Data.Colour.Matrix:determinant from colour-2.3.3, A

Percentage Accurate: 73.6% → 84.1%

Time: 18.3s

Alternatives: 16

Speedup: 0.5×

• 57.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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) - (t * i)))) + (j * ((c * a) - (y * i)));
}

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) - (t * i)))) + (j * ((c * a) - (y * i)))
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) - (t * i)))) + (j * ((c * a) - (y * i)));
}

Python

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

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(t * i)))) + Float64(j * Float64(Float64(c * a) - Float64(y * i))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
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[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

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) - (t * i)))) + (j * ((c * a) - (y * i)));
}

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) - (t * i)))) + (j * ((c * a) - (y * i)))
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) - (t * i)))) + (j * ((c * a) - (y * i)));
}

Python

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

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(t * i)))) + Float64(j * Float64(Float64(c * a) - Float64(y * i))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
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[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[(t * a), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(t * N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] + N[(z * N[(x * y + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i)))) < +inf.0

   1. Initial program 89.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. sub-neg N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 36.0%

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

   6. Taylor expanded in j around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 56.0

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

   8. Simplified 56.0%

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

4. Final simplification 82.9%

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

Alternative 2: 78.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[(t * a), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * t$95$1), $MachinePrecision]), $MachinePrecision], Infinity], N[(j * t$95$1 + N[(t * N[(a * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision] + N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] + N[(z * N[(x * y + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i)))) < +inf.0

   1. Initial program 89.4%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. sub-neg N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 82.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. sub-neg N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 36.0%

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

   6. Taylor expanded in j around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 56.0

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

   8. Simplified 56.0%

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

4. Final simplification 77.0%

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

Alternative 3: 51.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(a \cdot \mathsf{fma}\left(-i, \frac{y}{a}, c\right)\right)\\ \mathbf{if}\;j \leq -1.5 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 5.5 \cdot 10^{-287}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{elif}\;j \leq 2.7 \cdot 10^{-145}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{elif}\;j \leq 1.3 \cdot 10^{-39}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (* a (fma (- i) (/ y a) c)))))
   (if (<= j -1.5e+68)
     t_1
     (if (<= j 5.5e-287)
       (* t (fma a (- x) (* b i)))
       (if (<= j 2.7e-145)
         (* z (fma c (- b) (* x y)))
         (if (<= j 1.3e-39) (* i (fma j (- y) (* t b))) 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 = j * (a * fma(-i, (y / a), c));
	double tmp;
	if (j <= -1.5e+68) {
		tmp = t_1;
	} else if (j <= 5.5e-287) {
		tmp = t * fma(a, -x, (b * i));
	} else if (j <= 2.7e-145) {
		tmp = z * fma(c, -b, (x * y));
	} else if (j <= 1.3e-39) {
		tmp = i * fma(j, -y, (t * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(a * fma(Float64(-i), Float64(y / a), c)))
	tmp = 0.0
	if (j <= -1.5e+68)
		tmp = t_1;
	elseif (j <= 5.5e-287)
		tmp = Float64(t * fma(a, Float64(-x), Float64(b * i)));
	elseif (j <= 2.7e-145)
		tmp = Float64(z * fma(c, Float64(-b), Float64(x * y)));
	elseif (j <= 1.3e-39)
		tmp = Float64(i * fma(j, Float64(-y), Float64(t * b)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(a * N[((-i) * N[(y / a), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -1.5e+68], t$95$1, If[LessEqual[j, 5.5e-287], N[(t * N[(a * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 2.7e-145], N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 1.3e-39], N[(i * N[(j * (-y) + N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if j < -1.5000000000000001e68 or 1.3e-39 < j

