Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 73.4% → 82.5%

Time: 15.2s

Alternatives: 20

Speedup: 1.0×

• 58.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (-
          (* (- (* c t) (* i y)) j)
          (- (* (- (* a t) (* z y)) x) (* (- (* i a) (* c z)) b)))))
   (if (<= t_1 1e+292)
     t_1
     (if (<= t_1 INFINITY)
       (*
        (-
         (fma
          c
          j
          (/ (fma (fma (- j) i (* z x)) y (* (fma (- c) z (* i a)) b)) t))
         (* a x))
        t)
       (fma (fma (- b) c (* y x)) z (* (* j t) c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (((c * t) - (i * y)) * j) - ((((a * t) - (z * y)) * x) - (((i * a) - (c * z)) * b));
	double tmp;
	if (t_1 <= 1e+292) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (fma(c, j, (fma(fma(-j, i, (z * x)), y, (fma(-c, z, (i * a)) * b)) / t)) - (a * x)) * t;
	} else {
		tmp = fma(fma(-b, c, (y * x)), z, ((j * t) * c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(Float64(c * t) - Float64(i * y)) * j) - Float64(Float64(Float64(Float64(a * t) - Float64(z * y)) * x) - Float64(Float64(Float64(i * a) - Float64(c * z)) * b)))
	tmp = 0.0
	if (t_1 <= 1e+292)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(fma(c, j, Float64(fma(fma(Float64(-j), i, Float64(z * x)), y, Float64(fma(Float64(-c), z, Float64(i * a)) * b)) / t)) - Float64(a * x)) * t);
	else
		tmp = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(Float64(j * t) * c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision] - N[(N[(N[(N[(a * t), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - N[(N[(N[(i * a), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+292], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(N[(c * j + N[(N[(N[((-j) * i + N[(z * x), $MachinePrecision]), $MachinePrecision] * y + N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] - N[(a * x), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < 1e292

   1. Initial program 92.3%

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

   if 1e292 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < +inf.0

   1. Initial program 80.9%

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

   3. Taylor expanded in t around -inf

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

   5. Applied rewrites 91.1%

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

   if +inf.0 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y))))

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 0.0

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 3.9

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

   4. Applied rewrites 3.9%

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

   5. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 56.9%

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

   8. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-b, c, x \cdot y\right), z, c \cdot \left(j \cdot t\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 66.8%

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

4. Final simplification 86.9%

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

Alternative 2: 82.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < 2e285

   1. Initial program 92.3%

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

   if 2e285 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < +inf.0

   1. Initial program 81.2%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 89.6%

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

   if +inf.0 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y))))

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 0.0

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 3.9

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

   4. Applied rewrites 3.9%

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

   5. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 56.9%

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

   8. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-b, c, x \cdot y\right), z, c \cdot \left(j \cdot t\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 66.8%

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

4. Final simplification 86.5%

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

Alternative 3: 78.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right)\\ t_2 := \left(c \cdot t - i \cdot y\right) \cdot j - \left(\left(a \cdot t - z \cdot y\right) \cdot x - \left(i \cdot a - c \cdot z\right) \cdot b\right)\\ \mathbf{if}\;t\_2 \leq 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, i \cdot b\right), a, \mathsf{fma}\left(t\_1, z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, j \cdot c\right), t, \mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, \left(j \cdot t\right) \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- b) c (* y x)))
        (t_2
         (-
          (* (- (* c t) (* i y)) j)
          (- (* (- (* a t) (* z y)) x) (* (- (* i a) (* c z)) b)))))
   (if (<= t_2 1e-229)
     (fma (fma (- x) t (* i b)) a (fma t_1 z (* (fma (- i) y (* c t)) j)))
     (if (<= t_2 INFINITY)
       (fma
        (fma (- x) a (* j c))
        t
        (fma (fma (- j) i (* z x)) y (* (fma (- c) z (* i a)) b)))
       (fma t_1 z (* (* j t) c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x));
	double t_2 = (((c * t) - (i * y)) * j) - ((((a * t) - (z * y)) * x) - (((i * a) - (c * z)) * b));
	double tmp;
	if (t_2 <= 1e-229) {
		tmp = fma(fma(-x, t, (i * b)), a, fma(t_1, z, (fma(-i, y, (c * t)) * j)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma(fma(-x, a, (j * c)), t, fma(fma(-j, i, (z * x)), y, (fma(-c, z, (i * a)) * b)));
	} else {
		tmp = fma(t_1, z, ((j * t) * c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-b), c, Float64(y * x))
	t_2 = Float64(Float64(Float64(Float64(c * t) - Float64(i * y)) * j) - Float64(Float64(Float64(Float64(a * t) - Float64(z * y)) * x) - Float64(Float64(Float64(i * a) - Float64(c * z)) * b)))
	tmp = 0.0
	if (t_2 <= 1e-229)
		tmp = fma(fma(Float64(-x), t, Float64(i * b)), a, fma(t_1, z, Float64(fma(Float64(-i), y, Float64(c * t)) * j)));
	elseif (t_2 <= Inf)
		tmp = fma(fma(Float64(-x), a, Float64(j * c)), t, fma(fma(Float64(-j), i, Float64(z * x)), y, Float64(fma(Float64(-c), z, Float64(i * a)) * b)));
	else
		tmp = fma(t_1, z, Float64(Float64(j * t) * c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision] - N[(N[(N[(N[(a * t), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - N[(N[(N[(i * a), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-229], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a + N[(t$95$1 * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t + N[(N[((-j) * i + N[(z * x), $MachinePrecision]), $MachinePrecision] * y + N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * z + N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < 1.00000000000000007e-229

