Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 73.1% → 84.2%

Time: 14.5s

Alternatives: 24

Speedup: 0.9×

• 58.5% 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: 73.1% 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: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot t - i \cdot y\right) \cdot j - \left(\left(a \cdot t - z \cdot y\right) \cdot x - \left(i \cdot a - c \cdot z\right) \cdot b\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \mathsf{fma}\left(-\mathsf{fma}\left(-a, i, c \cdot z\right), b, \mathsf{fma}\left(-y, i, c \cdot t\right) \cdot j\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\mathsf{fma}\left(j, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)}{c}\right) - b \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, \mathsf{fma}\left(t, c, \frac{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z}{j}\right)\right) \cdot j\\ \end{array} \end{array} \]

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 5e+272)
     (fma
      (fma (- a) t (* z y))
      x
      (fma (- (fma (- a) i (* c z))) b (* (fma (- y) i (* c t)) j)))
     (if (<= t_1 INFINITY)
       (*
        (-
         (fma
          j
          t
          (/ (fma (fma (- y) j (* b a)) i (* (fma (- t) a (* z y)) x)) c))
         (* b z))
        c)
       (* (fma (- i) y (fma t c (/ (* (fma (- b) c (* y x)) z) j))) j)))))

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 <= 5e+272) {
		tmp = fma(fma(-a, t, (z * y)), x, fma(-fma(-a, i, (c * z)), b, (fma(-y, i, (c * t)) * j)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (fma(j, t, (fma(fma(-y, j, (b * a)), i, (fma(-t, a, (z * y)) * x)) / c)) - (b * z)) * c;
	} else {
		tmp = fma(-i, y, fma(t, c, ((fma(-b, c, (y * x)) * z) / j))) * j;
	}
	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 <= 5e+272)
		tmp = fma(fma(Float64(-a), t, Float64(z * y)), x, fma(Float64(-fma(Float64(-a), i, Float64(c * z))), b, Float64(fma(Float64(-y), i, Float64(c * t)) * j)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(fma(j, t, Float64(fma(fma(Float64(-y), j, Float64(b * a)), i, Float64(fma(Float64(-t), a, Float64(z * y)) * x)) / c)) - Float64(b * z)) * c);
	else
		tmp = Float64(fma(Float64(-i), y, fma(t, c, Float64(Float64(fma(Float64(-b), c, Float64(y * x)) * z) / j))) * j);
	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, 5e+272], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[((-N[((-a) * i + N[(c * z), $MachinePrecision]), $MachinePrecision]) * b + N[(N[((-y) * i + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(j * t + N[(N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] - N[(b * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[((-i) * y + N[(t * c + N[(N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] / j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * j), $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)))) < 4.99999999999999973e272

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 94.0%

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

   if 4.99999999999999973e272 < (+.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 81.3%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 91.4%

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

   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. 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. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 75.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot t - i \cdot y\right) \cdot j - \left(\left(a \cdot t - z \cdot y\right) \cdot x - \left(i \cdot a - c \cdot z\right) \cdot b\right) \leq 5 \cdot 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \mathsf{fma}\left(-\mathsf{fma}\left(-a, i, c \cdot z\right), b, \mathsf{fma}\left(-y, i, c \cdot t\right) \cdot j\right)\right)\\ \mathbf{elif}\;\left(c \cdot t - i \cdot y\right) \cdot j - \left(\left(a \cdot t - z \cdot y\right) \cdot x - \left(i \cdot a - c \cdot z\right) \cdot b\right) \leq \infty:\\ \;\;\;\;\left(\mathsf{fma}\left(j, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)}{c}\right) - b \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, \mathsf{fma}\left(t, c, \frac{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z}{j}\right)\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<=
      (-
       (* (- (* c t) (* i y)) j)
       (- (* (- (* a t) (* z y)) x) (* (- (* i a) (* c z)) b)))
      INFINITY)
   (fma
    (fma (- a) t (* z y))
    x
    (fma (- (fma (- a) i (* c z))) b (* (fma (- y) i (* c t)) j)))
   (* (fma (- i) y (fma t c (/ (* (fma (- b) c (* y x)) z) j))) 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 (((((c * t) - (i * y)) * j) - ((((a * t) - (z * y)) * x) - (((i * a) - (c * z)) * b))) <= ((double) INFINITY)) {
		tmp = fma(fma(-a, t, (z * y)), x, fma(-fma(-a, i, (c * z)), b, (fma(-y, i, (c * t)) * j)));
	} else {
		tmp = fma(-i, y, fma(t, c, ((fma(-b, c, (y * x)) * z) / j))) * j;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 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)))) < +inf.0

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 90.6%

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

   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. 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. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 75.3%

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

4. Final simplification 88.0%

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

Alternative 3: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{+179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.72 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, i \cdot b\right), a, \mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) a (* j c)) t)))
   (if (<= t -2.4e+179)
     t_1
     (if (<= t 1.72e+202)
       (fma
        (fma (- x) t (* i b))
        a
        (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(-x, a, (j * c)) * t;
	double tmp;
	if (t <= -2.4e+179) {
		tmp = t_1;
	} else if (t <= 1.72e+202) {
		tmp = fma(fma(-x, t, (i * b)), a, 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 = Float64(fma(Float64(-x), a, Float64(j * c)) * t)
	tmp = 0.0
	if (t <= -2.4e+179)
		tmp = t_1;
	elseif (t <= 1.72e+202)
		tmp = fma(fma(Float64(-x), t, Float64(i * b)), a, 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[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -2.4e+179], t$95$1, If[LessEqual[t, 1.72e+202], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.40000000000000013e179 or 1.71999999999999996e202 < t

