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

Percentage Accurate: 73.4% → 83.4%

Time: 20.2s

Alternatives: 20

Speedup: 0.9×

• 57.8% 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: 73.4% accurate, 1.0× speedup

Math

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

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: 83.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.6%

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

   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 x around 0

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

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

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 61.5%

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

4. Final simplification 86.2%

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

Alternative 2: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, y, c \cdot a\right)\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, j, \mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+234}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, j, \mathsf{fma}\left(\mathsf{fma}\left(-a, x, i \cdot b\right), t, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- i) y (* c a))))
   (if (<= b -3.5e+158)
     (fma t_1 j (* (fma (- c) z (* i t)) b))
     (if (<= b 1.3e+234)
       (fma t_1 j (fma (fma (- a) x (* i b)) t (* (fma (- b) c (* y x)) z)))
       (* (fma (- z) c (* i t)) b)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-i, y, (c * a));
	double tmp;
	if (b <= -3.5e+158) {
		tmp = fma(t_1, j, (fma(-c, z, (i * t)) * b));
	} else if (b <= 1.3e+234) {
		tmp = fma(t_1, j, fma(fma(-a, x, (i * b)), t, (fma(-b, c, (y * x)) * z)));
	} else {
		tmp = fma(-z, c, (i * t)) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-i), y, Float64(c * a))
	tmp = 0.0
	if (b <= -3.5e+158)
		tmp = fma(t_1, j, Float64(fma(Float64(-c), z, Float64(i * t)) * b));
	elseif (b <= 1.3e+234)
		tmp = fma(t_1, j, fma(fma(Float64(-a), x, Float64(i * b)), t, Float64(fma(Float64(-b), c, Float64(y * x)) * z)));
	else
		tmp = Float64(fma(Float64(-z), c, Float64(i * t)) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.5e+158], N[(t$95$1 * j + N[(N[((-c) * z + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3e+234], N[(t$95$1 * j + N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5000000000000001e158

   1. Initial program 78.0%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.9%

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

   if -3.5000000000000001e158 < b < 1.30000000000000008e234

   1. Initial program 77.4%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 84.5%

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

   if 1.30000000000000008e234 < b

   1. Initial program 64.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+234}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-a, x, i \cdot b\right), t, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, \mathsf{fma}\left(c, a, \frac{\mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b}{j}\right)\right) \cdot j\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -1.55e+23)
   (* (fma (- i) y (fma c a (/ (* (fma (- c) z (* i t)) b) j))) j)
   (if (<= c 7.2e+15)
     (fma (fma (- y) j (* b t)) i (* (fma (- a) t (* z y)) x))
     (fma (fma (- t) x (* j c)) a (* (fma (- b) c (* y x)) z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -1.55e+23) {
		tmp = fma(-i, y, fma(c, a, ((fma(-c, z, (i * t)) * b) / j))) * j;
	} else if (c <= 7.2e+15) {
		tmp = fma(fma(-y, j, (b * t)), i, (fma(-a, t, (z * y)) * x));
	} else {
		tmp = fma(fma(-t, x, (j * c)), a, (fma(-b, c, (y * x)) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -1.55e+23)
		tmp = Float64(fma(Float64(-i), y, fma(c, a, Float64(Float64(fma(Float64(-c), z, Float64(i * t)) * b) / j))) * j);
	elseif (c <= 7.2e+15)
		tmp = fma(fma(Float64(-y), j, Float64(b * t)), i, Float64(fma(Float64(-a), t, Float64(z * y)) * x));
	else
		tmp = fma(fma(Float64(-t), x, Float64(j * c)), a, Float64(fma(Float64(-b), c, Float64(y * x)) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -1.55e+23], N[(N[((-i) * y + N[(c * a + N[(N[(N[((-c) * z + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[c, 7.2e+15], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[((-t) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.54999999999999985e23