   1. Initial program 74.2%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 68.4

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

   5. Simplified 68.4%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 68.5

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

   8. Simplified 68.5%

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

   if -1.5000000000000001e68 < j < 5.4999999999999998e-287

   1. Initial program 71.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 57.8

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

   5. Simplified 57.8%

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

   if 5.4999999999999998e-287 < j < 2.7e-145

   1. Initial program 73.9%

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 66.0

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

   5. Simplified 66.0%

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

   if 2.7e-145 < j < 1.3e-39

   1. Initial program 58.9%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 66.9

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

   5. Simplified 66.9%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.5 \cdot 10^{+68}:\\ \;\;\;\;j \cdot \left(a \cdot \mathsf{fma}\left(-i, \frac{y}{a}, c\right)\right)\\ \mathbf{elif}\;j \leq 5.5 \cdot 10^{-287}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{elif}\;j \leq 2.7 \cdot 10^{-145}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{elif}\;j \leq 1.3 \cdot 10^{-39}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot \mathsf{fma}\left(-i, \frac{y}{a}, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot c - y \cdot i\\ \mathbf{if}\;j \leq -3.9 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(j, t\_1, b \cdot \left(t \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 1.5 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(i, t \cdot b, x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\right)\\ \mathbf{elif}\;j \leq 9.8 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i\right) + j \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (- (* a c) (* y i))))
   (if (<= j -3.9e+46)
     (fma j t_1 (* b (* t i)))
     (if (<= j 1.5e-287)
       (fma i (* t b) (* x (fma t (- a) (* y z))))
       (if (<= j 9.8e-133)
         (fma t (* b i) (* z (fma x y (* b (- c)))))
         (+ (* t (* b i)) (* 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 = (a * c) - (y * i);
	double tmp;
	if (j <= -3.9e+46) {
		tmp = fma(j, t_1, (b * (t * i)));
	} else if (j <= 1.5e-287) {
		tmp = fma(i, (t * b), (x * fma(t, -a, (y * z))));
	} else if (j <= 9.8e-133) {
		tmp = fma(t, (b * i), (z * fma(x, y, (b * -c))));
	} else {
		tmp = (t * (b * i)) + (j * t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(a * c) - Float64(y * i))
	tmp = 0.0
	if (j <= -3.9e+46)
		tmp = fma(j, t_1, Float64(b * Float64(t * i)));
	elseif (j <= 1.5e-287)
		tmp = fma(i, Float64(t * b), Float64(x * fma(t, Float64(-a), Float64(y * z))));
	elseif (j <= 9.8e-133)
		tmp = fma(t, Float64(b * i), Float64(z * fma(x, y, Float64(b * Float64(-c)))));
	else
		tmp = Float64(Float64(t * Float64(b * i)) + Float64(j * t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -3.9e+46], N[(j * t$95$1 + N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 1.5e-287], N[(i * N[(t * b), $MachinePrecision] + N[(x * N[(t * (-a) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 9.8e-133], N[(t * N[(b * i), $MachinePrecision] + N[(z * N[(x * y + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(b * i), $MachinePrecision]), $MachinePrecision] + N[(j * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if j < -3.89999999999999995e46

   1. Initial program 76.2%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. sub-neg N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 78.2%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.0

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

   8. Simplified 82.0%

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

   if -3.89999999999999995e46 < j < 1.49999999999999996e-287

   1. Initial program 70.1%

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

   3. Taylor expanded in c around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft-in N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

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

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

   5. Simplified 66.7%

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

   6. Taylor expanded in j around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 64.4

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

   8. Simplified 64.4%

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

   if 1.49999999999999996e-287 < j < 9.79999999999999992e-133

   1. Initial program 68.8%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. sub-neg N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 74.6%

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

   6. Taylor expanded in j around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 80.0

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

   8. Simplified 80.0%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 74.8

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

   11. Simplified 74.8%

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

   if 9.79999999999999992e-133 < j

   1. Initial program 72.1%

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

   3. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.9

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

   5. Simplified 67.9%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -3.9 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(j, a \cdot c - y \cdot i, b \cdot \left(t \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 1.5 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(i, t \cdot b, x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\right)\\ \mathbf{elif}\;j \leq 9.8 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i\right) + j \cdot \left(a \cdot c - y \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -4 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(j, a \cdot c - y \cdot i, b \cdot \left(t \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 9 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i - x \cdot a, z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot \mathsf{fma}\left(-i, \frac{y}{a}, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -4e+46)
   (fma j (- (* a c) (* y i)) (* b (* t i)))
   (if (<= j 9e+80)
     (fma t (- (* b i) (* x a)) (* z (fma x y (* b (- c)))))
     (* j (* a (fma (- i) (/ y a) 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 (j <= -4e+46) {
		tmp = fma(j, ((a * c) - (y * i)), (b * (t * i)));
	} else if (j <= 9e+80) {
		tmp = fma(t, ((b * i) - (x * a)), (z * fma(x, y, (b * -c))));
	} else {
		tmp = j * (a * fma(-i, (y / a), c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -4e+46)
		tmp = fma(j, Float64(Float64(a * c) - Float64(y * i)), Float64(b * Float64(t * i)));
	elseif (j <= 9e+80)
		tmp = fma(t, Float64(Float64(b * i) - Float64(x * a)), Float64(z * fma(x, y, Float64(b * Float64(-c)))));
	else
		tmp = Float64(j * Float64(a * fma(Float64(-i), Float64(y / a), c)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -4e+46], N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] + N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 9e+80], N[(t * N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] + N[(z * N[(x * y + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(j * N[(a * N[((-i) * N[(y / a), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if j < -4e46

