1. lift--.f64 N/A

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

      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 97.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.2

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

   7. Applied rewrites 96.2%

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

   if 2.00000000000000001e41 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e285

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if 2e285 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 72.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t, z, \left(i \cdot a\right) \cdot \left(-c\right)\right)\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, b, a\right) \cdot c\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(-i, t\_1, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(-i, t\_1, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma c b a) c)) (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -1e+197)
     (* (fma (- i) t_1 (* t z)) 2.0)
     (if (<= t_2 2e+41)
       (* (fma x y (* t z)) 2.0)
       (if (<= t_2 2e+285)
         (* (fma (- i) t_1 (* y x)) 2.0)
         (* (* -2.0 (* (fma c b a) i)) c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(c, b, a) * c;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -1e+197) {
		tmp = fma(-i, t_1, (t * z)) * 2.0;
	} else if (t_2 <= 2e+41) {
		tmp = fma(x, y, (t * z)) * 2.0;
	} else if (t_2 <= 2e+285) {
		tmp = fma(-i, t_1, (y * x)) * 2.0;
	} else {
		tmp = (-2.0 * (fma(c, b, a) * i)) * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(c, b, a) * c)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -1e+197)
		tmp = Float64(fma(Float64(-i), t_1, Float64(t * z)) * 2.0);
	elseif (t_2 <= 2e+41)
		tmp = Float64(fma(x, y, Float64(t * z)) * 2.0);
	elseif (t_2 <= 2e+285)
		tmp = Float64(fma(Float64(-i), t_1, Float64(y * x)) * 2.0);
	else
		tmp = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(N[(c * b), $MachinePrecision] + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+197], N[(N[((-i) * t$95$1 + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+41], N[(N[(x * y + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+285], N[(N[((-i) * t$95$1 + N[(y * x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.9999999999999995e196

   1. Initial program 82.1%

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

   3. Taylor expanded in x around 0

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

      1. 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 85.2

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

   5. Applied rewrites 85.2%

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

   if -9.9999999999999995e196 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000001e41

   1. Initial program 97.5%

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

      1. lift--.f64 N/A

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

      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 97.5

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

      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 98.3

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

      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 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.3

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

   7. Applied rewrites 90.3%

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

   if 2.00000000000000001e41 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e285

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if 2e285 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 72.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+197}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;t\_2 \leq 10^{+277}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma (- i) (* (fma c b a) c) (* t z)) 2.0))
        (t_2 (* i (* (+ (* c b) a) c))))
   (if (<= t_2 -1e+197)
     t_1
     (if (<= t_2 2e+90)
       (* (fma x y (* t z)) 2.0)
       (if (<= t_2 1e+277) t_1 (* (* -2.0 (* (fma c b a) i)) c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(-i, (fma(c, b, a) * c), (t * z)) * 2.0;
	double t_2 = i * (((c * b) + a) * c);
	double tmp;
	if (t_2 <= -1e+197) {
		tmp = t_1;
	} else if (t_2 <= 2e+90) {
		tmp = fma(x, y, (t * z)) * 2.0;
	} else if (t_2 <= 1e+277) {
		tmp = t_1;
	} else {
		tmp = (-2.0 * (fma(c, b, a) * i)) * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(Float64(-i), Float64(fma(c, b, a) * c), Float64(t * z)) * 2.0)
	t_2 = Float64(i * Float64(Float64(Float64(c * b) + a) * c))
	tmp = 0.0
	if (t_2 <= -1e+197)
		tmp = t_1;
	elseif (t_2 <= 2e+90)
		tmp = Float64(fma(x, y, Float64(t * z)) * 2.0);
	elseif (t_2 <= 1e+277)
		tmp = t_1;
	else
		tmp = Float64(Float64(-2.0 * Float64(fma(c, b, a) * i)) * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[((-i) * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] + N[(t * z), $MachinePrecision]), $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, -1e+197], t$95$1, If[LessEqual[t$95$2, 2e+90], N[(N[(x * y + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$2, 1e+277], t$95$1, N[(N[(-2.0 * N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.9999999999999995e196 or 1.99999999999999993e90 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e277

   1. Initial program 84.8%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 85.1

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

   5. Applied rewrites 85.1%

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

   if -9.9999999999999995e196 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999993e90

   1. Initial program 97.6%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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. 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 97.6

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

      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 98.4

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

      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 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.2

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

   7. Applied rewrites 89.2%

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

   if 1e277 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 72.8%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 87.1