   1. Initial program 59.7%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 69.7%

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

   if -1.54999999999999985e23 < c < 7.2e15

   1. Initial program 86.0%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. mul-1-neg N/A

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

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

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

      5. fp-cancel-sign-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-in N/A

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

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

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

   5. Applied rewrites 77.2%

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

   if 7.2e15 < c

   1. Initial program 67.4%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 82.1%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, \mathsf{fma}\left(c, a, \frac{\mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b}{j}\right)\right) \cdot j\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -1.25 \cdot 10^{-46} \lor \neg \left(j \leq 2.6 \cdot 10^{-83}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= j -1.25e-46) (not (<= j 2.6e-83)))
   (fma (fma (- i) y (* c a)) j (* (fma (- c) z (* i t)) b))
   (fma (fma (- t) x (* j c)) a (* (fma (- b) c (* y x)) z))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((j <= -1.25e-46) || !(j <= 2.6e-83)) {
		tmp = fma(fma(-i, y, (c * a)), j, (fma(-c, z, (i * t)) * b));
	} else {
		tmp = fma(fma(-t, x, (j * c)), a, (fma(-b, c, (y * x)) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((j <= -1.25e-46) || !(j <= 2.6e-83))
		tmp = fma(fma(Float64(-i), y, Float64(c * a)), j, Float64(fma(Float64(-c), z, Float64(i * t)) * b));
	else
		tmp = fma(fma(Float64(-t), x, Float64(j * c)), a, Float64(fma(Float64(-b), c, Float64(y * x)) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[j, -1.25e-46], N[Not[LessEqual[j, 2.6e-83]], $MachinePrecision]], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-c) * z + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[((-t) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if j < -1.24999999999999998e-46 or 2.60000000000000009e-83 < j

   1. Initial program 78.0%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.9%

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

   if -1.24999999999999998e-46 < j < 2.60000000000000009e-83

   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 t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 70.0%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.25 \cdot 10^{-46} \lor \neg \left(j \leq 2.6 \cdot 10^{-83}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -6.2 \cdot 10^{+72} \lor \neg \left(i \leq 3.9 \cdot 10^{+124}\right):\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= i -6.2e+72) (not (<= i 3.9e+124)))
   (* (fma (- y) j (* b t)) i)
   (fma (fma (- t) x (* j c)) a (* (fma (- b) c (* y x)) z))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((i <= -6.2e+72) || !(i <= 3.9e+124)) {
		tmp = fma(-y, j, (b * t)) * i;
	} else {
		tmp = fma(fma(-t, x, (j * c)), a, (fma(-b, c, (y * x)) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((i <= -6.2e+72) || !(i <= 3.9e+124))
		tmp = Float64(fma(Float64(-y), j, Float64(b * t)) * i);
	else
		tmp = fma(fma(Float64(-t), x, Float64(j * c)), a, Float64(fma(Float64(-b), c, Float64(y * x)) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[i, -6.2e+72], N[Not[LessEqual[i, 3.9e+124]], $MachinePrecision]], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], N[(N[((-t) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -6.19999999999999977e72 or 3.9e124 < i

   1. Initial program 71.4%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -6.19999999999999977e72 < i < 3.9e124

   1. Initial program 79.5%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 68.6%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6.2 \cdot 10^{+72} \lor \neg \left(i \leq 3.9 \cdot 10^{+124}\right):\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b\right)\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -1.55e+23)
   (fma (fma (- i) y (* c a)) j (* (fma (- c) z (* i t)) b))
   (if (<= c 7.2e+15)
     (fma (fma (- y) j (* b t)) i (* (fma (- a) t (* z y)) x))
     (fma (fma (- t) x (* j c)) a (* (fma (- b) c (* y x)) z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -1.55e+23) {
		tmp = fma(fma(-i, y, (c * a)), j, (fma(-c, z, (i * t)) * b));
	} else if (c <= 7.2e+15) {
		tmp = fma(fma(-y, j, (b * t)), i, (fma(-a, t, (z * y)) * x));
	} else {
		tmp = fma(fma(-t, x, (j * c)), a, (fma(-b, c, (y * x)) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -1.55e+23)
		tmp = fma(fma(Float64(-i), y, Float64(c * a)), j, Float64(fma(Float64(-c), z, Float64(i * t)) * b));
	elseif (c <= 7.2e+15)
		tmp = fma(fma(Float64(-y), j, Float64(b * t)), i, Float64(fma(Float64(-a), t, Float64(z * y)) * x));
	else
		tmp = fma(fma(Float64(-t), x, Float64(j * c)), a, Float64(fma(Float64(-b), c, Float64(y * x)) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -1.55e+23], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-c) * z + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.2e+15], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[((-t) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.54999999999999985e23