   1. Initial program 76.2%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. sub-neg N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 78.2%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.0

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

   8. Simplified 82.0%

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

   if -4e46 < j < 9.00000000000000013e80

   1. Initial program 70.5%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. sub-neg N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 69.9%

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

   6. Taylor expanded in j around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 70.0

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

   8. Simplified 70.0%

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

   if 9.00000000000000013e80 < j

   1. Initial program 71.6%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 80.9

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

   5. Simplified 80.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 81.0

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

   8. Simplified 81.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -4 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(j, a \cdot c - y \cdot i, b \cdot \left(t \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 9 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i - x \cdot a, z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot \mathsf{fma}\left(-i, \frac{y}{a}, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -3.9 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(j, a \cdot c - y \cdot i, b \cdot \left(t \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 3.6 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, t \cdot b\right), x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot \mathsf{fma}\left(-i, \frac{y}{a}, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -3.9e+46)
   (fma j (- (* a c) (* y i)) (* b (* t i)))
   (if (<= j 3.6e+111)
     (fma i (fma j (- y) (* t b)) (* x (fma t (- a) (* y z))))
     (* j (* a (fma (- i) (/ y a) 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 (j <= -3.9e+46) {
		tmp = fma(j, ((a * c) - (y * i)), (b * (t * i)));
	} else if (j <= 3.6e+111) {
		tmp = fma(i, fma(j, -y, (t * b)), (x * fma(t, -a, (y * z))));
	} else {
		tmp = j * (a * fma(-i, (y / a), c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -3.9e+46)
		tmp = fma(j, Float64(Float64(a * c) - Float64(y * i)), Float64(b * Float64(t * i)));
	elseif (j <= 3.6e+111)
		tmp = fma(i, fma(j, Float64(-y), Float64(t * b)), Float64(x * fma(t, Float64(-a), Float64(y * z))));
	else
		tmp = Float64(j * Float64(a * fma(Float64(-i), Float64(y / a), c)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -3.9e+46], N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] + N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 3.6e+111], N[(i * N[(j * (-y) + N[(t * b), $MachinePrecision]), $MachinePrecision] + N[(x * N[(t * (-a) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(j * N[(a * N[((-i) * N[(y / a), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if j < -3.89999999999999995e46

   1. Initial program 76.2%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. sub-neg N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 78.2%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.0

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

   8. Simplified 82.0%

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

   if -3.89999999999999995e46 < j < 3.6000000000000002e111

   1. Initial program 70.6%

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

   3. Taylor expanded in c around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft-in N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

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

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

   5. Simplified 65.3%

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

   if 3.6000000000000002e111 < j

   1. Initial program 71.3%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 81.5

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

   5. Simplified 81.5%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 81.6

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

   8. Simplified 81.6%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -3.9 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(j, a \cdot c - y \cdot i, b \cdot \left(t \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 3.6 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, t \cdot b\right), x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot \mathsf{fma}\left(-i, \frac{y}{a}, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j, a \cdot c - y \cdot i, b \cdot \left(t \cdot i\right)\right)\\ \mathbf{if}\;j \leq -2.7 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 4.2 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(i, t \cdot b, x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\right)\\ \mathbf{elif}\;j \leq 2.6 \cdot 10^{-134}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma j (- (* a c) (* y i)) (* b (* t i)))))
   (if (<= j -2.7e+47)
     t_1
     (if (<= j 4.2e-287)
       (fma i (* t b) (* x (fma t (- a) (* y z))))
       (if (<= j 2.6e-134) (fma t (* b i) (* z (fma x y (* b (- c))))) 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(j, ((a * c) - (y * i)), (b * (t * i)));
	double tmp;
	if (j <= -2.7e+47) {
		tmp = t_1;
	} else if (j <= 4.2e-287) {
		tmp = fma(i, (t * b), (x * fma(t, -a, (y * z))));
	} else if (j <= 2.6e-134) {
		tmp = fma(t, (b * i), (z * fma(x, y, (b * -c))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(j, Float64(Float64(a * c) - Float64(y * i)), Float64(b * Float64(t * i)))
	tmp = 0.0
	if (j <= -2.7e+47)
		tmp = t_1;
	elseif (j <= 4.2e-287)
		tmp = fma(i, Float64(t * b), Float64(x * fma(t, Float64(-a), Float64(y * z))));
	elseif (j <= 2.6e-134)
		tmp = fma(t, Float64(b * i), Float64(z * fma(x, y, Float64(b * Float64(-c)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] + N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -2.7e+47], t$95$1, If[LessEqual[j, 4.2e-287], N[(i * N[(t * b), $MachinePrecision] + N[(x * N[(t * (-a) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 2.6e-134], N[(t * N[(b * i), $MachinePrecision] + N[(z * N[(x * y + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -2.69999999999999996e47 or 2.60000000000000023e-134 < j