   1. Initial program 61.2%

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

   3. Taylor expanded in t around 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. *-commutative N/A

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

      9. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if -2.40000000000000013e179 < t < 1.71999999999999996e202

   1. Initial program 78.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 88.0%

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

Alternative 4: 74.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a\right)\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \mathsf{fma}\left(i \cdot a, b, \mathsf{fma}\left(-y, i, c \cdot t\right) \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- c) b (* y x)) z (* (fma (- t) x (* i b)) a))))
   (if (<= z -1.8e+144)
     (fma (fma (- b) c (* y x)) z (* (fma (- i) y (* c t)) j))
     (if (<= z -9.5e+16)
       t_1
       (if (<= z 5.7e+143)
         (fma
          (fma (- a) t (* z y))
          x
          (fma (* i a) b (* (fma (- y) i (* 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(-c, b, (y * x)), z, (fma(-t, x, (i * b)) * a));
	double tmp;
	if (z <= -1.8e+144) {
		tmp = fma(fma(-b, c, (y * x)), z, (fma(-i, y, (c * t)) * j));
	} else if (z <= -9.5e+16) {
		tmp = t_1;
	} else if (z <= 5.7e+143) {
		tmp = fma(fma(-a, t, (z * y)), x, fma((i * a), b, (fma(-y, i, (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(-c), b, Float64(y * x)), z, Float64(fma(Float64(-t), x, Float64(i * b)) * a))
	tmp = 0.0
	if (z <= -1.8e+144)
		tmp = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	elseif (z <= -9.5e+16)
		tmp = t_1;
	elseif (z <= 5.7e+143)
		tmp = fma(fma(Float64(-a), t, Float64(z * y)), x, fma(Float64(i * a), b, Float64(fma(Float64(-y), i, 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[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-t) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.8e+144], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -9.5e+16], t$95$1, If[LessEqual[z, 5.7e+143], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(i * a), $MachinePrecision] * b + N[(N[((-y) * i + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7999999999999999e144

   1. Initial program 65.1%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 86.6%

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

   if -1.7999999999999999e144 < z < -9.5e16 or 5.70000000000000022e143 < z

   1. Initial program 67.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 88.9%

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

   if -9.5e16 < z < 5.70000000000000022e143

   1. Initial program 79.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.7

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

   7. Applied rewrites 77.7%

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

Alternative 5: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a\right)\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{-68}:\\ \;\;\;\;\left(\mathsf{fma}\left(j, t, \frac{\mathsf{fma}\left(-j, y, b \cdot a\right) \cdot i}{c}\right) - b \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -3.7e-120)
   (fma (fma (- c) b (* y x)) z (* (fma (- t) x (* i b)) a))
   (if (<= x 6.7e-68)
     (* (- (fma j t (/ (* (fma (- j) y (* b a)) i) c)) (* b z)) c)
     (fma (fma (- y) j (* b a)) i (* (fma (- t) a (* 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 (x <= -3.7e-120) {
		tmp = fma(fma(-c, b, (y * x)), z, (fma(-t, x, (i * b)) * a));
	} else if (x <= 6.7e-68) {
		tmp = (fma(j, t, ((fma(-j, y, (b * a)) * i) / c)) - (b * z)) * c;
	} else {
		tmp = fma(fma(-y, j, (b * a)), i, (fma(-t, a, (z * y)) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -3.7e-120)
		tmp = fma(fma(Float64(-c), b, Float64(y * x)), z, Float64(fma(Float64(-t), x, Float64(i * b)) * a));
	elseif (x <= 6.7e-68)
		tmp = Float64(Float64(fma(j, t, Float64(Float64(fma(Float64(-j), y, Float64(b * a)) * i) / c)) - Float64(b * z)) * c);
	else
		tmp = fma(fma(Float64(-y), j, Float64(b * a)), i, Float64(fma(Float64(-t), a, Float64(z * y)) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -3.7e-120], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-t) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.7e-68], N[(N[(N[(j * t + N[(N[(N[((-j) * y + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] - N[(b * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.70000000000000001e-120

   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 z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 74.4%

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

   if -3.70000000000000001e-120 < x < 6.6999999999999996e-68

   1. Initial program 70.5%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\mathsf{fma}\left(j, t, \frac{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)}{c}\right) - b \cdot z\right) \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 79.2%