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

   5. Applied rewrites 87.1%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(-i, \mathsf{fma}\left(c, b, a\right) \cdot c, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+276}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -1e+276)
     t_1
     (if (<= t_2 2e+114) (* (fma x y (* t z)) 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 <= -1e+276) {
		tmp = t_1;
	} else if (t_2 <= 2e+114) {
		tmp = fma(x, y, (t * z)) * 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 <= -1e+276)
		tmp = t_1;
	elseif (t_2 <= 2e+114)
		tmp = Float64(fma(x, y, Float64(t * z)) * 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, -1e+276], t$95$1, If[LessEqual[t$95$2, 2e+114], N[(N[(x * y + N[(t * z), $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) < -1.0000000000000001e276 or 2e114 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 79.3%

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

   3. Taylor expanded in 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 84.2

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

   5. Applied rewrites 84.2%

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

   if -1.0000000000000001e276 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e114

   1. Initial program 97.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-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 97.7

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

      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 98.4

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

      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 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 86.5

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

   7. Applied rewrites 86.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+276}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+295}:\\ \;\;\;\;\left(\left(\left(i \cdot c\right) \cdot c\right) \cdot b\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(i \cdot c\right) \cdot b\right) \cdot -2\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -1e+295)
     (* (* (* (* i c) c) b) -2.0)
     (if (<= t_1 2e+114)
       (* (fma x y (* t z)) 2.0)
       (* (* (* (* i c) b) -2.0) c)))))

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+295) {
		tmp = (((i * c) * c) * b) * -2.0;
	} else if (t_1 <= 2e+114) {
		tmp = fma(x, y, (t * z)) * 2.0;
	} else {
		tmp = (((i * c) * b) * -2.0) * c;
	}
	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+295)
		tmp = Float64(Float64(Float64(Float64(i * c) * c) * b) * -2.0);
	elseif (t_1 <= 2e+114)
		tmp = Float64(fma(x, y, Float64(t * z)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(i * c) * b) * -2.0) * c);
	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+295], N[(N[(N[(N[(i * c), $MachinePrecision] * c), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+114], N[(N[(x * y + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(i * c), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision] * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.9999999999999998e294

   1. Initial program 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip3-+ N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}\right) \]

      7. clear-num N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{1}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      12. clear-num N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{i \cdot c}{\color{blue}{\frac{1}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}}}\right) \]

      13. flip3-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 88.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 88.8

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.2

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

   7. Applied rewrites 76.2%

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

   9. Applied rewrites 77.6%

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

   if -9.9999999999999998e294 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e114

   1. Initial program 97.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-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 97.7

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

      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 98.5

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

      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 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.2

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

   7. Applied rewrites 85.2%

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

   if 2e114 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 78.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      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 63.5

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

   5. Applied rewrites 63.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+295}:\\ \;\;\;\;\left(\left(\left(i \cdot c\right) \cdot c\right) \cdot b\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(i \cdot c\right) \cdot b\right) \cdot -2\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+295}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(i \cdot c\right) \cdot b\right) \cdot -2\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -1e+295)
     (* (* (* (* c c) i) b) -2.0)
     (if (<= t_1 2e+114)
       (* (fma x y (* t z)) 2.0)
       (* (* (* (* i c) b) -2.0) c)))))

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+295) {
		tmp = (((c * c) * i) * b) * -2.0;
	} else if (t_1 <= 2e+114) {
		tmp = fma(x, y, (t * z)) * 2.0;
	} else {
		tmp = (((i * c) * b) * -2.0) * c;
	}
	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+295)
		tmp = Float64(Float64(Float64(Float64(c * c) * i) * b) * -2.0);
	elseif (t_1 <= 2e+114)
		tmp = Float64(fma(x, y, Float64(t * z)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(i * c) * b) * -2.0) * c);
	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+295], N[(N[(N[(N[(c * c), $MachinePrecision] * i), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+114], N[(N[(x * y + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(i * c), $MachinePrecision] * b), $MachinePrecision] * -2.0), $MachinePrecision] * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.9999999999999998e294

   1. Initial program 79.5%

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

      1. lift--.f64 N/A

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

      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 88.9

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

      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 88.9

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

      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 88.9

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.2

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

   7. Applied rewrites 76.2%

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

   if -9.9999999999999998e294 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e114

   1. Initial program 97.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-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 97.7

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

      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 98.5

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

      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 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.2

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

   7. Applied rewrites 85.2%

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

   if 2e114 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 78.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      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 63.5

