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

Percentage Accurate: 73.2% → 84.6%

Time: 18.5s

Alternatives: 20

Speedup: 0.5×

• 57.9% 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: 73.2% 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: 84.6% accurate, 0.5× speedup

Math

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

FPCore

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

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 t_2 = (y * z) - (t * a);
	double tmp;
	if ((((x * t_2) + (b * ((t * i) - (z * c)))) + t_1) <= ((double) INFINITY)) {
		tmp = fma(t_2, x, fma(b, -fma(i, -t, (z * c)), t_1));
	} else {
		tmp = fma(z, fma(c, -b, (x * y)), (t * fma(i, b, (a * -x))));
	}
	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)))
	t_2 = Float64(Float64(y * z) - Float64(t * a))
	tmp = 0.0
	if (Float64(Float64(Float64(x * t_2) + Float64(b * Float64(Float64(t * i) - Float64(z * c)))) + t_1) <= Inf)
		tmp = fma(t_2, x, fma(b, Float64(-fma(i, Float64(-t), Float64(z * c))), t_1));
	else
		tmp = fma(z, fma(c, Float64(-b), Float64(x * y)), Float64(t * fma(i, b, Float64(a * Float64(-x)))));
	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]}, Block[{t$95$2 = N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x * t$95$2), $MachinePrecision] + N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], Infinity], N[(t$95$2 * x + N[(b * (-N[(i * (-t) + N[(z * c), $MachinePrecision]), $MachinePrecision]) + t$95$1), $MachinePrecision]), $MachinePrecision], N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(t * N[(i * b + N[(a * (-x)), $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 94.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 94.7%

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

   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 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 38.4

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

   5. Applied rewrites 38.4%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 63.0%

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

4. Final simplification 88.4%

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

Alternative 2: 54.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\\ \mathbf{if}\;x \leq -1.62 \cdot 10^{-28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-223}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, a \cdot j\right)\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-296}:\\ \;\;\;\;t \cdot \left(b \cdot i\right) + j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(c, -b, x \cdot y\right), b \cdot \left(t \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (fma t (- a) (* y z)))))
   (if (<= x -1.62e-28)
     t_1
     (if (<= x -3.8e-223)
       (* c (fma b (- z) (* a j)))
       (if (<= x -4.2e-296)
         (+ (* t (* b i)) (* j (* a c)))
         (if (<= x 3.6e+92)
           (fma z (fma c (- b) (* x y)) (* b (* t i)))
           t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * fma(t, -a, (y * z));
	double tmp;
	if (x <= -1.62e-28) {
		tmp = t_1;
	} else if (x <= -3.8e-223) {
		tmp = c * fma(b, -z, (a * j));
	} else if (x <= -4.2e-296) {
		tmp = (t * (b * i)) + (j * (a * c));
	} else if (x <= 3.6e+92) {
		tmp = fma(z, fma(c, -b, (x * y)), (b * (t * i)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(x * fma(t, Float64(-a), Float64(y * z)))
	tmp = 0.0
	if (x <= -1.62e-28)
		tmp = t_1;
	elseif (x <= -3.8e-223)
		tmp = Float64(c * fma(b, Float64(-z), Float64(a * j)));
	elseif (x <= -4.2e-296)
		tmp = Float64(Float64(t * Float64(b * i)) + Float64(j * Float64(a * c)));
	elseif (x <= 3.6e+92)
		tmp = fma(z, fma(c, Float64(-b), Float64(x * y)), Float64(b * Float64(t * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(t * (-a) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.62e-28], t$95$1, If[LessEqual[x, -3.8e-223], N[(c * N[(b * (-z) + N[(a * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.2e-296], N[(N[(t * N[(b * i), $MachinePrecision]), $MachinePrecision] + N[(j * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e+92], N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.62e-28 or 3.6e92 < x

   1. Initial program 78.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

   if -1.62e-28 < x < -3.80000000000000012e-223

   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 c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 66.2

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

   5. Applied rewrites 66.2%

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

   if -3.80000000000000012e-223 < x < -4.1999999999999999e-296

   1. Initial program 93.3%

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

   3. Taylor expanded in 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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in c around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 86.3

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

   8. Applied rewrites 86.3%

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

   10. Applied rewrites 93.2%

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

   if -4.1999999999999999e-296 < x < 3.6e92

   1. Initial program 68.2%

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

   3. Taylor expanded in 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 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 72.4%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 64.8%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.62 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-223}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, a \cdot j\right)\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-296}:\\ \;\;\;\;t \cdot \left(b \cdot i\right) + j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(c, -b, x \cdot y\right), b \cdot \left(t \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a \cdot c - y \cdot i, j, i \cdot \left(t \cdot b\right)\right)\\ \mathbf{if}\;j \leq -5.5 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 1.7 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(c, -b, x \cdot y\right), t \cdot \mathsf{fma}\left(i, b, a \cdot \left(-x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- (* a c) (* y i)) j (* i (* t b)))))
   (if (<= j -5.5e-55)
     t_1
     (if (<= j 1.7e+19)
       (fma z (fma c (- b) (* x y)) (* t (fma i b (* a (- x)))))
       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(((a * c) - (y * i)), j, (i * (t * b)));
	double tmp;
	if (j <= -5.5e-55) {
		tmp = t_1;
	} else if (j <= 1.7e+19) {
		tmp = fma(z, fma(c, -b, (x * y)), (t * fma(i, b, (a * -x))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(Float64(a * c) - Float64(y * i)), j, Float64(i * Float64(t * b)))
	tmp = 0.0
	if (j <= -5.5e-55)
		tmp = t_1;
	elseif (j <= 1.7e+19)
		tmp = fma(z, fma(c, Float64(-b), Float64(x * y)), Float64(t * fma(i, b, Float64(a * Float64(-x)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j + N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -5.5e-55], t$95$1, If[LessEqual[j, 1.7e+19], N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(t * N[(i * b + N[(a * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -5.4999999999999999e-55 or 1.7e19 < j