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

   if 6.6999999999999996e-68 < x

   1. Initial program 82.9%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 80.4%

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

Alternative 6: 51.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, j \cdot i\right) \cdot \left(-y\right)\\ \mathbf{elif}\;t \leq 8.3 \cdot 10^{-224}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+47}:\\ \;\;\;\;\left(\left(\frac{b \cdot a}{y} - j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) a (* j c)) t)))
   (if (<= t -4.2e+130)
     t_1
     (if (<= t -7.5e-46)
       (* (fma (- x) z (* j i)) (- y))
       (if (<= t 8.3e-224)
         (* (fma (- b) c (* y x)) z)
         (if (<= t 1.55e+47) (* (* (- (/ (* b a) y) j) y) 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(-x, a, (j * c)) * t;
	double tmp;
	if (t <= -4.2e+130) {
		tmp = t_1;
	} else if (t <= -7.5e-46) {
		tmp = fma(-x, z, (j * i)) * -y;
	} else if (t <= 8.3e-224) {
		tmp = fma(-b, c, (y * x)) * z;
	} else if (t <= 1.55e+47) {
		tmp = ((((b * a) / y) - j) * y) * i;
	} 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 <= -4.2e+130)
		tmp = t_1;
	elseif (t <= -7.5e-46)
		tmp = Float64(fma(Float64(-x), z, Float64(j * i)) * Float64(-y));
	elseif (t <= 8.3e-224)
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	elseif (t <= 1.55e+47)
		tmp = Float64(Float64(Float64(Float64(Float64(b * a) / y) - j) * y) * 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[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -4.2e+130], t$95$1, If[LessEqual[t, -7.5e-46], N[(N[((-x) * z + N[(j * i), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision], If[LessEqual[t, 8.3e-224], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t, 1.55e+47], N[(N[(N[(N[(N[(b * a), $MachinePrecision] / y), $MachinePrecision] - j), $MachinePrecision] * y), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.19999999999999981e130 or 1.55e47 < t

   1. Initial program 66.0%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      9. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if -4.19999999999999981e130 < t < -7.50000000000000027e-46

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 60.2

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

   7. Applied rewrites 60.2%

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

   if -7.50000000000000027e-46 < t < 8.29999999999999999e-224

   1. Initial program 77.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      11. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 8.29999999999999999e-224 < t < 1.55e47

   1. Initial program 85.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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 56.5

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 58.7%

      \[\leadsto \left(\left(\frac{b \cdot a}{y} - j\right) \cdot y\right) \cdot i \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 57.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(i \cdot b\right) \cdot a\right)\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+218}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -5.8e+75)
   (* (fma (- z) b (* j t)) c)
   (if (<= c 1.75e-47)
     (fma (fma (- a) t (* z y)) x (* (* i b) a))
     (if (<= c 8e+218)
       (+ (* (* z x) y) (* (- (* c t) (* i y)) j))
       (* (fma (- b) c (* y x)) 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 (c <= -5.8e+75) {
		tmp = fma(-z, b, (j * t)) * c;
	} else if (c <= 1.75e-47) {
		tmp = fma(fma(-a, t, (z * y)), x, ((i * b) * a));
	} else if (c <= 8e+218) {
		tmp = ((z * x) * y) + (((c * t) - (i * y)) * j);
	} else {
		tmp = fma(-b, c, (y * x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -5.8e+75)
		tmp = Float64(fma(Float64(-z), b, Float64(j * t)) * c);
	elseif (c <= 1.75e-47)
		tmp = fma(fma(Float64(-a), t, Float64(z * y)), x, Float64(Float64(i * b) * a));
	elseif (c <= 8e+218)
		tmp = Float64(Float64(Float64(z * x) * y) + Float64(Float64(Float64(c * t) - Float64(i * y)) * j));
	else
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -5.8e+75], N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 1.75e-47], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8e+218], N[(N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -5.7999999999999997e75

   1. Initial program 68.4%

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

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -5.7999999999999997e75 < c < 1.7499999999999999e-47

   1. Initial program 82.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.4

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

   7. Applied rewrites 74.4%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 64.0

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

   10. Applied rewrites 64.0%

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

   if 1.7499999999999999e-47 < c < 8.00000000000000066e218

   1. Initial program 69.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 61.9

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

   5. Applied rewrites 61.9%

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

   if 8.00000000000000066e218 < c

   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 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. *-commutative N/A

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

      11. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(i \cdot b\right) \cdot a\right)\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+218}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -1.45e-161)
   (fma (fma (- c) b (* y x)) z (* (fma (- t) x (* i b)) a))
   (if (<= x 1.45e-45)
     (fma (fma (- b) c (* y x)) z (* (fma (- i) y (* c t)) j))
     (fma (fma (- y) j (* b a)) i (* (fma (- t) a (* 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 (x <= -1.45e-161) {
		tmp = fma(fma(-c, b, (y * x)), z, (fma(-t, x, (i * b)) * a));
	} else if (x <= 1.45e-45) {
		tmp = fma(fma(-b, c, (y * x)), z, (fma(-i, y, (c * t)) * j));
	} else {
		tmp = fma(fma(-y, j, (b * a)), i, (fma(-t, a, (z * y)) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -1.45e-161)
		tmp = fma(fma(Float64(-c), b, Float64(y * x)), z, Float64(fma(Float64(-t), x, Float64(i * b)) * a));
	elseif (x <= 1.45e-45)
		tmp = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	else
		tmp = fma(fma(Float64(-y), j, Float64(b * a)), i, Float64(fma(Float64(-t), a, Float64(z * y)) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -1.45e-161], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-t) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-45], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e-161

   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 z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 73.2%