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

   5. Applied rewrites 63.5%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+295}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(i \cdot c\right) \cdot b\right) \cdot -2\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ t_2 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+295}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -1e+295)
     t_1
     (if (<= t_2 2e+114) (* (fma x y (* t z)) 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 <= -1e+295) {
		tmp = t_1;
	} else if (t_2 <= 2e+114) {
		tmp = fma(x, y, (t * z)) * 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 <= -1e+295)
		tmp = t_1;
	elseif (t_2 <= 2e+114)
		tmp = Float64(fma(x, y, Float64(t * z)) * 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, -1e+295], t$95$1, If[LessEqual[t$95$2, 2e+114], N[(N[(x * y + N[(t * z), $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) < -9.9999999999999998e294 or 2e114 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 78.9%

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

      1. lift--.f64 N/A

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

      2. 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 89.2

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

      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 89.2

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

      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 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.4

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

   7. Applied rewrites 68.4%

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

   if -9.9999999999999998e294 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e114

   1. Initial program 97.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-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 97.7

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

      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 98.5

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

      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 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.2

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

   7. Applied rewrites 85.2%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -1 \cdot 10^{+295}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+270}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot a\right) \cdot i\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* i (* (+ (* c b) a) c))))
   (if (<= t_1 -5e+270)
     (* (* (* i c) a) -2.0)
     (if (<= t_1 2e+90) (* (fma x y (* t z)) 2.0) (* (* (* c a) i) -2.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i * (((c * b) + a) * c);
	double tmp;
	if (t_1 <= -5e+270) {
		tmp = ((i * c) * a) * -2.0;
	} else if (t_1 <= 2e+90) {
		tmp = fma(x, y, (t * z)) * 2.0;
	} else {
		tmp = ((c * a) * i) * -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 <= -5e+270)
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	elseif (t_1 <= 2e+90)
		tmp = Float64(fma(x, y, Float64(t * z)) * 2.0);
	else
		tmp = Float64(Float64(Float64(c * a) * i) * -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, -5e+270], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+90], N[(N[(x * y + N[(t * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(c * a), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -4.99999999999999976e270

   1. Initial program 80.5%

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

   3. Taylor expanded in 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 41.3

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

   5. Applied rewrites 41.3%

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

   if -4.99999999999999976e270 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999993e90

   1. Initial program 97.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-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 97.7

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

      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 98.4

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

      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 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 87.6

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

   7. Applied rewrites 87.6%

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

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

   1. Initial program 78.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip3-+ N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}\right) \]

      7. clear-num N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{1}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      12. clear-num N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{i \cdot c}{\color{blue}{\frac{1}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}}}\right) \]

      13. flip3-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 88.1

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 88.1

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. 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. lower-*.f64 39.5

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

   7. Applied rewrites 39.5%

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

   9. Applied rewrites 39.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq -5 \cdot 10^{+270}:\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{elif}\;i \cdot \left(\left(c \cdot b + a\right) \cdot c\right) \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot a\right) \cdot i\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 4 \cdot 10^{+16}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{elif}\;t \cdot z \leq 10^{+110}:\\ \;\;\;\;\left(\left(c \cdot a\right) \cdot i\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (* t z))))
   (if (<= (* t z) -2e+86)
     t_1
     (if (<= (* t z) 4e+16)
       (* (* y x) 2.0)
       (if (<= (* t z) 1e+110) (* (* (* c a) i) -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 * (t * z);
	double tmp;
	if ((t * z) <= -2e+86) {
		tmp = t_1;
	} else if ((t * z) <= 4e+16) {
		tmp = (y * x) * 2.0;
	} else if ((t * z) <= 1e+110) {
		tmp = ((c * a) * i) * -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 = 2.0d0 * (t * z)
    if ((t * z) <= (-2d+86)) then
        tmp = t_1
    else if ((t * z) <= 4d+16) then
        tmp = (y * x) * 2.0d0
    else if ((t * z) <= 1d+110) then
        tmp = ((c * a) * i) * (-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 = 2.0 * (t * z);
	double tmp;
	if ((t * z) <= -2e+86) {
		tmp = t_1;
	} else if ((t * z) <= 4e+16) {
		tmp = (y * x) * 2.0;
	} else if ((t * z) <= 1e+110) {
		tmp = ((c * a) * i) * -2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = 2.0 * (t * z)
	tmp = 0
	if (t * z) <= -2e+86:
		tmp = t_1
	elif (t * z) <= 4e+16:
		tmp = (y * x) * 2.0
	elif (t * z) <= 1e+110:
		tmp = ((c * a) * i) * -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(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -2e+86)
		tmp = t_1;
	elseif (Float64(t * z) <= 4e+16)
		tmp = Float64(Float64(y * x) * 2.0);
	elseif (Float64(t * z) <= 1e+110)
		tmp = Float64(Float64(Float64(c * a) * i) * -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 = 2.0 * (t * z);
	tmp = 0.0;
	if ((t * z) <= -2e+86)
		tmp = t_1;
	elseif ((t * z) <= 4e+16)
		tmp = (y * x) * 2.0;
	elseif ((t * z) <= 1e+110)
		tmp = ((c * a) * i) * -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[(2.0 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -2e+86], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 4e+16], N[(N[(y * x), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 1e+110], N[(N[(N[(c * a), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2e86 or 1e110 < (*.f64 z t)