   1. Initial program 74.4%

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

   3. Taylor expanded in 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 64.1

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

   5. Applied rewrites 64.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 68.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 68.9

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

   7. Applied rewrites 71.8%

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

   if -5.4999999999999999e-55 < j < 1.7e19

   1. Initial program 77.3%

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

   3. Taylor expanded in 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 32.8

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

   5. Applied rewrites 32.8%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 78.5%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -5.5 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, i \cdot \left(t \cdot b\right)\right)\\ \mathbf{elif}\;j \leq 1.7 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(c, -b, x \cdot y\right), t \cdot \mathsf{fma}\left(i, b, a \cdot \left(-x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, i \cdot \left(t \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 29.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -21000000:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-67}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-224}:\\ \;\;\;\;c \cdot \left(z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-290}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+92}:\\ \;\;\;\;z \cdot \left(b \cdot \left(-c\right)\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+206}:\\ \;\;\;\;t \cdot \left(a \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -21000000.0)
   (* y (* x z))
   (if (<= x -2.45e-67)
     (* j (* a c))
     (if (<= x -3e-224)
       (* c (* z (- b)))
       (if (<= x -6.6e-290)
         (* b (* t i))
         (if (<= x 1.4e+92)
           (* z (* b (- c)))
           (if (<= x 5.4e+206) (* t (* a (- x))) (* z (* x y)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -21000000.0) {
		tmp = y * (x * z);
	} else if (x <= -2.45e-67) {
		tmp = j * (a * c);
	} else if (x <= -3e-224) {
		tmp = c * (z * -b);
	} else if (x <= -6.6e-290) {
		tmp = b * (t * i);
	} else if (x <= 1.4e+92) {
		tmp = z * (b * -c);
	} else if (x <= 5.4e+206) {
		tmp = t * (a * -x);
	} else {
		tmp = z * (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (x <= (-21000000.0d0)) then
        tmp = y * (x * z)
    else if (x <= (-2.45d-67)) then
        tmp = j * (a * c)
    else if (x <= (-3d-224)) then
        tmp = c * (z * -b)
    else if (x <= (-6.6d-290)) then
        tmp = b * (t * i)
    else if (x <= 1.4d+92) then
        tmp = z * (b * -c)
    else if (x <= 5.4d+206) then
        tmp = t * (a * -x)
    else
        tmp = z * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -21000000.0) {
		tmp = y * (x * z);
	} else if (x <= -2.45e-67) {
		tmp = j * (a * c);
	} else if (x <= -3e-224) {
		tmp = c * (z * -b);
	} else if (x <= -6.6e-290) {
		tmp = b * (t * i);
	} else if (x <= 1.4e+92) {
		tmp = z * (b * -c);
	} else if (x <= 5.4e+206) {
		tmp = t * (a * -x);
	} else {
		tmp = z * (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if x <= -21000000.0:
		tmp = y * (x * z)
	elif x <= -2.45e-67:
		tmp = j * (a * c)
	elif x <= -3e-224:
		tmp = c * (z * -b)
	elif x <= -6.6e-290:
		tmp = b * (t * i)
	elif x <= 1.4e+92:
		tmp = z * (b * -c)
	elif x <= 5.4e+206:
		tmp = t * (a * -x)
	else:
		tmp = z * (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -21000000.0)
		tmp = Float64(y * Float64(x * z));
	elseif (x <= -2.45e-67)
		tmp = Float64(j * Float64(a * c));
	elseif (x <= -3e-224)
		tmp = Float64(c * Float64(z * Float64(-b)));
	elseif (x <= -6.6e-290)
		tmp = Float64(b * Float64(t * i));
	elseif (x <= 1.4e+92)
		tmp = Float64(z * Float64(b * Float64(-c)));
	elseif (x <= 5.4e+206)
		tmp = Float64(t * Float64(a * Float64(-x)));
	else
		tmp = Float64(z * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (x <= -21000000.0)
		tmp = y * (x * z);
	elseif (x <= -2.45e-67)
		tmp = j * (a * c);
	elseif (x <= -3e-224)
		tmp = c * (z * -b);
	elseif (x <= -6.6e-290)
		tmp = b * (t * i);
	elseif (x <= 1.4e+92)
		tmp = z * (b * -c);
	elseif (x <= 5.4e+206)
		tmp = t * (a * -x);
	else
		tmp = z * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -21000000.0], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.45e-67], N[(j * N[(a * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3e-224], N[(c * N[(z * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.6e-290], N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e+92], N[(z * N[(b * (-c)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.4e+206], N[(t * N[(a * (-x)), $MachinePrecision]), $MachinePrecision], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 7 regimes

2. if x < -2.1e7

   1. Initial program 78.6%

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

   3. Taylor expanded in 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 35.3

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

   5. Applied rewrites 35.3%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 65.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 54.0%