   1. Initial program 90.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 88.6%

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

   if 1.00000000000000007e-229 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < +inf.0

   1. Initial program 86.7%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 88.5%

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

   if +inf.0 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y))))

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 0.0

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 3.9

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

   4. Applied rewrites 3.9%

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

   5. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 56.9%

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

   8. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-b, c, x \cdot y\right), z, c \cdot \left(j \cdot t\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 66.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot t - i \cdot y\right) \cdot j - \left(\left(a \cdot t - z \cdot y\right) \cdot x - \left(i \cdot a - c \cdot z\right) \cdot b\right) \leq 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, i \cdot b\right), a, \mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\right)\\ \mathbf{elif}\;\left(c \cdot t - i \cdot y\right) \cdot j - \left(\left(a \cdot t - z \cdot y\right) \cdot x - \left(i \cdot a - c \cdot z\right) \cdot b\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, j \cdot c\right), t, \mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \left(j \cdot t\right) \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, j \cdot c\right), t, \mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z 5.8e+45)
   (fma
    (fma (- x) a (* j c))
    t
    (fma (fma (- j) i (* z x)) y (* (fma (- c) z (* i a)) b)))
   (fma (fma (- b) c (* y x)) z (* (fma (- i) y (* c t)) j))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= 5.8e+45) {
		tmp = fma(fma(-x, a, (j * c)), t, fma(fma(-j, i, (z * x)), y, (fma(-c, z, (i * a)) * b)));
	} else {
		tmp = fma(fma(-b, c, (y * x)), z, (fma(-i, y, (c * t)) * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= 5.8e+45)
		tmp = fma(fma(Float64(-x), a, Float64(j * c)), t, fma(fma(Float64(-j), i, Float64(z * x)), y, Float64(fma(Float64(-c), z, Float64(i * a)) * b)));
	else
		tmp = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, 5.8e+45], N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t + N[(N[((-j) * i + N[(z * x), $MachinePrecision]), $MachinePrecision] * y + N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.7999999999999994e45

   1. Initial program 73.3%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 80.0%

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

   if 5.7999999999999994e45 < z

   1. Initial program 61.3%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 89.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, j \cdot c\right), t, \mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\\ \mathbf{if}\;i \leq -1.8 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 8.5 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- y) j (* b a)) i (* (fma (- c) b (* y x)) z))))
   (if (<= i -1.8e+102)
     t_1
     (if (<= i 8.5e+74)
       (fma (fma (- b) c (* y x)) z (* (fma (- i) y (* c t)) j))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(fma(-y, j, (b * a)), i, (fma(-c, b, (y * x)) * z));
	double tmp;
	if (i <= -1.8e+102) {
		tmp = t_1;
	} else if (i <= 8.5e+74) {
		tmp = fma(fma(-b, c, (y * x)), z, (fma(-i, y, (c * t)) * j));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-y), j, Float64(b * a)), i, Float64(fma(Float64(-c), b, Float64(y * x)) * z))
	tmp = 0.0
	if (i <= -1.8e+102)
		tmp = t_1;
	elseif (i <= 8.5e+74)
		tmp = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.8e+102], t$95$1, If[LessEqual[i, 8.5e+74], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.8000000000000001e102 or 8.50000000000000028e74 < i

   1. Initial program 69.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in z around 0

      \[\leadsto -1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \color{blue}{\left(a \cdot \left(b \cdot i\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 79.8%