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

   if -1.45e-161 < x < 1.45e-45

   1. Initial program 73.0%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 76.0%

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

   if 1.45e-45 < x

   1. Initial program 83.2%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 80.8%

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

Alternative 9: 69.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-111}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -6.8e+175)
   (* (fma (- x) t (* i b)) a)
   (if (<= a 1.05e-111)
     (fma (fma (- b) c (* y x)) z (* (fma (- i) y (* c t)) j))
     (fma (fma (- c) b (* y x)) z (* (fma (- t) x (* i b)) a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -6.8e+175) {
		tmp = fma(-x, t, (i * b)) * a;
	} else if (a <= 1.05e-111) {
		tmp = fma(fma(-b, c, (y * x)), z, (fma(-i, y, (c * t)) * j));
	} else {
		tmp = fma(fma(-c, b, (y * x)), z, (fma(-t, x, (i * b)) * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -6.8e+175)
		tmp = Float64(fma(Float64(-x), t, Float64(i * b)) * a);
	elseif (a <= 1.05e-111)
		tmp = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	else
		tmp = fma(fma(Float64(-c), b, Float64(y * x)), z, Float64(fma(Float64(-t), x, Float64(i * b)) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -6.8e+175], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, 1.05e-111], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-t) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.80000000000000056e175

   1. Initial program 55.1%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if -6.80000000000000056e175 < a < 1.0499999999999999e-111

   1. Initial program 79.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)} \]
   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. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 75.6%

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

   if 1.0499999999999999e-111 < a

   1. Initial program 72.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 72.8%

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

Alternative 10: 64.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, j \cdot i\right) \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -4.8e+22)
   (fma (* y x) z (* (fma (- i) y (* c t)) j))
   (if (<= y 8.5e+43)
     (fma (fma (- c) b (* y x)) z (* (fma (- t) x (* i b)) a))
     (* (fma (- x) z (* j i)) (- y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -4.8e+22) {
		tmp = fma((y * x), z, (fma(-i, y, (c * t)) * j));
	} else if (y <= 8.5e+43) {
		tmp = fma(fma(-c, b, (y * x)), z, (fma(-t, x, (i * b)) * a));
	} else {
		tmp = fma(-x, z, (j * i)) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -4.8e+22)
		tmp = fma(Float64(y * x), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	elseif (y <= 8.5e+43)
		tmp = fma(fma(Float64(-c), b, Float64(y * x)), z, Float64(fma(Float64(-t), x, Float64(i * b)) * a));
	else
		tmp = Float64(fma(Float64(-x), z, Float64(j * i)) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -4.8e+22], N[(N[(y * x), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e+43], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-t) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * z + N[(j * i), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8e22

   1. Initial program 66.2%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 63.4%

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

   if -4.8e22 < y < 8.5e43

   1. Initial program 81.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 74.2%

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

   if 8.5e43 < y

   1. Initial program 64.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 70.6%

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

   5. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 74.4

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

   7. Applied rewrites 74.4%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, j \cdot i\right) \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(i \cdot b\right) \cdot a\right)\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -5.8e+75)
   (* (fma (- z) b (* j t)) c)
   (if (<= c 1.5e-47)
     (fma (fma (- a) t (* z y)) x (* (* i b) a))
     (if (<= c 8e+218)
       (fma (* y x) z (* (fma (- i) y (* c t)) j))
       (* (fma (- b) c (* y x)) 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 (c <= -5.8e+75) {
		tmp = fma(-z, b, (j * t)) * c;
	} else if (c <= 1.5e-47) {
		tmp = fma(fma(-a, t, (z * y)), x, ((i * b) * a));
	} else if (c <= 8e+218) {
		tmp = fma((y * x), z, (fma(-i, y, (c * t)) * j));
	} else {
		tmp = fma(-b, c, (y * x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -5.8e+75)
		tmp = Float64(fma(Float64(-z), b, Float64(j * t)) * c);
	elseif (c <= 1.5e-47)
		tmp = fma(fma(Float64(-a), t, Float64(z * y)), x, Float64(Float64(i * b) * a));
	elseif (c <= 8e+218)
		tmp = fma(Float64(y * x), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	else
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -5.8e+75], N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 1.5e-47], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8e+218], N[(N[(y * x), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -5.7999999999999997e75

   1. Initial program 68.4%

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

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -5.7999999999999997e75 < c < 1.50000000000000008e-47

   1. Initial program 82.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.4

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

   7. Applied rewrites 74.4%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 64.0

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

   10. Applied rewrites 64.0%

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

   if 1.50000000000000008e-47 < c < 8.00000000000000066e218

   1. Initial program 69.1%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 61.9%

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

   if 8.00000000000000066e218 < c

   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 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. *-commutative N/A