   1. Initial program 73.3%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. sub-neg N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 76.6%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 71.8

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

   8. Simplified 71.8%

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

   if -2.69999999999999996e47 < j < 4.1999999999999998e-287

   1. Initial program 70.1%

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

   3. Taylor expanded in c around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft-in N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

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

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

   5. Simplified 66.7%

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

   6. Taylor expanded in j around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 64.4

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

   8. Simplified 64.4%

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

   if 4.1999999999999998e-287 < j < 2.60000000000000023e-134

   1. Initial program 70.6%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. sub-neg N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 76.8%

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

   6. Taylor expanded in j around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 82.4

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

   8. Simplified 82.4%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 77.1

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

   11. Simplified 77.1%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.7 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(j, a \cdot c - y \cdot i, b \cdot \left(t \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 4.2 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(i, t \cdot b, x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\right)\\ \mathbf{elif}\;j \leq 2.6 \cdot 10^{-134}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(j, a \cdot c - y \cdot i, b \cdot \left(t \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{if}\;j \leq -1.5 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 5.6 \cdot 10^{-287}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{elif}\;j \leq 1.15 \cdot 10^{-146}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{elif}\;j \leq 1.3 \cdot 10^{-39}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* a c) (* y i)))))
   (if (<= j -1.5e+70)
     t_1
     (if (<= j 5.6e-287)
       (* t (fma a (- x) (* b i)))
       (if (<= j 1.15e-146)
         (* z (fma c (- b) (* x y)))
         (if (<= j 1.3e-39) (* i (fma j (- y) (* t b))) 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 = j * ((a * c) - (y * i));
	double tmp;
	if (j <= -1.5e+70) {
		tmp = t_1;
	} else if (j <= 5.6e-287) {
		tmp = t * fma(a, -x, (b * i));
	} else if (j <= 1.15e-146) {
		tmp = z * fma(c, -b, (x * y));
	} else if (j <= 1.3e-39) {
		tmp = i * fma(j, -y, (t * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(a * c) - Float64(y * i)))
	tmp = 0.0
	if (j <= -1.5e+70)
		tmp = t_1;
	elseif (j <= 5.6e-287)
		tmp = Float64(t * fma(a, Float64(-x), Float64(b * i)));
	elseif (j <= 1.15e-146)
		tmp = Float64(z * fma(c, Float64(-b), Float64(x * y)));
	elseif (j <= 1.3e-39)
		tmp = Float64(i * fma(j, Float64(-y), Float64(t * b)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -1.5e+70], t$95$1, If[LessEqual[j, 5.6e-287], N[(t * N[(a * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 1.15e-146], N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 1.3e-39], N[(i * N[(j * (-y) + N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if j < -1.49999999999999988e70 or 1.3e-39 < j