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

   if -2.1e7 < x < -2.44999999999999997e-67

   1. Initial program 94.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 36.7

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 42.1%

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

   if -2.44999999999999997e-67 < x < -2.99999999999999982e-224

   1. Initial program 72.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 47.3%

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

   if -2.99999999999999982e-224 < x < -6.59999999999999972e-290

   1. Initial program 92.3%

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

   3. Taylor expanded in 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 25.4

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

   5. Applied rewrites 25.4%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 69.9%

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

   if -6.59999999999999972e-290 < x < 1.4e92

   1. Initial program 69.1%

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

   3. Taylor expanded in 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 26.6

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

   5. Applied rewrites 26.6%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 70.6%

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

   9. Taylor expanded in z around inf

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. +-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. mul-1-neg N/A

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

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

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

       10. mul-1-neg N/A

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

       11. lower-*.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 48.3

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

   11. Applied rewrites 48.3%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 38.8%

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

   if 1.4e92 < x < 5.40000000000000007e206

   1. Initial program 78.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 53.5%

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

   if 5.40000000000000007e206 < x

   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 x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.6%

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -21000000:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-67}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-224}:\\ \;\;\;\;c \cdot \left(z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-290}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+92}:\\ \;\;\;\;z \cdot \left(b \cdot \left(-c\right)\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+206}:\\ \;\;\;\;t \cdot \left(a \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(z \cdot \left(-b\right)\right)\\ \mathbf{if}\;x \leq -21000000:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-67}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-290}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+206}:\\ \;\;\;\;t \cdot \left(a \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* z (- b)))))
   (if (<= x -21000000.0)
     (* y (* x z))
     (if (<= x -2.45e-67)
       (* j (* a c))
       (if (<= x -3e-224)
         t_1
         (if (<= x -6.6e-290)
           (* b (* t i))
           (if (<= x 1.35e+92)
             t_1
             (if (<= x 5.4e+206) (* t (* a (- x))) (* z (* x y))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (z * -b);
	double tmp;
	if (x <= -21000000.0) {
		tmp = y * (x * z);
	} else if (x <= -2.45e-67) {
		tmp = j * (a * c);
	} else if (x <= -3e-224) {
		tmp = t_1;
	} else if (x <= -6.6e-290) {
		tmp = b * (t * i);
	} else if (x <= 1.35e+92) {
		tmp = t_1;
	} else if (x <= 5.4e+206) {
		tmp = t * (a * -x);
	} else {
		tmp = z * (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (z * -b)
    if (x <= (-21000000.0d0)) then
        tmp = y * (x * z)
    else if (x <= (-2.45d-67)) then
        tmp = j * (a * c)
    else if (x <= (-3d-224)) then
        tmp = t_1
    else if (x <= (-6.6d-290)) then
        tmp = b * (t * i)
    else if (x <= 1.35d+92) then
        tmp = t_1
    else if (x <= 5.4d+206) then
        tmp = t * (a * -x)
    else
        tmp = z * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (z * -b);
	double tmp;
	if (x <= -21000000.0) {
		tmp = y * (x * z);
	} else if (x <= -2.45e-67) {
		tmp = j * (a * c);
	} else if (x <= -3e-224) {
		tmp = t_1;
	} else if (x <= -6.6e-290) {
		tmp = b * (t * i);
	} else if (x <= 1.35e+92) {
		tmp = t_1;
	} else if (x <= 5.4e+206) {
		tmp = t * (a * -x);
	} else {
		tmp = z * (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (z * -b)
	tmp = 0
	if x <= -21000000.0:
		tmp = y * (x * z)
	elif x <= -2.45e-67:
		tmp = j * (a * c)
	elif x <= -3e-224:
		tmp = t_1
	elif x <= -6.6e-290:
		tmp = b * (t * i)
	elif x <= 1.35e+92:
		tmp = t_1
	elif x <= 5.4e+206:
		tmp = t * (a * -x)
	else:
		tmp = z * (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(z * Float64(-b)))
	tmp = 0.0
	if (x <= -21000000.0)
		tmp = Float64(y * Float64(x * z));
	elseif (x <= -2.45e-67)
		tmp = Float64(j * Float64(a * c));
	elseif (x <= -3e-224)
		tmp = t_1;
	elseif (x <= -6.6e-290)
		tmp = Float64(b * Float64(t * i));
	elseif (x <= 1.35e+92)
		tmp = t_1;
	elseif (x <= 5.4e+206)
		tmp = Float64(t * Float64(a * Float64(-x)));
	else
		tmp = Float64(z * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (z * -b);
	tmp = 0.0;
	if (x <= -21000000.0)
		tmp = y * (x * z);
	elseif (x <= -2.45e-67)
		tmp = j * (a * c);
	elseif (x <= -3e-224)
		tmp = t_1;
	elseif (x <= -6.6e-290)
		tmp = b * (t * i);
	elseif (x <= 1.35e+92)
		tmp = t_1;
	elseif (x <= 5.4e+206)
		tmp = t * (a * -x);
	else
		tmp = z * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(z * (-b)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -21000000.0], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.45e-67], N[(j * N[(a * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3e-224], t$95$1, If[LessEqual[x, -6.6e-290], N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+92], t$95$1, If[LessEqual[x, 5.4e+206], N[(t * N[(a * (-x)), $MachinePrecision]), $MachinePrecision], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -2.1e7

   1. Initial program 78.6%

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

   3. Taylor expanded in 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 35.3

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

   5. Applied rewrites 35.3%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 65.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 54.0%

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

   if -2.1e7 < x < -2.44999999999999997e-67

   1. Initial program 94.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 36.7

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 42.1%

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

   if -2.44999999999999997e-67 < x < -2.99999999999999982e-224 or -6.59999999999999972e-290 < x < 1.35e92

   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 c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 41.4%