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

      11. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(i \cdot b\right) \cdot a\right)\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, j \cdot i\right) \cdot \left(-y\right)\\ \mathbf{elif}\;t \leq 8.3 \cdot 10^{-224}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) a (* j c)) t)))
   (if (<= t -4.2e+130)
     t_1
     (if (<= t -7.5e-46)
       (* (fma (- x) z (* j i)) (- y))
       (if (<= t 8.3e-224)
         (* (fma (- b) c (* y x)) z)
         (if (<= t 1.55e+47) (* (fma (- y) j (* b a)) 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(-x, a, (j * c)) * t;
	double tmp;
	if (t <= -4.2e+130) {
		tmp = t_1;
	} else if (t <= -7.5e-46) {
		tmp = fma(-x, z, (j * i)) * -y;
	} else if (t <= 8.3e-224) {
		tmp = fma(-b, c, (y * x)) * z;
	} else if (t <= 1.55e+47) {
		tmp = fma(-y, j, (b * a)) * i;
	} 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 <= -4.2e+130)
		tmp = t_1;
	elseif (t <= -7.5e-46)
		tmp = Float64(fma(Float64(-x), z, Float64(j * i)) * Float64(-y));
	elseif (t <= 8.3e-224)
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	elseif (t <= 1.55e+47)
		tmp = Float64(fma(Float64(-y), j, Float64(b * a)) * 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[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -4.2e+130], t$95$1, If[LessEqual[t, -7.5e-46], N[(N[((-x) * z + N[(j * i), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision], If[LessEqual[t, 8.3e-224], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t, 1.55e+47], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.19999999999999981e130 or 1.55e47 < t

   1. Initial program 66.0%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      9. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if -4.19999999999999981e130 < t < -7.50000000000000027e-46

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 60.2

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

   7. Applied rewrites 60.2%

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

   if -7.50000000000000027e-46 < t < 8.29999999999999999e-224

   1. Initial program 77.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      11. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 8.29999999999999999e-224 < t < 1.55e47

   1. Initial program 85.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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 56.5

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

   5. Applied rewrites 56.5%

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

Alternative 13: 52.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;t \leq 8.3 \cdot 10^{-224}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) a (* j c)) t)))
   (if (<= t -4.2e+130)
     t_1
     (if (<= t -7.5e-46)
       (* (fma (- i) j (* z x)) y)
       (if (<= t 8.3e-224)
         (* (fma (- b) c (* y x)) z)
         (if (<= t 1.55e+47) (* (fma (- y) j (* b a)) 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(-x, a, (j * c)) * t;
	double tmp;
	if (t <= -4.2e+130) {
		tmp = t_1;
	} else if (t <= -7.5e-46) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (t <= 8.3e-224) {
		tmp = fma(-b, c, (y * x)) * z;
	} else if (t <= 1.55e+47) {
		tmp = fma(-y, j, (b * a)) * i;
	} 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 <= -4.2e+130)
		tmp = t_1;
	elseif (t <= -7.5e-46)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (t <= 8.3e-224)
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	elseif (t <= 1.55e+47)
		tmp = Float64(fma(Float64(-y), j, Float64(b * a)) * 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[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -4.2e+130], t$95$1, If[LessEqual[t, -7.5e-46], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t, 8.3e-224], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t, 1.55e+47], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.19999999999999981e130 or 1.55e47 < t

   1. Initial program 66.0%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      9. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if -4.19999999999999981e130 < t < -7.50000000000000027e-46

   1. Initial program 76.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 57.8

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

   5. Applied rewrites 57.8%

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

   if -7.50000000000000027e-46 < t < 8.29999999999999999e-224

   1. Initial program 77.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      11. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 8.29999999999999999e-224 < t < 1.55e47

   1. Initial program 85.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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 56.5

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

   5. Applied rewrites 56.5%

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

Alternative 14: 51.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ t_2 := \mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+130}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-224}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- i) j (* z x)) y)) (t_2 (* (fma (- x) a (* j c)) t)))
   (if (<= t -4.2e+130)
     t_2
     (if (<= t -7.5e-46)
       t_1
       (if (<= t 1.3e-224)
         (* (fma (- b) c (* y x)) z)
         (if (<= t 3e-14) 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 = fma(-i, j, (z * x)) * y;
	double t_2 = fma(-x, a, (j * c)) * t;
	double tmp;
	if (t <= -4.2e+130) {
		tmp = t_2;
	} else if (t <= -7.5e-46) {
		tmp = t_1;
	} else if (t <= 1.3e-224) {
		tmp = fma(-b, c, (y * x)) * z;
	} else if (t <= 3e-14) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-i), j, Float64(z * x)) * y)
	t_2 = Float64(fma(Float64(-x), a, Float64(j * c)) * t)
	tmp = 0.0
	if (t <= -4.2e+130)
		tmp = t_2;
	elseif (t <= -7.5e-46)
		tmp = t_1;
	elseif (t <= 1.3e-224)
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	elseif (t <= 3e-14)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -4.2e+130], t$95$2, If[LessEqual[t, -7.5e-46], t$95$1, If[LessEqual[t, 1.3e-224], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t, 3e-14], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.19999999999999981e130 or 2.9999999999999998e-14 < t

   1. Initial program 67.8%

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

   3. Taylor expanded in t around 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. *-commutative N/A

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

      9. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   if -4.19999999999999981e130 < t < -7.50000000000000027e-46 or 1.3000000000000001e-224 < t < 2.9999999999999998e-14

   1. Initial program 81.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   if -7.50000000000000027e-46 < t < 1.3000000000000001e-224

   1. Initial program 77.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      11. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