   1. Initial program 59.7%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.6%

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

   if -1.54999999999999985e23 < c < 7.2e15

   1. Initial program 86.0%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. mul-1-neg N/A

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

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

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

      5. fp-cancel-sign-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-in N/A

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

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

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

   5. Applied rewrites 77.2%

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

   if 7.2e15 < c

   1. Initial program 67.4%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 82.1%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b\right)\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot t\right), b, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-80}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot t, i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- c) z (* i t)) b (* (* j c) a))))
   (if (<= b -1.1e+97)
     t_1
     (if (<= b 2.5e-80)
       (+ (* (* z x) y) (* j (- (* c a) (* y i))))
       (if (<= b 1.75e+58) (fma (* b t) i (* (fma (- a) t (* z y)) 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(fma(-c, z, (i * t)), b, ((j * c) * a));
	double tmp;
	if (b <= -1.1e+97) {
		tmp = t_1;
	} else if (b <= 2.5e-80) {
		tmp = ((z * x) * y) + (j * ((c * a) - (y * i)));
	} else if (b <= 1.75e+58) {
		tmp = fma((b * t), i, (fma(-a, t, (z * y)) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-c), z, Float64(i * t)), b, Float64(Float64(j * c) * a))
	tmp = 0.0
	if (b <= -1.1e+97)
		tmp = t_1;
	elseif (b <= 2.5e-80)
		tmp = Float64(Float64(Float64(z * x) * y) + Float64(j * Float64(Float64(c * a) - Float64(y * i))));
	elseif (b <= 1.75e+58)
		tmp = fma(Float64(b * t), i, Float64(fma(Float64(-a), t, Float64(z * y)) * 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[((-c) * z + N[(i * t), $MachinePrecision]), $MachinePrecision] * b + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.1e+97], t$95$1, If[LessEqual[b, 2.5e-80], N[(N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.75e+58], N[(N[(b * t), $MachinePrecision] * i + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1e97 or 1.7499999999999999e58 < b

   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 x around 0

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

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

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 72.7%

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

   if -1.1e97 < b < 2.5e-80

   1. Initial program 76.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if 2.5e-80 < b < 1.7499999999999999e58

   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 c around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. mul-1-neg N/A

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

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

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

      5. fp-cancel-sign-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-in N/A

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

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

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 72.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot t\right), b, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-80}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot t, i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot t\right), b, \left(j \cdot c\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot t\right), b, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{if}\;b \leq -3.8 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, j \cdot c\right), a, \left(y \cdot x\right) \cdot z\right)\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot t, i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- c) z (* i t)) b (* (* j c) a))))
   (if (<= b -3.8e+53)
     t_1
     (if (<= b 7.2e-79)
       (fma (fma (- x) t (* j c)) a (* (* y x) z))
       (if (<= b 1.75e+58) (fma (* b t) i (* (fma (- a) t (* z y)) 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(fma(-c, z, (i * t)), b, ((j * c) * a));
	double tmp;
	if (b <= -3.8e+53) {
		tmp = t_1;
	} else if (b <= 7.2e-79) {
		tmp = fma(fma(-x, t, (j * c)), a, ((y * x) * z));
	} else if (b <= 1.75e+58) {
		tmp = fma((b * t), i, (fma(-a, t, (z * y)) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-c), z, Float64(i * t)), b, Float64(Float64(j * c) * a))
	tmp = 0.0
	if (b <= -3.8e+53)
		tmp = t_1;
	elseif (b <= 7.2e-79)
		tmp = fma(fma(Float64(-x), t, Float64(j * c)), a, Float64(Float64(y * x) * z));
	elseif (b <= 1.75e+58)
		tmp = fma(Float64(b * t), i, Float64(fma(Float64(-a), t, Float64(z * y)) * 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[((-c) * z + N[(i * t), $MachinePrecision]), $MachinePrecision] * b + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.8e+53], t$95$1, If[LessEqual[b, 7.2e-79], N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.75e+58], N[(N[(b * t), $MachinePrecision] * i + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.79999999999999997e53 or 1.7499999999999999e58 < b

   1. Initial program 76.1%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 70.4%

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

   if -3.79999999999999997e53 < b < 7.2000000000000005e-79

   1. Initial program 75.8%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 64.9%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 61.2%

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

   if 7.2000000000000005e-79 < b < 1.7499999999999999e58

   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 c around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. mul-1-neg N/A

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

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

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

      5. fp-cancel-sign-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-in N/A

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

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

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 72.9%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot t\right), b, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, j \cdot c\right), a, \left(y \cdot x\right) \cdot z\right)\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot t, i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot t\right), b, \left(j \cdot c\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -6.5 \cdot 10^{+156} \lor \neg \left(j \leq 10^{-27} \lor \neg \left(j \leq 5.9 \cdot 10^{+161} \lor \neg \left(j \leq 2.4 \cdot 10^{+205}\right)\right)\right):\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= j -6.5e+156)
         (not
          (or (<= j 1e-27) (not (or (<= j 5.9e+161) (not (<= j 2.4e+205)))))))
   (* (* (- j) y) i)
   (* (fma (- a) x (* i b)) t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((j <= -6.5e+156) || !((j <= 1e-27) || !((j <= 5.9e+161) || !(j <= 2.4e+205)))) {
		tmp = (-j * y) * i;
	} else {
		tmp = fma(-a, x, (i * b)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((j <= -6.5e+156) || !((j <= 1e-27) || !((j <= 5.9e+161) || !(j <= 2.4e+205))))
		tmp = Float64(Float64(Float64(-j) * y) * i);
	else
		tmp = Float64(fma(Float64(-a), x, Float64(i * b)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[j, -6.5e+156], N[Not[Or[LessEqual[j, 1e-27], N[Not[Or[LessEqual[j, 5.9e+161], N[Not[LessEqual[j, 2.4e+205]], $MachinePrecision]]], $MachinePrecision]]], $MachinePrecision]], N[(N[((-j) * y), $MachinePrecision] * i), $MachinePrecision], N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if j < -6.50000000000000027e156 or 1e-27 < j < 5.9000000000000003e161 or 2.39999999999999986e205 < j