   1. Initial program 74.2%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 68.4

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

   5. Simplified 68.4%

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

   if -1.49999999999999988e70 < j < 5.6000000000000005e-287

   1. Initial program 71.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 57.8

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

   5. Simplified 57.8%

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

   if 5.6000000000000005e-287 < j < 1.15e-146

   1. Initial program 73.9%

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 66.0

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

   5. Simplified 66.0%

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

   if 1.15e-146 < j < 1.3e-39

   1. Initial program 58.9%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 66.9

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

   5. Simplified 66.9%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.5 \cdot 10^{+70}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;j \leq 5.6 \cdot 10^{-287}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{elif}\;j \leq 1.15 \cdot 10^{-146}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{elif}\;j \leq 1.3 \cdot 10^{-39}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j, a \cdot c - y \cdot i, b \cdot \left(t \cdot i\right)\right)\\ \mathbf{if}\;j \leq -5.2 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 7.5 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(i, t \cdot b, x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma j (- (* a c) (* y i)) (* b (* t i)))))
   (if (<= j -5.2e+46)
     t_1
     (if (<= j 7.5e-150) (fma i (* t b) (* x (fma t (- a) (* y z)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(j, ((a * c) - (y * i)), (b * (t * i)));
	double tmp;
	if (j <= -5.2e+46) {
		tmp = t_1;
	} else if (j <= 7.5e-150) {
		tmp = fma(i, (t * b), (x * fma(t, -a, (y * z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(j, Float64(Float64(a * c) - Float64(y * i)), Float64(b * Float64(t * i)))
	tmp = 0.0
	if (j <= -5.2e+46)
		tmp = t_1;
	elseif (j <= 7.5e-150)
		tmp = fma(i, Float64(t * b), Float64(x * fma(t, Float64(-a), Float64(y * z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] + N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -5.2e+46], t$95$1, If[LessEqual[j, 7.5e-150], N[(i * N[(t * b), $MachinePrecision] + N[(x * N[(t * (-a) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -5.20000000000000027e46 or 7.5000000000000004e-150 < j

   1. Initial program 72.7%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. sub-neg N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 76.0%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.6

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

   8. Simplified 70.6%

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

   if -5.20000000000000027e46 < j < 7.5000000000000004e-150

   1. Initial program 70.9%

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

   3. Taylor expanded in c around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft-in N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

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

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

   5. Simplified 67.5%

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

   6. Taylor expanded in j around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.8

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

   8. Simplified 65.8%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -5.2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(j, a \cdot c - y \cdot i, b \cdot \left(t \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 7.5 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(i, t \cdot b, x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(j, a \cdot c - y \cdot i, b \cdot \left(t \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(a \cdot \mathsf{fma}\left(-i, \frac{y}{a}, c\right)\right)\\ \mathbf{if}\;j \leq -7.6 \cdot 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 5 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(i, t \cdot b, x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (* a (fma (- i) (/ y a) c)))))
   (if (<= j -7.6e+117)
     t_1
     (if (<= j 5e+70) (fma i (* t b) (* x (fma t (- a) (* y 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 = j * (a * fma(-i, (y / a), c));
	double tmp;
	if (j <= -7.6e+117) {
		tmp = t_1;
	} else if (j <= 5e+70) {
		tmp = fma(i, (t * b), (x * fma(t, -a, (y * z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(a * fma(Float64(-i), Float64(y / a), c)))
	tmp = 0.0
	if (j <= -7.6e+117)
		tmp = t_1;
	elseif (j <= 5e+70)
		tmp = fma(i, Float64(t * b), Float64(x * fma(t, Float64(-a), Float64(y * z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(a * N[((-i) * N[(y / a), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -7.6e+117], t$95$1, If[LessEqual[j, 5e+70], N[(i * N[(t * b), $MachinePrecision] + N[(x * N[(t * (-a) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -7.6000000000000003e117 or 5.0000000000000002e70 < j

   1. Initial program 73.6%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 75.5

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

   5. Simplified 75.5%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 75.6

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

   8. Simplified 75.6%

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

   if -7.6000000000000003e117 < j < 5.0000000000000002e70

   1. Initial program 71.0%

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

   3. Taylor expanded in c around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft-in N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

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

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

   5. Simplified 65.3%

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

   6. Taylor expanded in j around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.2

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

   8. Simplified 61.2%

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

Alternative 11: 41.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(t \cdot b\right)\\ \mathbf{if}\;b \leq -7 \cdot 10^{+230}:\\ \;\;\;\;b \cdot \left(c \cdot \left(-z\right)\right)\\ \mathbf{elif}\;b \leq -3.3 \cdot 10^{+66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+120}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (* t b))))
   (if (<= b -7e+230)
     (* b (* c (- z)))
     (if (<= b -3.3e+66)
       t_1
       (if (<= b 1.8e+120) (* a (fma j c (* x (- t)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * (t * b);
	double tmp;
	if (b <= -7e+230) {
		tmp = b * (c * -z);
	} else if (b <= -3.3e+66) {
		tmp = t_1;
	} else if (b <= 1.8e+120) {
		tmp = a * fma(j, c, (x * -t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * Float64(t * b))
	tmp = 0.0
	if (b <= -7e+230)
		tmp = Float64(b * Float64(c * Float64(-z)));
	elseif (b <= -3.3e+66)
		tmp = t_1;
	elseif (b <= 1.8e+120)
		tmp = Float64(a * fma(j, c, Float64(x * Float64(-t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -7e+230], N[(b * N[(c * (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3.3e+66], t$95$1, If[LessEqual[b, 1.8e+120], N[(a * N[(j * c + N[(x * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.0000000000000001e230