Alternative 15: 55.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -6.8e+34)
   (* (fma (- z) b (* j t)) c)
   (if (<= b 1.5e-39)
     (fma (* y x) z (* (fma (- i) y (* c t)) j))
     (if (<= b 1.85e+101)
       (* (fma (- x) t (* i b)) a)
       (* (fma (- z) c (* i a)) 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 (b <= -6.8e+34) {
		tmp = fma(-z, b, (j * t)) * c;
	} else if (b <= 1.5e-39) {
		tmp = fma((y * x), z, (fma(-i, y, (c * t)) * j));
	} else if (b <= 1.85e+101) {
		tmp = fma(-x, t, (i * b)) * a;
	} else {
		tmp = fma(-z, c, (i * a)) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -6.8e+34)
		tmp = Float64(fma(Float64(-z), b, Float64(j * t)) * c);
	elseif (b <= 1.5e-39)
		tmp = fma(Float64(y * x), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	elseif (b <= 1.85e+101)
		tmp = Float64(fma(Float64(-x), t, Float64(i * b)) * a);
	else
		tmp = Float64(fma(Float64(-z), c, Float64(i * a)) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -6.8e+34], N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[b, 1.5e-39], N[(N[(y * x), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.85e+101], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.7999999999999999e34

   1. Initial program 75.5%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 60.1

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

   5. Applied rewrites 60.1%

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

   if -6.7999999999999999e34 < b < 1.50000000000000014e-39

   1. Initial program 75.2%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.7%

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

   if 1.50000000000000014e-39 < b < 1.8499999999999999e101

   1. Initial program 83.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if 1.8499999999999999e101 < b

   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 b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-z\right) \cdot c\right) \cdot b\\ \mathbf{if}\;c \leq -6.2 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-116}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{+195}:\\ \;\;\;\;\left(c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- z) c) b)))
   (if (<= c -6.2e+75)
     t_1
     (if (<= c 1.05e-116)
       (* (* z x) y)
       (if (<= c 2e-35)
         (* (* i b) a)
         (if (<= c 3.4e+195) (* (* 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 = (-z * c) * b;
	double tmp;
	if (c <= -6.2e+75) {
		tmp = t_1;
	} else if (c <= 1.05e-116) {
		tmp = (z * x) * y;
	} else if (c <= 2e-35) {
		tmp = (i * b) * a;
	} else if (c <= 3.4e+195) {
		tmp = (c * t) * j;
	} 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 * c) * b
    if (c <= (-6.2d+75)) then
        tmp = t_1
    else if (c <= 1.05d-116) then
        tmp = (z * x) * y
    else if (c <= 2d-35) then
        tmp = (i * b) * a
    else if (c <= 3.4d+195) then
        tmp = (c * t) * j
    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 * c) * b;
	double tmp;
	if (c <= -6.2e+75) {
		tmp = t_1;
	} else if (c <= 1.05e-116) {
		tmp = (z * x) * y;
	} else if (c <= 2e-35) {
		tmp = (i * b) * a;
	} else if (c <= 3.4e+195) {
		tmp = (c * t) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-z * c) * b
	tmp = 0
	if c <= -6.2e+75:
		tmp = t_1
	elif c <= 1.05e-116:
		tmp = (z * x) * y
	elif c <= 2e-35:
		tmp = (i * b) * a
	elif c <= 3.4e+195:
		tmp = (c * t) * j
	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 (c <= -6.2e+75)
		tmp = t_1;
	elseif (c <= 1.05e-116)
		tmp = Float64(Float64(z * x) * y);
	elseif (c <= 2e-35)
		tmp = Float64(Float64(i * b) * a);
	elseif (c <= 3.4e+195)
		tmp = Float64(Float64(c * t) * j);
	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 * c) * b;
	tmp = 0.0;
	if (c <= -6.2e+75)
		tmp = t_1;
	elseif (c <= 1.05e-116)
		tmp = (z * x) * y;
	elseif (c <= 2e-35)
		tmp = (i * b) * a;
	elseif (c <= 3.4e+195)
		tmp = (c * t) * j;
	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) * c), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[c, -6.2e+75], t$95$1, If[LessEqual[c, 1.05e-116], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[c, 2e-35], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[c, 3.4e+195], N[(N[(c * t), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -6.2000000000000002e75 or 3.4000000000000001e195 < c

   1. Initial program 67.5%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      11. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.7%

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

   if -6.2000000000000002e75 < c < 1.05e-116

   1. Initial program 82.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      11. lower-*.f64 42.8

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 32.2%

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

   10. Applied rewrites 33.0%

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

   if 1.05e-116 < c < 2.00000000000000002e-35

   1. Initial program 76.5%

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

   3. Taylor expanded in 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 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 46.2%

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

   if 2.00000000000000002e-35 < c < 3.4000000000000001e195

   1. Initial program 67.6%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      11. lower-*.f64 24.0

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

   5. Applied rewrites 24.0%

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

   6. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 61.3

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

   8. Applied rewrites 61.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 41.6%

       \[\leadsto \left(c \cdot t\right) \cdot j \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{+75}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-116}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{+195}:\\ \;\;\;\;\left(c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{-65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)))
   (if (<= z -1.05e-65)
     t_1
     (if (<= z 4e-9)
       (* (fma (- i) y (* c t)) j)
       (if (<= z 1.25e+93) (* (fma (- i) j (* z x)) y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x)) * z;
	double tmp;
	if (z <= -1.05e-65) {
		tmp = t_1;
	} else if (z <= 4e-9) {
		tmp = fma(-i, y, (c * t)) * j;
	} else if (z <= 1.25e+93) {
		tmp = fma(-i, j, (z * x)) * y;
	} 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 <= -1.05e-65)
		tmp = t_1;
	elseif (z <= 4e-9)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	elseif (z <= 1.25e+93)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	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, -1.05e-65], t$95$1, If[LessEqual[z, 4e-9], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, 1.25e+93], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.05000000000000001e-65 or 1.25e93 < z