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

   if -1.8000000000000001e102 < i < 8.50000000000000028e74

   1. Initial program 71.7%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 72.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.8 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\\ \mathbf{elif}\;i \leq 8.5 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{if}\;t \leq -5 \cdot 10^{+190}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) a (* j c)) t)))
   (if (<= t -5e+190)
     t_1
     (if (<= t 2.9e+19)
       (fma (fma (- y) j (* b a)) i (* (fma (- c) b (* y x)) z))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-x, a, (j * c)) * t;
	double tmp;
	if (t <= -5e+190) {
		tmp = t_1;
	} else if (t <= 2.9e+19) {
		tmp = fma(fma(-y, j, (b * a)), i, (fma(-c, b, (y * x)) * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-x), a, Float64(j * c)) * t)
	tmp = 0.0
	if (t <= -5e+190)
		tmp = t_1;
	elseif (t <= 2.9e+19)
		tmp = fma(fma(Float64(-y), j, Float64(b * a)), i, Float64(fma(Float64(-c), b, Float64(y * x)) * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -5e+190], t$95$1, If[LessEqual[t, 2.9e+19], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.00000000000000036e190 or 2.9e19 < t

   1. Initial program 57.1%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, c \cdot j\right)} \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, c \cdot j\right) \cdot t \]

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   if -5.00000000000000036e190 < t < 2.9e19

   1. Initial program 77.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in z around 0

      \[\leadsto -1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \color{blue}{\left(a \cdot \left(b \cdot i\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.2%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+190}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \left(j \cdot t\right) \cdot c\right)\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 400000000:\\ \;\;\;\;\left(i \cdot b\right) \cdot a + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- b) c (* y x)) z (* (* j t) c))))
   (if (<= z -1.75e-44)
     t_1
     (if (<= z 400000000.0) (+ (* (* i b) a) (* (- (* c t) (* i y)) j)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(fma(-b, c, (y * x)), z, ((j * t) * c));
	double tmp;
	if (z <= -1.75e-44) {
		tmp = t_1;
	} else if (z <= 400000000.0) {
		tmp = ((i * b) * a) + (((c * t) - (i * y)) * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(Float64(j * t) * c))
	tmp = 0.0
	if (z <= -1.75e-44)
		tmp = t_1;
	elseif (z <= 400000000.0)
		tmp = Float64(Float64(Float64(i * b) * a) + Float64(Float64(Float64(c * t) - Float64(i * y)) * j));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.75e-44], t$95$1, If[LessEqual[z, 400000000.0], N[(N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7499999999999999e-44 or 4e8 < z

   1. Initial program 64.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 64.8

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 65.5

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

   4. Applied rewrites 65.5%

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

   5. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 77.2%

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

   8. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-b, c, x \cdot y\right), z, c \cdot \left(j \cdot t\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.5%

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

   if -1.7499999999999999e-44 < z < 4e8

   1. Initial program 77.7%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \left(j \cdot t\right) \cdot c\right)\\ \mathbf{elif}\;z \leq 400000000:\\ \;\;\;\;\left(i \cdot b\right) \cdot a + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \left(j \cdot t\right) \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -6 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-167}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{elif}\;z \leq 128000000:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)))
   (if (<= z -6e+151)
     t_1
     (if (<= z -1.5e+71)
       (* (fma (- b) z (* j t)) c)
       (if (<= z -1.8e-167)
         (* (fma b a (* (- y) j)) i)
         (if (<= z 128000000.0) (* (fma (- i) y (* c t)) j) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x)) * z;
	double tmp;
	if (z <= -6e+151) {
		tmp = t_1;
	} else if (z <= -1.5e+71) {
		tmp = fma(-b, z, (j * t)) * c;
	} else if (z <= -1.8e-167) {
		tmp = fma(b, a, (-y * j)) * i;
	} else if (z <= 128000000.0) {
		tmp = fma(-i, y, (c * t)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-b), c, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -6e+151)
		tmp = t_1;
	elseif (z <= -1.5e+71)
		tmp = Float64(fma(Float64(-b), z, Float64(j * t)) * c);
	elseif (z <= -1.8e-167)
		tmp = Float64(fma(b, a, Float64(Float64(-y) * j)) * i);
	elseif (z <= 128000000.0)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -6e+151], t$95$1, If[LessEqual[z, -1.5e+71], N[(N[((-b) * z + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[z, -1.8e-167], N[(N[(b * a + N[((-y) * j), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[z, 128000000.0], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.9999999999999998e151 or 1.28e8 < z

   1. Initial program 64.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if -5.9999999999999998e151 < z < -1.50000000000000006e71

   1. Initial program 70.2%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot t - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(j \cdot t - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -1.50000000000000006e71 < z < -1.8e-167