   1. Initial program 77.2%

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.5

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

   5. Simplified 49.5%

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

   6. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 49.0

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

   8. Simplified 49.0%

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

   if -7.0000000000000001e230 < b < -3.3000000000000001e66 or 1.80000000000000008e120 < b

   1. Initial program 76.1%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. sub-neg N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 76.4%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 56.3

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

   8. Simplified 56.3%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 57.7

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

   11. Simplified 57.7%

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

   if -3.3000000000000001e66 < b < 1.80000000000000008e120

   1. Initial program 69.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 51.2

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

   5. Simplified 51.2%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+230}:\\ \;\;\;\;b \cdot \left(c \cdot \left(-z\right)\right)\\ \mathbf{elif}\;b \leq -3.3 \cdot 10^{+66}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+120}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{if}\;j \leq -4.1 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 4.8 \cdot 10^{-49}:\\ \;\;\;\;b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* a c) (* y i)))))
   (if (<= j -4.1e+74) t_1 (if (<= j 4.8e-49) (* b (- (* t i) (* z c))) 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 = j * ((a * c) - (y * i));
	double tmp;
	if (j <= -4.1e+74) {
		tmp = t_1;
	} else if (j <= 4.8e-49) {
		tmp = b * ((t * i) - (z * c));
	} 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 = j * ((a * c) - (y * i))
    if (j <= (-4.1d+74)) then
        tmp = t_1
    else if (j <= 4.8d-49) then
        tmp = b * ((t * i) - (z * c))
    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 = j * ((a * c) - (y * i));
	double tmp;
	if (j <= -4.1e+74) {
		tmp = t_1;
	} else if (j <= 4.8e-49) {
		tmp = b * ((t * i) - (z * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((a * c) - (y * i))
	tmp = 0
	if j <= -4.1e+74:
		tmp = t_1
	elif j <= 4.8e-49:
		tmp = b * ((t * i) - (z * c))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(a * c) - Float64(y * i)))
	tmp = 0.0
	if (j <= -4.1e+74)
		tmp = t_1;
	elseif (j <= 4.8e-49)
		tmp = Float64(b * Float64(Float64(t * i) - Float64(z * c)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * ((a * c) - (y * i));
	tmp = 0.0;
	if (j <= -4.1e+74)
		tmp = t_1;
	elseif (j <= 4.8e-49)
		tmp = b * ((t * i) - (z * c));
	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[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -4.1e+74], t$95$1, If[LessEqual[j, 4.8e-49], N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -4.1e74 or 4.79999999999999985e-49 < j

   1. Initial program 74.1%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 67.6

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

   5. Simplified 67.6%

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

   if -4.1e74 < j < 4.79999999999999985e-49

   1. Initial program 70.0%

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

   3. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. remove-double-neg N/A

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

      12. +-commutative N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 55.0

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

   5. Simplified 55.0%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -4.1 \cdot 10^{+74}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;j \leq 4.8 \cdot 10^{-49}:\\ \;\;\;\;b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{if}\;b \leq -6 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{+119}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* t i) (* z c)))))
   (if (<= b -6e+65)
     t_1
     (if (<= b 8.8e+119) (* a (fma j c (* x (- t)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * ((t * i) - (z * c));
	double tmp;
	if (b <= -6e+65) {
		tmp = t_1;
	} else if (b <= 8.8e+119) {
		tmp = a * fma(j, c, (x * -t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(t * i) - Float64(z * c)))
	tmp = 0.0
	if (b <= -6e+65)
		tmp = t_1;
	elseif (b <= 8.8e+119)
		tmp = Float64(a * fma(j, c, Float64(x * Float64(-t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6e+65], t$95$1, If[LessEqual[b, 8.8e+119], N[(a * N[(j * c + N[(x * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.0000000000000004e65 or 8.8000000000000005e119 < b