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

   if -2.99999999999999982e-224 < x < -6.59999999999999972e-290

   1. Initial program 92.3%

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

   3. Taylor expanded in 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 25.4

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

   5. Applied rewrites 25.4%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 69.9%

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

   if 1.35e92 < x < 5.40000000000000007e206

   1. Initial program 78.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 53.5%

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

   if 5.40000000000000007e206 < x

   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 x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.6%

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -21000000:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-67}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-224}:\\ \;\;\;\;c \cdot \left(z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-290}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+92}:\\ \;\;\;\;c \cdot \left(z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+206}:\\ \;\;\;\;t \cdot \left(a \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \mathsf{fma}\left(c, -b, x \cdot y\right), b \cdot \left(t \cdot i\right)\right)\\ \mathbf{if}\;z \leq -7 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-261}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(i, b, a \cdot \left(-x\right)\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, i \cdot \left(t \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma z (fma c (- b) (* x y)) (* b (* t i)))))
   (if (<= z -7e-41)
     t_1
     (if (<= z -1.3e-261)
       (* t (fma i b (* a (- x))))
       (if (<= z 3.8e-16) (fma (- (* a c) (* y i)) j (* 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 = fma(z, fma(c, -b, (x * y)), (b * (t * i)));
	double tmp;
	if (z <= -7e-41) {
		tmp = t_1;
	} else if (z <= -1.3e-261) {
		tmp = t * fma(i, b, (a * -x));
	} else if (z <= 3.8e-16) {
		tmp = fma(((a * c) - (y * i)), j, (i * (t * b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(z, fma(c, Float64(-b), Float64(x * y)), Float64(b * Float64(t * i)))
	tmp = 0.0
	if (z <= -7e-41)
		tmp = t_1;
	elseif (z <= -1.3e-261)
		tmp = Float64(t * fma(i, b, Float64(a * Float64(-x))));
	elseif (z <= 3.8e-16)
		tmp = fma(Float64(Float64(a * c) - Float64(y * i)), j, Float64(i * 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[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7e-41], t$95$1, If[LessEqual[z, -1.3e-261], N[(t * N[(i * b + N[(a * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e-16], N[(N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j + N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.9999999999999999e-41 or 3.80000000000000012e-16 < z

   1. Initial program 68.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 29.1

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 78.0%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 74.2%

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

   if -6.9999999999999999e-41 < z < -1.3000000000000001e-261

   1. Initial program 84.7%

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

   3. Taylor expanded in 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 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   7. 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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. cancel-sign-sub N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 66.0

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

   8. Applied rewrites 66.0%

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

   if -1.3000000000000001e-261 < z < 3.80000000000000012e-16

   1. Initial program 86.4%

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

   3. Taylor expanded in 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 67.6

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

   5. Applied rewrites 67.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 69.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.1

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

   7. Applied rewrites 69.1%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(c, -b, x \cdot y\right), b \cdot \left(t \cdot i\right)\right)\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-261}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(i, b, a \cdot \left(-x\right)\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, i \cdot \left(t \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(c, -b, x \cdot y\right), b \cdot \left(t \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -21000000:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-201}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+73}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+206}:\\ \;\;\;\;t \cdot \left(a \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -21000000.0)
   (* y (* x z))
   (if (<= x -2.2e-201)
     (* j (* a c))
     (if (<= x 3.2e+73)
       (* i (* t b))
       (if (<= x 5.4e+206) (* t (* a (- x))) (* z (* x y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -21000000.0) {
		tmp = y * (x * z);
	} else if (x <= -2.2e-201) {
		tmp = j * (a * c);
	} else if (x <= 3.2e+73) {
		tmp = i * (t * b);
	} else if (x <= 5.4e+206) {
		tmp = t * (a * -x);
	} else {
		tmp = z * (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (x <= (-21000000.0d0)) then
        tmp = y * (x * z)
    else if (x <= (-2.2d-201)) then
        tmp = j * (a * c)
    else if (x <= 3.2d+73) then
        tmp = i * (t * b)
    else if (x <= 5.4d+206) then
        tmp = t * (a * -x)
    else
        tmp = z * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -21000000.0) {
		tmp = y * (x * z);
	} else if (x <= -2.2e-201) {
		tmp = j * (a * c);
	} else if (x <= 3.2e+73) {
		tmp = i * (t * b);
	} else if (x <= 5.4e+206) {
		tmp = t * (a * -x);
	} else {
		tmp = z * (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if x <= -21000000.0:
		tmp = y * (x * z)
	elif x <= -2.2e-201:
		tmp = j * (a * c)
	elif x <= 3.2e+73:
		tmp = i * (t * b)
	elif x <= 5.4e+206:
		tmp = t * (a * -x)
	else:
		tmp = z * (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -21000000.0)
		tmp = Float64(y * Float64(x * z));
	elseif (x <= -2.2e-201)
		tmp = Float64(j * Float64(a * c));
	elseif (x <= 3.2e+73)
		tmp = Float64(i * Float64(t * b));
	elseif (x <= 5.4e+206)
		tmp = Float64(t * Float64(a * Float64(-x)));
	else
		tmp = Float64(z * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (x <= -21000000.0)
		tmp = y * (x * z);
	elseif (x <= -2.2e-201)
		tmp = j * (a * c);
	elseif (x <= 3.2e+73)
		tmp = i * (t * b);
	elseif (x <= 5.4e+206)
		tmp = t * (a * -x);
	else
		tmp = z * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -21000000.0], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.2e-201], N[(j * N[(a * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e+73], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.4e+206], N[(t * N[(a * (-x)), $MachinePrecision]), $MachinePrecision], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.1e7