   1. Initial program 72.9%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 57.2

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

   5. Applied rewrites 57.2%

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

   7. Applied rewrites 57.2%

      \[\leadsto \mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i \]

   if -1.8e-167 < z < 1.28e8

   1. Initial program 76.7%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 54.2

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

   5. Applied rewrites 54.2%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-167}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{elif}\;z \leq 128000000:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{if}\;j \leq -2.5 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 5.9 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, i, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- i) y (* c t)) j)))
   (if (<= j -2.5e-19)
     t_1
     (if (<= j 5.9e+134) (fma (* b a) i (* (fma (- c) b (* y x)) z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-i, y, (c * t)) * j;
	double tmp;
	if (j <= -2.5e-19) {
		tmp = t_1;
	} else if (j <= 5.9e+134) {
		tmp = fma((b * a), i, (fma(-c, b, (y * x)) * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-i), y, Float64(c * t)) * j)
	tmp = 0.0
	if (j <= -2.5e-19)
		tmp = t_1;
	elseif (j <= 5.9e+134)
		tmp = fma(Float64(b * a), i, Float64(fma(Float64(-c), b, Float64(y * x)) * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -2.5e-19], t$95$1, If[LessEqual[j, 5.9e+134], N[(N[(b * a), $MachinePrecision] * i + N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -2.5000000000000002e-19 or 5.90000000000000008e134 < j

   1. Initial program 68.9%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if -2.5000000000000002e-19 < j < 5.90000000000000008e134

   1. Initial program 72.6%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 52.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \left(a \cdot i\right) \cdot b\right) \]

   9. Taylor expanded in c around inf

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 28.3%

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

   12. Taylor expanded in j around 0

       \[\leadsto b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right) + \color{blue}{x \cdot \left(y \cdot z\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 65.4%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.5 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{elif}\;j \leq 5.9 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, i, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+151}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-27}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-166}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;z \leq 420000000:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -6e+151)
   (* (* z x) y)
   (if (<= z -7.2e-27)
     (* (* j c) t)
     (if (<= z -2.8e-166)
       (* (* b a) i)
       (if (<= z 420000000.0) (* (* j t) c) (* (* (- z) c) b))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -6e+151) {
		tmp = (z * x) * y;
	} else if (z <= -7.2e-27) {
		tmp = (j * c) * t;
	} else if (z <= -2.8e-166) {
		tmp = (b * a) * i;
	} else if (z <= 420000000.0) {
		tmp = (j * t) * c;
	} else {
		tmp = (-z * c) * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    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), intent (in) :: j
    real(8) :: tmp
    if (z <= (-6d+151)) then
        tmp = (z * x) * y
    else if (z <= (-7.2d-27)) then
        tmp = (j * c) * t
    else if (z <= (-2.8d-166)) then
        tmp = (b * a) * i
    else if (z <= 420000000.0d0) then
        tmp = (j * t) * c
    else
        tmp = (-z * c) * b
    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 j) {
	double tmp;
	if (z <= -6e+151) {
		tmp = (z * x) * y;
	} else if (z <= -7.2e-27) {
		tmp = (j * c) * t;
	} else if (z <= -2.8e-166) {
		tmp = (b * a) * i;
	} else if (z <= 420000000.0) {
		tmp = (j * t) * c;
	} else {
		tmp = (-z * c) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -6e+151:
		tmp = (z * x) * y
	elif z <= -7.2e-27:
		tmp = (j * c) * t
	elif z <= -2.8e-166:
		tmp = (b * a) * i
	elif z <= 420000000.0:
		tmp = (j * t) * c
	else:
		tmp = (-z * c) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -6e+151)
		tmp = Float64(Float64(z * x) * y);
	elseif (z <= -7.2e-27)
		tmp = Float64(Float64(j * c) * t);
	elseif (z <= -2.8e-166)
		tmp = Float64(Float64(b * a) * i);
	elseif (z <= 420000000.0)
		tmp = Float64(Float64(j * t) * c);
	else
		tmp = Float64(Float64(Float64(-z) * c) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -6e+151)
		tmp = (z * x) * y;
	elseif (z <= -7.2e-27)
		tmp = (j * c) * t;
	elseif (z <= -2.8e-166)
		tmp = (b * a) * i;
	elseif (z <= 420000000.0)
		tmp = (j * t) * c;
	else
		tmp = (-z * c) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -6e+151], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -7.2e-27], N[(N[(j * c), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, -2.8e-166], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[z, 420000000.0], N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision], N[(N[((-z) * c), $MachinePrecision] * b), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -5.9999999999999998e151

   1. Initial program 68.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{a \cdot \left(t \cdot x\right)}{y} + x \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

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

   9. Taylor expanded in a around 0

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 57.7%

       \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -5.9999999999999998e151 < z < -7.1999999999999997e-27

   1. Initial program 69.0%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 41.6

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 38.2%

      \[\leadsto \left(c \cdot j\right) \cdot \color{blue}{t} \]

   if -7.1999999999999997e-27 < z < -2.7999999999999999e-166

   1. Initial program 79.1%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(a \cdot b\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 51.7%

      \[\leadsto \left(b \cdot a\right) \cdot i \]

   if -2.7999999999999999e-166 < z < 4.2e8

   1. Initial program 76.1%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 53.7

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 32.1%

      \[\leadsto \left(c \cdot j\right) \cdot \color{blue}{t} \]