   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 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. *-commutative N/A

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

      11. lower-*.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if -1.05000000000000001e-65 < z < 4.00000000000000025e-9

   1. Initial program 76.5%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      11. lower-*.f64 19.6

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

   5. Applied rewrites 19.6%

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

   6. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.1

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

   8. Applied rewrites 52.1%

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

   if 4.00000000000000025e-9 < z < 1.25e93

   1. Initial program 94.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 65.7

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

   5. Applied rewrites 65.7%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)))
   (if (<= z -3.2e-59)
     t_1
     (if (<= z 1.25e+93) (* (fma (- i) j (* z x)) y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x)) * z;
	double tmp;
	if (z <= -3.2e-59) {
		tmp = t_1;
	} else if (z <= 1.25e+93) {
		tmp = fma(-i, j, (z * x)) * y;
	} 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 <= -3.2e-59)
		tmp = t_1;
	elseif (z <= 1.25e+93)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	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, -3.2e-59], t$95$1, If[LessEqual[z, 1.25e+93], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1999999999999999e-59 or 1.25e93 < z

   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 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. *-commutative N/A

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

      11. lower-*.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if -3.1999999999999999e-59 < z < 1.25e93

   1. Initial program 78.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 42.2

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

   5. Applied rewrites 42.2%

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

Alternative 19: 41.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{+181}:\\ \;\;\;\;\left(c \cdot t\right) \cdot j\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+274}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -1.7e+181)
   (* (* c t) j)
   (if (<= t 1.05e+274) (* (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 tmp;
	if (t <= -1.7e+181) {
		tmp = (c * t) * j;
	} else if (t <= 1.05e+274) {
		tmp = fma(-b, c, (y * x)) * z;
	} else {
		tmp = (j * t) * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -1.7e+181)
		tmp = Float64(Float64(c * t) * j);
	elseif (t <= 1.05e+274)
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	else
		tmp = Float64(Float64(j * t) * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -1.7e+181], N[(N[(c * t), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[t, 1.05e+274], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.70000000000000015e181

   1. Initial program 71.4%

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

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

      11. lower-*.f64 15.6

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

   5. Applied rewrites 15.6%

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

   6. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 72.0

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 72.0%

       \[\leadsto \left(c \cdot t\right) \cdot j \]

   if -1.70000000000000015e181 < t < 1.05000000000000008e274

   1. Initial program 76.2%

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

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

      11. lower-*.f64 45.7

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

   5. Applied rewrites 45.7%

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

   if 1.05000000000000008e274 < t

   1. Initial program 59.2%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 100.0%

      \[\leadsto \left(\left(j - \frac{x \cdot a}{c}\right) \cdot t\right) \cdot c \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 51.9%

       \[\leadsto \left(j \cdot t\right) \cdot c \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 29.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot t\right) \cdot j\\ \mathbf{if}\;c \leq -3.6 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-116}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* c t) j)))
   (if (<= c -3.6e+70)
     t_1
     (if (<= c 1.05e-116) (* (* z x) y) (if (<= c 2e-35) (* (* i b) a) 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 = (c * t) * j;
	double tmp;
	if (c <= -3.6e+70) {
		tmp = t_1;
	} else if (c <= 1.05e-116) {
		tmp = (z * x) * y;
	} else if (c <= 2e-35) {
		tmp = (i * b) * a;
	} 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 = (c * t) * j
    if (c <= (-3.6d+70)) then
        tmp = t_1
    else if (c <= 1.05d-116) then
        tmp = (z * x) * y
    else if (c <= 2d-35) then
        tmp = (i * b) * a
    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 = (c * t) * j;
	double tmp;
	if (c <= -3.6e+70) {
		tmp = t_1;
	} else if (c <= 1.05e-116) {
		tmp = (z * x) * y;
	} else if (c <= 2e-35) {
		tmp = (i * b) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (c * t) * j
	tmp = 0
	if c <= -3.6e+70:
		tmp = t_1
	elif c <= 1.05e-116:
		tmp = (z * x) * y
	elif c <= 2e-35:
		tmp = (i * b) * a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(c * t) * j)
	tmp = 0.0
	if (c <= -3.6e+70)
		tmp = t_1;
	elseif (c <= 1.05e-116)
		tmp = Float64(Float64(z * x) * y);
	elseif (c <= 2e-35)
		tmp = Float64(Float64(i * b) * a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (c * t) * j;
	tmp = 0.0;
	if (c <= -3.6e+70)
		tmp = t_1;
	elseif (c <= 1.05e-116)
		tmp = (z * x) * y;
	elseif (c <= 2e-35)
		tmp = (i * b) * a;
	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[(c * t), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[c, -3.6e+70], t$95$1, If[LessEqual[c, 1.05e-116], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[c, 2e-35], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.6e70 or 2.00000000000000002e-35 < c