   1. Initial program 78.6%

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

   3. Taylor expanded in 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 35.3

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

   5. Applied rewrites 35.3%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 65.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 54.0%

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

   if -2.1e7 < x < -2.2e-201

   1. Initial program 78.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 37.2%

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

   if -2.2e-201 < x < 3.19999999999999982e73

   1. Initial program 75.2%

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

   3. Taylor expanded in 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 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 36.8%

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

   if 3.19999999999999982e73 < x < 5.40000000000000007e206

   1. Initial program 72.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 49.8%

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

   if 5.40000000000000007e206 < x

   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 x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.6%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -21000000:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-201}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+73}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+206}:\\ \;\;\;\;t \cdot \left(a \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\\ \mathbf{if}\;x \leq -1.62 \cdot 10^{-28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-201}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, a \cdot j\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, t \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (fma t (- a) (* y z)))))
   (if (<= x -1.62e-28)
     t_1
     (if (<= x -2.6e-201)
       (* c (fma b (- z) (* a j)))
       (if (<= x 1.25e+34) (* b (fma c (- z) (* t i))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * fma(t, -a, (y * z));
	double tmp;
	if (x <= -1.62e-28) {
		tmp = t_1;
	} else if (x <= -2.6e-201) {
		tmp = c * fma(b, -z, (a * j));
	} else if (x <= 1.25e+34) {
		tmp = b * fma(c, -z, (t * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(x * fma(t, Float64(-a), Float64(y * z)))
	tmp = 0.0
	if (x <= -1.62e-28)
		tmp = t_1;
	elseif (x <= -2.6e-201)
		tmp = Float64(c * fma(b, Float64(-z), Float64(a * j)));
	elseif (x <= 1.25e+34)
		tmp = Float64(b * fma(c, Float64(-z), Float64(t * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(t * (-a) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.62e-28], t$95$1, If[LessEqual[x, -2.6e-201], N[(c * N[(b * (-z) + N[(a * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e+34], N[(b * N[(c * (-z) + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.62e-28 or 1.25e34 < x

   1. Initial program 77.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if -1.62e-28 < x < -2.59999999999999982e-201

   1. Initial program 73.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   if -2.59999999999999982e-201 < x < 1.25e34

   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 24.5

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

   5. Applied rewrites 24.5%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   7. 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. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 60.1

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

   8. Applied rewrites 60.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.62 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-201}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, a \cdot j\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, t \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, -a, y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+22}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-186}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+60}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, a \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(i, b, a \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -2.3e+22)
   (* t (fma a (- x) (* b i)))
   (if (<= t 2.4e-186)
     (* j (- (* a c) (* y i)))
     (if (<= t 1.7e+60)
       (* c (fma b (- z) (* a j)))
       (* t (fma i b (* a (- x))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= -2.3e+22) {
		tmp = t * fma(a, -x, (b * i));
	} else if (t <= 2.4e-186) {
		tmp = j * ((a * c) - (y * i));
	} else if (t <= 1.7e+60) {
		tmp = c * fma(b, -z, (a * j));
	} else {
		tmp = t * fma(i, b, (a * -x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -2.3e+22)
		tmp = Float64(t * fma(a, Float64(-x), Float64(b * i)));
	elseif (t <= 2.4e-186)
		tmp = Float64(j * Float64(Float64(a * c) - Float64(y * i)));
	elseif (t <= 1.7e+60)
		tmp = Float64(c * fma(b, Float64(-z), Float64(a * j)));
	else
		tmp = Float64(t * fma(i, b, Float64(a * Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -2.3e+22], N[(t * N[(a * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e-186], N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.7e+60], N[(c * N[(b * (-z) + N[(a * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(i * b + N[(a * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.3000000000000002e22

   1. Initial program 74.6%

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

   3. Taylor expanded in t around 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 64.3

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

   5. Applied rewrites 64.3%

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

   if -2.3000000000000002e22 < t < 2.40000000000000003e-186

   1. Initial program 80.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 51.5

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

   5. Applied rewrites 51.5%

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

   if 2.40000000000000003e-186 < t < 1.7e60

   1. Initial program 72.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 59.5

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

   5. Applied rewrites 59.5%

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

   if 1.7e60 < t

   1. Initial program 71.2%

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

   3. Taylor expanded in 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 46.6

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

   5. Applied rewrites 46.6%

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

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   7. 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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. cancel-sign-sub N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 77.8

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

   8. Applied rewrites 77.8%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+22}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-186}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+60}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, a \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(i, b, a \cdot \left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-186}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+60}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, a \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* t (fma a (- x) (* b i)))))
   (if (<= t -2.3e+22)
     t_1
     (if (<= t 2.4e-186)
       (* j (- (* a c) (* y i)))
       (if (<= t 1.7e+60) (* c (fma b (- z) (* a 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 = t * fma(a, -x, (b * i));
	double tmp;
	if (t <= -2.3e+22) {
		tmp = t_1;
	} else if (t <= 2.4e-186) {
		tmp = j * ((a * c) - (y * i));
	} else if (t <= 1.7e+60) {
		tmp = c * fma(b, -z, (a * j));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(t * fma(a, Float64(-x), Float64(b * i)))
	tmp = 0.0
	if (t <= -2.3e+22)
		tmp = t_1;
	elseif (t <= 2.4e-186)
		tmp = Float64(j * Float64(Float64(a * c) - Float64(y * i)));
	elseif (t <= 1.7e+60)
		tmp = Float64(c * fma(b, Float64(-z), Float64(a * j)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(t * N[(a * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.3e+22], t$95$1, If[LessEqual[t, 2.4e-186], N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.7e+60], N[(c * N[(b * (-z) + N[(a * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.3000000000000002e22 or 1.7e60 < t