   9. Taylor expanded in c around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 32.5%

       \[\leadsto \left(t \cdot j\right) \cdot \color{blue}{c} \]

   if 4.2e8 < z

   1. Initial program 61.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 45.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \left(a \cdot i\right) \cdot b\right) \]

   9. Taylor expanded in c around inf

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.3%

       \[\leadsto \left(\left(-z\right) \cdot c\right) \cdot \color{blue}{b} \]
3. Recombined 5 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+151}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-27}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-166}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;z \leq 420000000:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+151}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-27}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-166}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;z \leq 420000000:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -6e+151)
   (* (* z x) y)
   (if (<= z -7.2e-27)
     (* (* j c) t)
     (if (<= z -2.8e-166)
       (* (* b a) i)
       (if (<= z 420000000.0) (* (* j t) c) (* (* (- b) c) z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -6e+151) {
		tmp = (z * x) * y;
	} else if (z <= -7.2e-27) {
		tmp = (j * c) * t;
	} else if (z <= -2.8e-166) {
		tmp = (b * a) * i;
	} else if (z <= 420000000.0) {
		tmp = (j * t) * c;
	} else {
		tmp = (-b * c) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    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), intent (in) :: j
    real(8) :: tmp
    if (z <= (-6d+151)) then
        tmp = (z * x) * y
    else if (z <= (-7.2d-27)) then
        tmp = (j * c) * t
    else if (z <= (-2.8d-166)) then
        tmp = (b * a) * i
    else if (z <= 420000000.0d0) then
        tmp = (j * t) * c
    else
        tmp = (-b * c) * z
    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 j) {
	double tmp;
	if (z <= -6e+151) {
		tmp = (z * x) * y;
	} else if (z <= -7.2e-27) {
		tmp = (j * c) * t;
	} else if (z <= -2.8e-166) {
		tmp = (b * a) * i;
	} else if (z <= 420000000.0) {
		tmp = (j * t) * c;
	} else {
		tmp = (-b * c) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -6e+151:
		tmp = (z * x) * y
	elif z <= -7.2e-27:
		tmp = (j * c) * t
	elif z <= -2.8e-166:
		tmp = (b * a) * i
	elif z <= 420000000.0:
		tmp = (j * t) * c
	else:
		tmp = (-b * c) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -6e+151)
		tmp = Float64(Float64(z * x) * y);
	elseif (z <= -7.2e-27)
		tmp = Float64(Float64(j * c) * t);
	elseif (z <= -2.8e-166)
		tmp = Float64(Float64(b * a) * i);
	elseif (z <= 420000000.0)
		tmp = Float64(Float64(j * t) * c);
	else
		tmp = Float64(Float64(Float64(-b) * c) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -6e+151)
		tmp = (z * x) * y;
	elseif (z <= -7.2e-27)
		tmp = (j * c) * t;
	elseif (z <= -2.8e-166)
		tmp = (b * a) * i;
	elseif (z <= 420000000.0)
		tmp = (j * t) * c;
	else
		tmp = (-b * c) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -6e+151], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -7.2e-27], N[(N[(j * c), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, -2.8e-166], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[z, 420000000.0], N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision], N[(N[((-b) * c), $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -5.9999999999999998e151

   1. Initial program 68.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{a \cdot \left(t \cdot x\right)}{y} + x \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

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

   9. Taylor expanded in a around 0

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 57.7%

       \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -5.9999999999999998e151 < z < -7.1999999999999997e-27

   1. Initial program 69.0%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 41.6

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 38.2%

      \[\leadsto \left(c \cdot j\right) \cdot \color{blue}{t} \]

   if -7.1999999999999997e-27 < z < -2.7999999999999999e-166

   1. Initial program 79.1%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(a \cdot b\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 51.7%

      \[\leadsto \left(b \cdot a\right) \cdot i \]

   if -2.7999999999999999e-166 < z < 4.2e8

   1. Initial program 76.1%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 53.7

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 32.1%

      \[\leadsto \left(c \cdot j\right) \cdot \color{blue}{t} \]