   1. Initial program 67.1%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      11. lower-*.f64 40.9

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

   5. Applied rewrites 40.9%

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

   6. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.1

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

   8. Applied rewrites 52.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 35.2%

       \[\leadsto \left(c \cdot t\right) \cdot j \]

   if -3.6e70 < c < 1.05e-116

   1. Initial program 83.4%

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

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

      11. lower-*.f64 43.2

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 32.4%

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

   10. Applied rewrites 33.2%

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

   if 1.05e-116 < c < 2.00000000000000002e-35

   1. Initial program 76.5%

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

   3. Taylor expanded in 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 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 46.2%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.6 \cdot 10^{+70}:\\ \;\;\;\;\left(c \cdot t\right) \cdot j\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-116}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot t\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 28.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(i \cdot b\right) \cdot a\\ \mathbf{if}\;i \leq -3.5 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 7.8 \cdot 10^{+140}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* i b) a)))
   (if (<= i -3.5e+107) t_1 (if (<= i 7.8e+140) (* (* 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 = (i * b) * a;
	double tmp;
	if (i <= -3.5e+107) {
		tmp = t_1;
	} else if (i <= 7.8e+140) {
		tmp = (y * x) * z;
	} 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 = (i * b) * a
    if (i <= (-3.5d+107)) then
        tmp = t_1
    else if (i <= 7.8d+140) then
        tmp = (y * x) * z
    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 = (i * b) * a;
	double tmp;
	if (i <= -3.5e+107) {
		tmp = t_1;
	} else if (i <= 7.8e+140) {
		tmp = (y * x) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (i * b) * a
	tmp = 0
	if i <= -3.5e+107:
		tmp = t_1
	elif i <= 7.8e+140:
		tmp = (y * x) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(i * b) * a)
	tmp = 0.0
	if (i <= -3.5e+107)
		tmp = t_1;
	elseif (i <= 7.8e+140)
		tmp = Float64(Float64(y * x) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (i * b) * a;
	tmp = 0.0;
	if (i <= -3.5e+107)
		tmp = t_1;
	elseif (i <= 7.8e+140)
		tmp = (y * x) * z;
	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[(i * b), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[i, -3.5e+107], t$95$1, If[LessEqual[i, 7.8e+140], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -3.4999999999999997e107 or 7.79999999999999949e140 < i

   1. Initial program 59.3%

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

   3. Taylor expanded in 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 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 36.5%

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

   if -3.4999999999999997e107 < i < 7.79999999999999949e140

   1. Initial program 80.7%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      11. lower-*.f64 46.3

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 26.5%

      \[\leadsto \left(x \cdot y\right) \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.5 \cdot 10^{+107}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;i \leq 7.8 \cdot 10^{+140}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 29.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-20}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-54}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -4.3e-20)
   (* (* z x) y)
   (if (<= x 3.8e-54) (* (* i b) a) (* (* 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 (x <= -4.3e-20) {
		tmp = (z * x) * y;
	} else if (x <= 3.8e-54) {
		tmp = (i * b) * a;
	} 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 (x <= (-4.3d-20)) then
        tmp = (z * x) * y
    else if (x <= 3.8d-54) then
        tmp = (i * b) * a
    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 (x <= -4.3e-20) {
		tmp = (z * x) * y;
	} else if (x <= 3.8e-54) {
		tmp = (i * b) * a;
	} else {
		tmp = (z * y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if x <= -4.3e-20:
		tmp = (z * x) * y
	elif x <= 3.8e-54:
		tmp = (i * b) * a
	else:
		tmp = (z * y) * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -4.3e-20)
		tmp = Float64(Float64(z * x) * y);
	elseif (x <= 3.8e-54)
		tmp = Float64(Float64(i * b) * a);
	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 (x <= -4.3e-20)
		tmp = (z * x) * y;
	elseif (x <= 3.8e-54)
		tmp = (i * b) * a;
	else
		tmp = (z * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -4.3e-20], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 3.8e-54], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.30000000000000011e-20

   1. Initial program 73.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      11. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 36.8%

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

   10. Applied rewrites 39.7%

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

   if -4.30000000000000011e-20 < x < 3.8000000000000002e-54

   1. Initial program 70.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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 43.3

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 23.1%

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

   if 3.8000000000000002e-54 < x

   1. Initial program 83.9%

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

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

      11. lower-*.f64 40.3

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

   5. Applied rewrites 40.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 27.7%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-20}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-54}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 22.4% 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 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 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. *-commutative N/A

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

   11. lower-*.f64 41.1

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

5. Applied rewrites 41.1%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 20.4%

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

10. Applied rewrites 21.1%

    \[\leadsto \left(z \cdot x\right) \cdot y \]
11. Add Preprocessing

Alternative 24: 22.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot y\right) \cdot x \end{array} \]

FPCore

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

C

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

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 * y) * x
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 * y) * x;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 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. *-commutative N/A

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

   11. lower-*.f64 41.1

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

5. Applied rewrites 41.1%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 20.4%

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

9. Final simplification 20.4%

   \[\leadsto \left(z \cdot y\right) \cdot x \]
10. Add Preprocessing

Developer Target 1: 68.2% 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 2024331 
(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)))))