   1. Initial program 72.9%

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

   3. Taylor expanded in 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 71.1

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

   5. Applied rewrites 71.1%

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

   if -2.3000000000000002e22 < t < 2.40000000000000003e-186

   1. Initial program 80.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 51.5

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

   5. Applied rewrites 51.5%

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

   if 2.40000000000000003e-186 < t < 1.7e60

   1. Initial program 72.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 59.5

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

   5. Applied rewrites 59.5%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+22}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-186}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+60}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, a \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(c, -z, t \cdot i\right)\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-275}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+14}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma c (- z) (* t i)))))
   (if (<= b -5.8e-30)
     t_1
     (if (<= b 4.5e-275)
       (* a (fma j c (* x (- t))))
       (if (<= b 1.2e+14) (* j (- (* a c) (* y i))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(c, -z, (t * i));
	double tmp;
	if (b <= -5.8e-30) {
		tmp = t_1;
	} else if (b <= 4.5e-275) {
		tmp = a * fma(j, c, (x * -t));
	} else if (b <= 1.2e+14) {
		tmp = j * ((a * c) - (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(c, Float64(-z), Float64(t * i)))
	tmp = 0.0
	if (b <= -5.8e-30)
		tmp = t_1;
	elseif (b <= 4.5e-275)
		tmp = Float64(a * fma(j, c, Float64(x * Float64(-t))));
	elseif (b <= 1.2e+14)
		tmp = Float64(j * Float64(Float64(a * c) - Float64(y * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(c * (-z) + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.8e-30], t$95$1, If[LessEqual[b, 4.5e-275], N[(a * N[(j * c + N[(x * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.2e+14], N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.79999999999999978e-30 or 1.2e14 < b

   1. Initial program 77.3%

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

   3. Taylor expanded in 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 26.5

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

   5. Applied rewrites 26.5%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   7. 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. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 66.5

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

   8. Applied rewrites 66.5%

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

   if -5.79999999999999978e-30 < b < 4.49999999999999978e-275

   1. Initial program 73.2%

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

   3. Taylor expanded in 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 56.3

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

   5. Applied rewrites 56.3%

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

   if 4.49999999999999978e-275 < b < 1.2e14

   1. Initial program 75.9%

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

   3. Taylor expanded in 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 49.4

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

   5. Applied rewrites 49.4%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-30}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, t \cdot i\right)\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-275}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+14}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, t \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{if}\;a \leq -2.5 \cdot 10^{-110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -1.9 \cdot 10^{-285}:\\ \;\;\;\;c \cdot \left(z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;a \leq 0.00018:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (fma j c (* x (- t))))))
   (if (<= a -2.5e-110)
     t_1
     (if (<= a -1.9e-285)
       (* c (* z (- b)))
       (if (<= a 0.00018) (* y (* x z)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * fma(j, c, (x * -t));
	double tmp;
	if (a <= -2.5e-110) {
		tmp = t_1;
	} else if (a <= -1.9e-285) {
		tmp = c * (z * -b);
	} else if (a <= 0.00018) {
		tmp = y * (x * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * fma(j, c, Float64(x * Float64(-t))))
	tmp = 0.0
	if (a <= -2.5e-110)
		tmp = t_1;
	elseif (a <= -1.9e-285)
		tmp = Float64(c * Float64(z * Float64(-b)));
	elseif (a <= 0.00018)
		tmp = Float64(y * Float64(x * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(j * c + N[(x * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.5e-110], t$95$1, If[LessEqual[a, -1.9e-285], N[(c * N[(z * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.00018], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.5e-110 or 1.80000000000000011e-4 < a

   1. Initial program 70.8%

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

   3. Taylor expanded in 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 53.6

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

   5. Applied rewrites 53.6%

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

   if -2.5e-110 < a < -1.9000000000000001e-285

   1. Initial program 86.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 47.1%

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

   if -1.9000000000000001e-285 < a < 1.80000000000000011e-4

   1. Initial program 83.3%

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

   3. Taylor expanded in 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 10.7

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

   5. Applied rewrites 10.7%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 66.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 38.3%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{-110}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{elif}\;a \leq -1.9 \cdot 10^{-285}:\\ \;\;\;\;c \cdot \left(z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;a \leq 0.00018:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(c, -z, t \cdot i\right)\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-100}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma c (- z) (* t i)))))
   (if (<= b -5.8e-30)
     t_1
     (if (<= b 1.26e-100) (* 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 * fma(c, -z, (t * i));
	double tmp;
	if (b <= -5.8e-30) {
		tmp = t_1;
	} else if (b <= 1.26e-100) {
		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 * fma(c, Float64(-z), Float64(t * i)))
	tmp = 0.0
	if (b <= -5.8e-30)
		tmp = t_1;
	elseif (b <= 1.26e-100)
		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[(c * (-z) + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.8e-30], t$95$1, If[LessEqual[b, 1.26e-100], N[(a * N[(j * c + N[(x * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.79999999999999978e-30 or 1.2599999999999999e-100 < b

   1. Initial program 75.1%

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

   3. Taylor expanded in 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 27.3

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

   5. Applied rewrites 27.3%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   7. 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. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if -5.79999999999999978e-30 < b < 1.2599999999999999e-100