   9. Taylor expanded in c around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 32.5%

       \[\leadsto \left(t \cdot j\right) \cdot \color{blue}{c} \]

   if 4.2e8 < z

   1. Initial program 61.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in c around inf

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.2%

      \[\leadsto \left(\left(-c\right) \cdot b\right) \cdot \color{blue}{z} \]
3. Recombined 5 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+151}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-27}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-166}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;z \leq 420000000:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -6 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-267}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{elif}\;z \leq 128000000:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)))
   (if (<= z -6e+151)
     t_1
     (if (<= z -4e-267)
       (* (fma (- x) a (* j c)) t)
       (if (<= z 128000000.0) (* (fma (- i) y (* c t)) j) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x)) * z;
	double tmp;
	if (z <= -6e+151) {
		tmp = t_1;
	} else if (z <= -4e-267) {
		tmp = fma(-x, a, (j * c)) * t;
	} else if (z <= 128000000.0) {
		tmp = fma(-i, y, (c * t)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-b), c, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -6e+151)
		tmp = t_1;
	elseif (z <= -4e-267)
		tmp = Float64(fma(Float64(-x), a, Float64(j * c)) * t);
	elseif (z <= 128000000.0)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -6e+151], t$95$1, If[LessEqual[z, -4e-267], N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 128000000.0], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.9999999999999998e151 or 1.28e8 < z

   1. Initial program 64.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if -5.9999999999999998e151 < z < -3.9999999999999999e-267

   1. Initial program 75.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, c \cdot j\right)} \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, c \cdot j\right) \cdot t \]

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 54.2

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

   5. Applied rewrites 54.2%

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

   if -3.9999999999999999e-267 < z < 1.28e8

   1. Initial program 74.3%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 59.6

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

   5. Applied rewrites 59.6%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-267}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{elif}\;z \leq 128000000:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ \mathbf{if}\;c \leq -0.0018:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -1.15 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) z (* j t)) c)))
   (if (<= c -0.0018)
     t_1
     (if (<= c -1.15e-289)
       (* (fma (- a) t (* z y)) x)
       (if (<= c 4.2e-60) (* (fma b a (* (- y) j)) i) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, z, (j * t)) * c;
	double tmp;
	if (c <= -0.0018) {
		tmp = t_1;
	} else if (c <= -1.15e-289) {
		tmp = fma(-a, t, (z * y)) * x;
	} else if (c <= 4.2e-60) {
		tmp = fma(b, a, (-y * j)) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-b), z, Float64(j * t)) * c)
	tmp = 0.0
	if (c <= -0.0018)
		tmp = t_1;
	elseif (c <= -1.15e-289)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	elseif (c <= 4.2e-60)
		tmp = Float64(fma(b, a, Float64(Float64(-y) * j)) * i);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * z + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[c, -0.0018], t$95$1, If[LessEqual[c, -1.15e-289], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[c, 4.2e-60], N[(N[(b * a + N[((-y) * j), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if c < -0.0018 or 4.19999999999999982e-60 < c

   1. Initial program 63.0%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot t - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(j \cdot t - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 65.2

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

   5. Applied rewrites 65.2%

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

   if -0.0018 < c < -1.1500000000000001e-289

   1. Initial program 78.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 50.9

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

   5. Applied rewrites 50.9%

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

   if -1.1500000000000001e-289 < c < 4.19999999999999982e-60

   1. Initial program 81.7%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

   7. Applied rewrites 58.8%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -0.0018:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ \mathbf{elif}\;c \leq -1.15 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq -35000000000000:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -2.2e+76)
   (* (fma (- a) t (* z y)) x)
   (if (<= x -35000000000000.0)
     (* (* j c) t)
     (if (<= x 2.4e-64)
       (* (fma b a (* (- y) j)) i)
       (* (fma z y (* (- a) t)) x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -2.2e+76) {
		tmp = fma(-a, t, (z * y)) * x;
	} else if (x <= -35000000000000.0) {
		tmp = (j * c) * t;
	} else if (x <= 2.4e-64) {
		tmp = fma(b, a, (-y * j)) * i;
	} else {
		tmp = fma(z, y, (-a * t)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -2.2e+76)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	elseif (x <= -35000000000000.0)
		tmp = Float64(Float64(j * c) * t);
	elseif (x <= 2.4e-64)
		tmp = Float64(fma(b, a, Float64(Float64(-y) * j)) * i);
	else
		tmp = Float64(fma(z, y, Float64(Float64(-a) * t)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -2.2e+76], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -35000000000000.0], N[(N[(j * c), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[x, 2.4e-64], N[(N[(b * a + N[((-y) * j), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], N[(N[(z * y + N[((-a) * t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.2e76