   1. Initial program 76.4%

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

   3. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. remove-double-neg N/A

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

      12. +-commutative N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 72.6

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

   5. Simplified 72.6%

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

   if -6.0000000000000004e65 < b < 8.8000000000000005e119

   1. Initial program 69.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 51.2

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

   5. Simplified 51.2%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+65}:\\ \;\;\;\;b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{+119}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(t \cdot i - z \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(a \cdot c\right)\\ \mathbf{if}\;j \leq -3.2 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 8.5 \cdot 10^{-41}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (* a c))))
   (if (<= j -3.2e+118) t_1 (if (<= j 8.5e-41) (* i (* t b)) 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 = j * (a * c);
	double tmp;
	if (j <= -3.2e+118) {
		tmp = t_1;
	} else if (j <= 8.5e-41) {
		tmp = i * (t * b);
	} 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 = j * (a * c)
    if (j <= (-3.2d+118)) then
        tmp = t_1
    else if (j <= 8.5d-41) then
        tmp = i * (t * b)
    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 = j * (a * c);
	double tmp;
	if (j <= -3.2e+118) {
		tmp = t_1;
	} else if (j <= 8.5e-41) {
		tmp = i * (t * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * (a * c)
	tmp = 0
	if j <= -3.2e+118:
		tmp = t_1
	elif j <= 8.5e-41:
		tmp = i * (t * b)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(a * c))
	tmp = 0.0
	if (j <= -3.2e+118)
		tmp = t_1;
	elseif (j <= 8.5e-41)
		tmp = Float64(i * Float64(t * b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * (a * c);
	tmp = 0.0;
	if (j <= -3.2e+118)
		tmp = t_1;
	elseif (j <= 8.5e-41)
		tmp = i * (t * b);
	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[(j * N[(a * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -3.2e+118], t$95$1, If[LessEqual[j, 8.5e-41], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -3.20000000000000016e118 or 8.4999999999999996e-41 < j

   1. Initial program 74.9%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 59.3

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

   5. Simplified 59.3%

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

   6. Taylor expanded in j around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 54.8

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

   8. Simplified 54.8%

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

   if -3.20000000000000016e118 < j < 8.4999999999999996e-41

   1. Initial program 69.8%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. associate-+r+ N/A

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

      4. sub-neg N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 69.2%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(t, b, \color{blue}{\mathsf{neg}\left(j \cdot y\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto i \cdot \mathsf{fma}\left(t, b, \mathsf{neg}\left(\color{blue}{y \cdot j}\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \mathsf{fma}\left(t, b, \color{blue}{y \cdot \left(\mathsf{neg}\left(j\right)\right)}\right) \]

      8. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(t, b, y \cdot \color{blue}{\left(-1 \cdot j\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto i \cdot \mathsf{fma}\left(t, b, \color{blue}{y \cdot \left(-1 \cdot j\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto i \cdot \mathsf{fma}\left(t, b, y \cdot \color{blue}{\left(\mathsf{neg}\left(j\right)\right)}\right) \]

      11. lower-neg.f64 43.1

          \[\leadsto i \cdot \mathsf{fma}\left(t, b, y \cdot \color{blue}{\left(-j\right)}\right) \]

   8. Simplified 43.1%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(t, b, y \cdot \left(-j\right)\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

       2. lower-*.f64 34.6

          \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   11. Simplified 34.6%

       \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -3.2 \cdot 10^{+118}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;j \leq 8.5 \cdot 10^{-41}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(c \cdot j\right)\\ \mathbf{if}\;j \leq -3.2 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 7.2 \cdot 10^{-40}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (* c j))))
   (if (<= j -3.2e+118) t_1 (if (<= j 7.2e-40) (* i (* t b)) 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 = a * (c * j);
	double tmp;
	if (j <= -3.2e+118) {
		tmp = t_1;
	} else if (j <= 7.2e-40) {
		tmp = i * (t * b);
	} 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 = a * (c * j)
    if (j <= (-3.2d+118)) then
        tmp = t_1
    else if (j <= 7.2d-40) then
        tmp = i * (t * b)
    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 = a * (c * j);
	double tmp;
	if (j <= -3.2e+118) {
		tmp = t_1;
	} else if (j <= 7.2e-40) {
		tmp = i * (t * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = a * (c * j)
	tmp = 0
	if j <= -3.2e+118:
		tmp = t_1
	elif j <= 7.2e-40:
		tmp = i * (t * b)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * Float64(c * j))
	tmp = 0.0
	if (j <= -3.2e+118)
		tmp = t_1;
	elseif (j <= 7.2e-40)
		tmp = Float64(i * Float64(t * b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = a * (c * j);
	tmp = 0.0;
	if (j <= -3.2e+118)
		tmp = t_1;
	elseif (j <= 7.2e-40)
		tmp = i * (t * b);
	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[(a * N[(c * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -3.2e+118], t$95$1, If[LessEqual[j, 7.2e-40], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -3.20000000000000016e118 or 7.2e-40 < j

   1. Initial program 74.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. lower-neg.f64 59.3

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Simplified 59.3%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 52.2

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]

   8. Simplified 52.2%

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]

   if -3.20000000000000016e118 < j < 7.2e-40

   1. Initial program 69.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \left(a \cdot c - i \cdot y\right) + \color{blue}{\left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto j \cdot \left(a \cdot c - i \cdot y\right) + \color{blue}{\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)\right)} \]