   1. Initial program 76.8%

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

   3. Taylor expanded in 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 50.6

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

   5. Applied rewrites 50.6%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-30}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, t \cdot i\right)\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-100}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, t \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-100}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* t i) (* z c)))))
   (if (<= b -5.8e-30)
     t_1
     (if (<= b 1.26e-100) (* 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 <= -5.8e-30) {
		tmp = t_1;
	} else if (b <= 1.26e-100) {
		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 <= -5.8e-30)
		tmp = t_1;
	elseif (b <= 1.26e-100)
		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, -5.8e-30], t$95$1, If[LessEqual[b, 1.26e-100], N[(a * N[(j * c + N[(x * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.79999999999999978e-30 or 1.2599999999999999e-100 < b

   1. Initial program 75.1%

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

   3. Taylor expanded in 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 62.2

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

   5. Applied rewrites 62.2%

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

   if -5.79999999999999978e-30 < b < 1.2599999999999999e-100

   1. Initial program 76.8%

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

   3. Taylor expanded in 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 50.6

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

   5. Applied rewrites 50.6%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-30}:\\ \;\;\;\;b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-100}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(t \cdot i - z \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot z\right)\\ \mathbf{if}\;x \leq -21000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-201}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+33}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* y (* x z))))
   (if (<= x -21000000.0)
     t_1
     (if (<= x -2.2e-201)
       (* j (* a c))
       (if (<= x 6.4e+33) (* 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 = y * (x * z);
	double tmp;
	if (x <= -21000000.0) {
		tmp = t_1;
	} else if (x <= -2.2e-201) {
		tmp = j * (a * c);
	} else if (x <= 6.4e+33) {
		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 = y * (x * z)
    if (x <= (-21000000.0d0)) then
        tmp = t_1
    else if (x <= (-2.2d-201)) then
        tmp = j * (a * c)
    else if (x <= 6.4d+33) 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 = y * (x * z);
	double tmp;
	if (x <= -21000000.0) {
		tmp = t_1;
	} else if (x <= -2.2e-201) {
		tmp = j * (a * c);
	} else if (x <= 6.4e+33) {
		tmp = i * (t * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = y * (x * z)
	tmp = 0
	if x <= -21000000.0:
		tmp = t_1
	elif x <= -2.2e-201:
		tmp = j * (a * c)
	elif x <= 6.4e+33:
		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(y * Float64(x * z))
	tmp = 0.0
	if (x <= -21000000.0)
		tmp = t_1;
	elseif (x <= -2.2e-201)
		tmp = Float64(j * Float64(a * c));
	elseif (x <= 6.4e+33)
		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 = y * (x * z);
	tmp = 0.0;
	if (x <= -21000000.0)
		tmp = t_1;
	elseif (x <= -2.2e-201)
		tmp = j * (a * c);
	elseif (x <= 6.4e+33)
		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[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -21000000.0], t$95$1, If[LessEqual[x, -2.2e-201], N[(j * N[(a * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.4e+33], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1e7 or 6.40000000000000034e33 < x

   1. Initial program 75.6%

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

   3. Taylor expanded in 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 42.6

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

   5. Applied rewrites 42.6%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 49.5%

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

   if -2.1e7 < x < -2.2e-201

   1. Initial program 78.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 37.2%

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

   if -2.2e-201 < x < 6.40000000000000034e33

   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 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 48.4

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

   5. Applied rewrites 48.4%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 37.7%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -21000000:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-201}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+33}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot z\right)\\ \mathbf{if}\;x \leq -21000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-201}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+33}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* y (* x z))))
   (if (<= x -21000000.0)
     t_1
     (if (<= x -2.8e-201)
       (* j (* a c))
       (if (<= x 6.4e+33) (* b (* t i)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = y * (x * z);
	double tmp;
	if (x <= -21000000.0) {
		tmp = t_1;
	} else if (x <= -2.8e-201) {
		tmp = j * (a * c);
	} else if (x <= 6.4e+33) {
		tmp = b * (t * i);
	} 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 = y * (x * z)
    if (x <= (-21000000.0d0)) then
        tmp = t_1
    else if (x <= (-2.8d-201)) then
        tmp = j * (a * c)
    else if (x <= 6.4d+33) then
        tmp = b * (t * i)
    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 = y * (x * z);
	double tmp;
	if (x <= -21000000.0) {
		tmp = t_1;
	} else if (x <= -2.8e-201) {
		tmp = j * (a * c);
	} else if (x <= 6.4e+33) {
		tmp = b * (t * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = y * (x * z)
	tmp = 0
	if x <= -21000000.0:
		tmp = t_1
	elif x <= -2.8e-201:
		tmp = j * (a * c)
	elif x <= 6.4e+33:
		tmp = b * (t * i)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(y * Float64(x * z))
	tmp = 0.0
	if (x <= -21000000.0)
		tmp = t_1;
	elseif (x <= -2.8e-201)
		tmp = Float64(j * Float64(a * c));
	elseif (x <= 6.4e+33)
		tmp = Float64(b * Float64(t * i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = y * (x * z);
	tmp = 0.0;
	if (x <= -21000000.0)
		tmp = t_1;
	elseif (x <= -2.8e-201)
		tmp = j * (a * c);
	elseif (x <= 6.4e+33)
		tmp = b * (t * i);
	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[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -21000000.0], t$95$1, If[LessEqual[x, -2.8e-201], N[(j * N[(a * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.4e+33], N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1e7 or 6.40000000000000034e33 < x