   1. Initial program 61.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

   if -2.2e76 < x < -3.5e13

   1. Initial program 81.0%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 63.1%

      \[\leadsto \left(c \cdot j\right) \cdot \color{blue}{t} \]

   if -3.5e13 < x < 2.39999999999999998e-64

   1. Initial program 69.2%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 46.3

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

   5. Applied rewrites 46.3%

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

   7. Applied rewrites 46.3%

      \[\leadsto \mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i \]

   if 2.39999999999999998e-64 < x

   1. Initial program 75.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 55.5

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 55.5%

      \[\leadsto \mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq -35000000000000:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -35000000000000:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma z y (* (- a) t)) x)))
   (if (<= x -2.2e+76)
     t_1
     (if (<= x -35000000000000.0)
       (* (* j c) t)
       (if (<= x 2.4e-64) (* (fma b a (* (- y) j)) i) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(z, y, (-a * t)) * x;
	double tmp;
	if (x <= -2.2e+76) {
		tmp = t_1;
	} else if (x <= -35000000000000.0) {
		tmp = (j * c) * t;
	} else if (x <= 2.4e-64) {
		tmp = fma(b, a, (-y * j)) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(z, y, Float64(Float64(-a) * t)) * x)
	tmp = 0.0
	if (x <= -2.2e+76)
		tmp = t_1;
	elseif (x <= -35000000000000.0)
		tmp = Float64(Float64(j * c) * t);
	elseif (x <= 2.4e-64)
		tmp = Float64(fma(b, a, Float64(Float64(-y) * j)) * i);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * y + N[((-a) * t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.2e+76], t$95$1, If[LessEqual[x, -35000000000000.0], N[(N[(j * c), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[x, 2.4e-64], N[(N[(b * a + N[((-y) * j), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2e76 or 2.39999999999999998e-64 < x

   1. Initial program 71.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

   7. Applied rewrites 56.0%

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

   if -2.2e76 < x < -3.5e13

   1. Initial program 81.0%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 63.1%

      \[\leadsto \left(c \cdot j\right) \cdot \color{blue}{t} \]

   if -3.5e13 < x < 2.39999999999999998e-64

   1. Initial program 69.2%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 46.3

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

   5. Applied rewrites 46.3%

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

   7. Applied rewrites 46.3%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;x \leq -35000000000000:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 41.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-z\right) \cdot c\right) \cdot b\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{+151}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{+133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 490000000:\\ \;\;\;\;\mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- z) c) b)))
   (if (<= z -2.05e+151)
     (* (* z x) y)
     (if (<= z -2.1e+133)
       t_1
       (if (<= z 490000000.0) (* (fma b a (* (- y) j)) i) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-z * c) * b;
	double tmp;
	if (z <= -2.05e+151) {
		tmp = (z * x) * y;
	} else if (z <= -2.1e+133) {
		tmp = t_1;
	} else if (z <= 490000000.0) {
		tmp = fma(b, a, (-y * j)) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-z) * c) * b)
	tmp = 0.0
	if (z <= -2.05e+151)
		tmp = Float64(Float64(z * x) * y);
	elseif (z <= -2.1e+133)
		tmp = t_1;
	elseif (z <= 490000000.0)
		tmp = Float64(fma(b, a, Float64(Float64(-y) * j)) * i);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-z) * c), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[z, -2.05e+151], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -2.1e+133], t$95$1, If[LessEqual[z, 490000000.0], N[(N[(b * a + N[((-y) * j), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.0499999999999999e151

   1. Initial program 68.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{a \cdot \left(t \cdot x\right)}{y} + x \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.1%

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

   9. Taylor expanded in a around 0

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 56.2%

       \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -2.0499999999999999e151 < z < -2.1e133 or 4.9e8 < z

   1. Initial program 62.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 40.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \left(a \cdot i\right) \cdot b\right) \]

   9. Taylor expanded in c around inf

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.0%

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

   if -2.1e133 < z < 4.9e8

   1. Initial program 75.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 44.4

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

   5. Applied rewrites 44.4%

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

   7. Applied rewrites 44.4%

      \[\leadsto \mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 52.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -2.45 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)))
   (if (<= z -2.45e+45)
     t_1
     (if (<= z 2.1e-9) (* (fma b a (* (- y) j)) i) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x)) * z;
	double tmp;
	if (z <= -2.45e+45) {
		tmp = t_1;
	} else if (z <= 2.1e-9) {
		tmp = fma(b, a, (-y * j)) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-b), c, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -2.45e+45)
		tmp = t_1;
	elseif (z <= 2.1e-9)
		tmp = Float64(fma(b, a, Float64(Float64(-y) * j)) * i);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.45e+45], t$95$1, If[LessEqual[z, 2.1e-9], N[(N[(b * a + N[((-y) * j), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4500000000000001e45 or 2.10000000000000019e-9 < z

   1. Initial program 65.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if -2.4500000000000001e45 < z < 2.10000000000000019e-9