      4. sub-neg N/A

         \[\leadsto j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \color{blue}{\left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(j, a \cdot c - i \cdot y, t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(j, \color{blue}{a \cdot c - i \cdot y}, t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(j, \color{blue}{a \cdot c} - i \cdot y, t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(j, a \cdot c - \color{blue}{i \cdot y}, t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(j, a \cdot c - i \cdot y, \color{blue}{\mathsf{fma}\left(t, -1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right), z \cdot \left(x \cdot y - b \cdot c\right)\right)}\right) \]

   5. Simplified 69.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, a \cdot c - i \cdot y, \mathsf{fma}\left(t, \mathsf{fma}\left(a, -x, i \cdot b\right), z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)\right)\right)} \]

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]

      2. +-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot t + -1 \cdot \left(j \cdot y\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto i \cdot \left(\color{blue}{t \cdot b} + -1 \cdot \left(j \cdot y\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(t, b, -1 \cdot \left(j \cdot y\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(t, b, \color{blue}{\mathsf{neg}\left(j \cdot y\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto i \cdot \mathsf{fma}\left(t, b, \mathsf{neg}\left(\color{blue}{y \cdot j}\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \mathsf{fma}\left(t, b, \color{blue}{y \cdot \left(\mathsf{neg}\left(j\right)\right)}\right) \]

      8. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(t, b, y \cdot \color{blue}{\left(-1 \cdot j\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto i \cdot \mathsf{fma}\left(t, b, \color{blue}{y \cdot \left(-1 \cdot j\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto i \cdot \mathsf{fma}\left(t, b, y \cdot \color{blue}{\left(\mathsf{neg}\left(j\right)\right)}\right) \]

      11. lower-neg.f64 43.1

          \[\leadsto i \cdot \mathsf{fma}\left(t, b, y \cdot \color{blue}{\left(-j\right)}\right) \]

   8. Simplified 43.1%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(t, b, y \cdot \left(-j\right)\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

       2. lower-*.f64 34.6

          \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   11. Simplified 34.6%

       \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 23.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(c \cdot j\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* a (* c j)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (c * j);
}

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 = a * (c * j)
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 a * (c * j);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return a * (c * j)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(a * Float64(c * j))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = a * (c * j);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(a * N[(c * j), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.9%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

   2. +-commutative N/A

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

   5. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

   7. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

   9. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

   10. lower-neg.f64 41.5

       \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

5. Simplified 41.5%

   \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

6. Taylor expanded in j around inf

   \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 28.2

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]

8. Simplified 28.2%

   \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
9. Add Preprocessing

Developer Target 1: 59.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_2 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + t\_1\\ \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* c a) (* y i))))
        (t_2
         (+
          (-
           (* x (- (* y z) (* t a)))
           (/
            (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0)))
            (+ (* c z) (* t i))))
          t_1)))
   (if (< x -1.469694296777705e-64)
     t_2
     (if (< x 3.2113527362226803e-147)
       (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) 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 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (pow((c * z), 2.0) - pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - 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 = j * ((c * a) - (y * i))
    t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ** 2.0d0) - ((t * i) ** 2.0d0))) / ((c * z) + (t * i)))) + t_1
    if (x < (-1.469694296777705d-64)) then
        tmp = t_2
    else if (x < 3.2113527362226803d-147) then
        tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - 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 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (Math.pow((c * z), 2.0) - Math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((c * a) - (y * i))
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (math.pow((c * z), 2.0) - math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1
	tmp = 0
	if x < -1.469694296777705e-64:
		tmp = t_2
	elif x < 3.2113527362226803e-147:
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_2 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(b * Float64((Float64(c * z) ^ 2.0) - (Float64(t * i) ^ 2.0))) / Float64(Float64(c * z) + Float64(t * i)))) + t_1)
	tmp = 0.0
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = Float64(Float64(Float64(Float64(b * i) - Float64(x * a)) * t) - Float64(Float64(z * Float64(c * b)) - 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 = j * ((c * a) - (y * i));
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ^ 2.0) - ((t * i) ^ 2.0))) / ((c * z) + (t * i)))) + t_1;
	tmp = 0.0;
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - 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[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[Power[N[(c * z), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(t * i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Less[x, -1.469694296777705e-64], t$95$2, If[Less[x, 3.2113527362226803e-147], N[(N[(N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] - N[(N[(z * N[(c * b), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -293938859355541/2000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 32113527362226803/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))