   1. Initial program 75.6%

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

   3. Taylor expanded in 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 42.6

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

   5. Applied rewrites 42.6%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 49.5%

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

   if -2.1e7 < x < -2.7999999999999999e-201

   1. Initial program 78.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 37.2%

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

   if -2.7999999999999999e-201 < x < 6.40000000000000034e33

   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 24.5

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

   5. Applied rewrites 24.5%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 69.2%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 35.7%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -21000000:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-201}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+33}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 29.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -21000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-201}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+33}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (* x y))))
   (if (<= x -21000000.0)
     t_1
     (if (<= x -2.8e-201)
       (* j (* a c))
       (if (<= x 6.4e+33) (* b (* t i)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (x <= -21000000.0) {
		tmp = t_1;
	} else if (x <= -2.8e-201) {
		tmp = j * (a * c);
	} else if (x <= 6.4e+33) {
		tmp = b * (t * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (x * y)
    if (x <= (-21000000.0d0)) then
        tmp = t_1
    else if (x <= (-2.8d-201)) then
        tmp = j * (a * c)
    else if (x <= 6.4d+33) then
        tmp = b * (t * i)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (x <= -21000000.0) {
		tmp = t_1;
	} else if (x <= -2.8e-201) {
		tmp = j * (a * c);
	} else if (x <= 6.4e+33) {
		tmp = b * (t * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = z * (x * y)
	tmp = 0
	if x <= -21000000.0:
		tmp = t_1
	elif x <= -2.8e-201:
		tmp = j * (a * c)
	elif x <= 6.4e+33:
		tmp = b * (t * i)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * Float64(x * y))
	tmp = 0.0
	if (x <= -21000000.0)
		tmp = t_1;
	elseif (x <= -2.8e-201)
		tmp = Float64(j * Float64(a * c));
	elseif (x <= 6.4e+33)
		tmp = Float64(b * Float64(t * i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = z * (x * y);
	tmp = 0.0;
	if (x <= -21000000.0)
		tmp = t_1;
	elseif (x <= -2.8e-201)
		tmp = j * (a * c);
	elseif (x <= 6.4e+33)
		tmp = b * (t * i);
	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[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -21000000.0], t$95$1, If[LessEqual[x, -2.8e-201], N[(j * N[(a * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.4e+33], N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1e7 or 6.40000000000000034e33 < x

   1. Initial program 75.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 44.3%

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

   if -2.1e7 < x < -2.7999999999999999e-201

   1. Initial program 78.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 37.2%

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

   if -2.7999999999999999e-201 < x < 6.40000000000000034e33

   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 24.5

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

   5. Applied rewrites 24.5%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 69.2%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 35.7%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -21000000:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-201}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+33}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 28.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -21000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{+205}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (* x y))))
   (if (<= x -21000000.0) t_1 (if (<= x 6.3e+205) (* j (* a 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 = z * (x * y);
	double tmp;
	if (x <= -21000000.0) {
		tmp = t_1;
	} else if (x <= 6.3e+205) {
		tmp = j * (a * 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 = z * (x * y)
    if (x <= (-21000000.0d0)) then
        tmp = t_1
    else if (x <= 6.3d+205) then
        tmp = j * (a * 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 = z * (x * y);
	double tmp;
	if (x <= -21000000.0) {
		tmp = t_1;
	} else if (x <= 6.3e+205) {
		tmp = j * (a * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = z * (x * y)
	tmp = 0
	if x <= -21000000.0:
		tmp = t_1
	elif x <= 6.3e+205:
		tmp = j * (a * c)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * Float64(x * y))
	tmp = 0.0
	if (x <= -21000000.0)
		tmp = t_1;
	elseif (x <= 6.3e+205)
		tmp = Float64(j * Float64(a * 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 = z * (x * y);
	tmp = 0.0;
	if (x <= -21000000.0)
		tmp = t_1;
	elseif (x <= 6.3e+205)
		tmp = j * (a * 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[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -21000000.0], t$95$1, If[LessEqual[x, 6.3e+205], N[(j * N[(a * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e7 or 6.30000000000000014e205 < x

   1. Initial program 76.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 52.4%

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

   if -2.1e7 < x < 6.30000000000000014e205

   1. Initial program 75.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 27.2%

      \[\leadsto j \cdot \color{blue}{\left(a \cdot c\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 22.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-168}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t 1.1e-168) (* j (* a c)) (* c (* a j))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= 1.1e-168) {
		tmp = j * (a * c);
	} else {
		tmp = c * (a * j);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (t <= 1.1d-168) then
        tmp = j * (a * c)
    else
        tmp = c * (a * j)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= 1.1e-168) {
		tmp = j * (a * c);
	} else {
		tmp = c * (a * j);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if t <= 1.1e-168:
		tmp = j * (a * c)
	else:
		tmp = c * (a * j)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= 1.1e-168)
		tmp = Float64(j * Float64(a * c));
	else
		tmp = Float64(c * Float64(a * j));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (t <= 1.1e-168)
		tmp = j * (a * c);
	else
		tmp = c * (a * j);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, 1.1e-168], N[(j * N[(a * c), $MachinePrecision]), $MachinePrecision], N[(c * N[(a * j), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.0999999999999999e-168

   1. Initial program 79.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 43.5

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

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 25.4%

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

   if 1.0999999999999999e-168 < t

   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 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 40.4

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 22.9%

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 22.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(a \cdot j\right) \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(c * N[(a * j), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.9%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
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 37.5

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

5. Applied rewrites 37.5%

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

6. Taylor expanded in j around inf

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

8. Applied rewrites 22.6%

   \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
9. Add Preprocessing

Developer Target 1: 59.3% 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 2024233 
(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)))))