   1. Initial program 84.6%

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

   3. Taylor expanded in z around 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 60.9

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

   5. Applied rewrites 60.9%

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

   if -2e86 < (*.f64 z t) < 4e16

   1. Initial program 90.7%

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

   3. Taylor expanded in x around 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 43.1

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

   5. Applied rewrites 43.1%

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

   if 4e16 < (*.f64 z t) < 1e110

   1. Initial program 94.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip3-+ N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}\right) \]

      7. clear-num N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot i\right) \cdot \color{blue}{\frac{1}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      8. un-div-inv N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\frac{c \cdot i}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}}\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{\color{blue}{i \cdot c}}{\frac{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}{{a}^{3} + {\left(b \cdot c\right)}^{3}}}\right) \]

      12. clear-num N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \frac{i \cdot c}{\color{blue}{\frac{1}{\frac{{a}^{3} + {\left(b \cdot c\right)}^{3}}{a \cdot a + \left(\left(b \cdot c\right) \cdot \left(b \cdot c\right) - a \cdot \left(b \cdot c\right)\right)}}}}\right) \]

      13. flip3-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. 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. lower-*.f64 44.9

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

   7. Applied rewrites 44.9%

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

   9. Applied rewrites 50.3%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+86}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 4 \cdot 10^{+16}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{elif}\;t \cdot z \leq 10^{+110}:\\ \;\;\;\;\left(\left(c \cdot a\right) \cdot i\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.8% 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 89.0%

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

   1. lift--.f64 N/A

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

   2. 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 93.0%

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

5. Final simplification 93.0%

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

Alternative 13: 44.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 10^{+110}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (* t z))))
   (if (<= (* t z) -2e+86) t_1 (if (<= (* t z) 1e+110) (* (* y x) 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 * (t * z);
	double tmp;
	if ((t * z) <= -2e+86) {
		tmp = t_1;
	} else if ((t * z) <= 1e+110) {
		tmp = (y * x) * 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 = 2.0d0 * (t * z)
    if ((t * z) <= (-2d+86)) then
        tmp = t_1
    else if ((t * z) <= 1d+110) then
        tmp = (y * x) * 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 = 2.0 * (t * z);
	double tmp;
	if ((t * z) <= -2e+86) {
		tmp = t_1;
	} else if ((t * z) <= 1e+110) {
		tmp = (y * x) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = 2.0 * (t * z)
	tmp = 0
	if (t * z) <= -2e+86:
		tmp = t_1
	elif (t * z) <= 1e+110:
		tmp = (y * x) * 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(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -2e+86)
		tmp = t_1;
	elseif (Float64(t * z) <= 1e+110)
		tmp = Float64(Float64(y * x) * 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 = 2.0 * (t * z);
	tmp = 0.0;
	if ((t * z) <= -2e+86)
		tmp = t_1;
	elseif ((t * z) <= 1e+110)
		tmp = (y * x) * 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[(2.0 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -2e+86], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 1e+110], N[(N[(y * x), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e86 or 1e110 < (*.f64 z t)

   1. Initial program 84.6%

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

   3. Taylor expanded in z around 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 60.9

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

   5. Applied rewrites 60.9%

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

   if -2e86 < (*.f64 z t) < 1e110

   1. Initial program 91.1%

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

   3. Taylor expanded in x around 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 41.2

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

   5. Applied rewrites 41.2%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+86}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 10^{+110}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 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 26.8

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

5. Applied rewrites 26.8%

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

Developer Target 1: 94.7% 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 2024331 
(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))))