   1. Initial program 75.9%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 47.2

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

   5. Applied rewrites 47.2%

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

   7. Applied rewrites 47.2%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+151}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+105}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -6e+151)
   (* (* z x) y)
   (if (<= z 8e+105) (* (* j c) t) (* (* z y) x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -6e+151) {
		tmp = (z * x) * y;
	} else if (z <= 8e+105) {
		tmp = (j * c) * t;
	} else {
		tmp = (z * y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    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), intent (in) :: j
    real(8) :: tmp
    if (z <= (-6d+151)) then
        tmp = (z * x) * y
    else if (z <= 8d+105) then
        tmp = (j * c) * t
    else
        tmp = (z * y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -6e+151) {
		tmp = (z * x) * y;
	} else if (z <= 8e+105) {
		tmp = (j * c) * t;
	} else {
		tmp = (z * y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -6e+151:
		tmp = (z * x) * y
	elif z <= 8e+105:
		tmp = (j * c) * t
	else:
		tmp = (z * y) * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -6e+151)
		tmp = Float64(Float64(z * x) * y);
	elseif (z <= 8e+105)
		tmp = Float64(Float64(j * c) * t);
	else
		tmp = Float64(Float64(z * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -6e+151)
		tmp = (z * x) * y;
	elseif (z <= 8e+105)
		tmp = (j * c) * t;
	else
		tmp = (z * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -6e+151], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 8e+105], N[(N[(j * c), $MachinePrecision] * t), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.9999999999999998e151

   1. Initial program 68.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{a \cdot \left(t \cdot x\right)}{y} + x \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

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

   9. Taylor expanded in a around 0

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 57.7%

       \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -5.9999999999999998e151 < z < 7.9999999999999995e105

   1. Initial program 73.7%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 31.8%

      \[\leadsto \left(c \cdot j\right) \cdot \color{blue}{t} \]

   if 7.9999999999999995e105 < z

   1. Initial program 61.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 49.9%

      \[\leadsto \left(z \cdot y\right) \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+151}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+105}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 29.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot y\\ \mathbf{if}\;z \leq -6 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+105}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z x) y)))
   (if (<= z -6e+151) t_1 (if (<= z 7.8e+105) (* (* j c) t) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (z <= -6e+151) {
		tmp = t_1;
	} else if (z <= 7.8e+105) {
		tmp = (j * c) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    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), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * x) * y
    if (z <= (-6d+151)) then
        tmp = t_1
    else if (z <= 7.8d+105) then
        tmp = (j * c) * t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (z <= -6e+151) {
		tmp = t_1;
	} else if (z <= 7.8e+105) {
		tmp = (j * c) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * x) * y
	tmp = 0
	if z <= -6e+151:
		tmp = t_1
	elif z <= 7.8e+105:
		tmp = (j * c) * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * x) * y)
	tmp = 0.0
	if (z <= -6e+151)
		tmp = t_1;
	elseif (z <= 7.8e+105)
		tmp = Float64(Float64(j * c) * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * x) * y;
	tmp = 0.0;
	if (z <= -6e+151)
		tmp = t_1;
	elseif (z <= 7.8e+105)
		tmp = (j * c) * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -6e+151], t$95$1, If[LessEqual[z, 7.8e+105], N[(N[(j * c), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.9999999999999998e151 or 7.79999999999999957e105 < z

   1. Initial program 64.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{a \cdot \left(t \cdot x\right)}{y} + x \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.6%

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

   9. Taylor expanded in a around 0

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 51.3%

       \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -5.9999999999999998e151 < z < 7.79999999999999957e105

   1. Initial program 73.7%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, c \cdot t\right) \cdot j \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, y, c \cdot t\right) \cdot j \]

      9. lower-*.f64 46.0

         \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot t}\right) \cdot j \]

   5. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]

   6. Taylor expanded in c around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.8%

      \[\leadsto \left(c \cdot j\right) \cdot \color{blue}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+151}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+105}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 23.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot x\right) \cdot y \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* z x) y))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * x) * y;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    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), intent (in) :: j
    code = (z * x) * y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * x) * y;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (z * x) * y

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(z * x) * y)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (z * x) * y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 71.0%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

   5. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

   6. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

   9. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   11. lower-*.f64 35.0

       \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

5. Applied rewrites 35.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

6. Taylor expanded in y around inf

   \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{a \cdot \left(t \cdot x\right)}{y} + x \cdot z\right)} \]
7. Step-by-step derivation

8. Applied rewrites 32.0%

   \[\leadsto \mathsf{fma}\left(-a, \frac{t \cdot x}{y}, z \cdot x\right) \cdot \color{blue}{y} \]

9. Taylor expanded in a around 0

   \[\leadsto \left(x \cdot z\right) \cdot y \]
10. Step-by-step derivation

11. Applied rewrites 20.5%

    \[\leadsto \left(z \cdot x\right) \cdot y \]
12. Add Preprocessing

Developer Target 1: 68.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    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), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    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 j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2024268 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1015122364899489/125000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -942510763643697/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -238547917063487/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 10535888557455487/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))