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 58.0%

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

   if -6.50000000000000027e156 < j < 1e-27 or 5.9000000000000003e161 < j < 2.39999999999999986e205

   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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 51.8

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

   5. Applied rewrites 51.8%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -6.5 \cdot 10^{+156} \lor \neg \left(j \leq 10^{-27} \lor \neg \left(j \leq 5.9 \cdot 10^{+161} \lor \neg \left(j \leq 2.4 \cdot 10^{+205}\right)\right)\right):\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -1.06 \cdot 10^{+203}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot t\right), b, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+240}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- t) a (* z y)) x)))
   (if (<= x -1.06e+203)
     t_1
     (if (<= x 8.2e-50)
       (fma (fma (- c) z (* i t)) b (* (* j c) a))
       (if (<= x 6.6e+173)
         (* (fma (- j) i (* z x)) y)
         (if (<= x 9e+240) t_1 (* (fma (- b) c (* y x)) z)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-t, a, (z * y)) * x;
	double tmp;
	if (x <= -1.06e+203) {
		tmp = t_1;
	} else if (x <= 8.2e-50) {
		tmp = fma(fma(-c, z, (i * t)), b, ((j * c) * a));
	} else if (x <= 6.6e+173) {
		tmp = fma(-j, i, (z * x)) * y;
	} else if (x <= 9e+240) {
		tmp = t_1;
	} else {
		tmp = fma(-b, c, (y * x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-t), a, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -1.06e+203)
		tmp = t_1;
	elseif (x <= 8.2e-50)
		tmp = fma(fma(Float64(-c), z, Float64(i * t)), b, Float64(Float64(j * c) * a));
	elseif (x <= 6.6e+173)
		tmp = Float64(fma(Float64(-j), i, Float64(z * x)) * y);
	elseif (x <= 9e+240)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.06e+203], t$95$1, If[LessEqual[x, 8.2e-50], N[(N[((-c) * z + N[(i * t), $MachinePrecision]), $MachinePrecision] * b + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e+173], N[(N[((-j) * i + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 9e+240], t$95$1, N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.05999999999999994e203 or 6.59999999999999993e173 < x < 8.99999999999999957e240

   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 t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 90.1

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

   8. Applied rewrites 90.1%

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

   if -1.05999999999999994e203 < x < 8.19999999999999971e-50

   1. Initial program 72.5%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 60.3%

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

   if 8.19999999999999971e-50 < x < 6.59999999999999993e173

   1. Initial program 90.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

   if 8.99999999999999957e240 < x

   1. Initial program 84.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06 \cdot 10^{+203}:\\ \;\;\;\;\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot t\right), b, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+240}:\\ \;\;\;\;\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+75}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;t \leq -1.18 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(-a\right) \cdot x\right) \cdot t\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-177}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+96}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -5e+75)
   (* (* i t) b)
   (if (<= t -1.18e-21)
     (* (* (- a) x) t)
     (if (<= t -9e-177)
       (* (* z y) x)
       (if (<= t 9e+96) (* (* (- j) y) i) (* (* (- t) x) a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= -5e+75) {
		tmp = (i * t) * b;
	} else if (t <= -1.18e-21) {
		tmp = (-a * x) * t;
	} else if (t <= -9e-177) {
		tmp = (z * y) * x;
	} else if (t <= 9e+96) {
		tmp = (-j * y) * i;
	} else {
		tmp = (-t * x) * a;
	}
	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 <= (-5d+75)) then
        tmp = (i * t) * b
    else if (t <= (-1.18d-21)) then
        tmp = (-a * x) * t
    else if (t <= (-9d-177)) then
        tmp = (z * y) * x
    else if (t <= 9d+96) then
        tmp = (-j * y) * i
    else
        tmp = (-t * x) * a
    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 <= -5e+75) {
		tmp = (i * t) * b;
	} else if (t <= -1.18e-21) {
		tmp = (-a * x) * t;
	} else if (t <= -9e-177) {
		tmp = (z * y) * x;
	} else if (t <= 9e+96) {
		tmp = (-j * y) * i;
	} else {
		tmp = (-t * x) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if t <= -5e+75:
		tmp = (i * t) * b
	elif t <= -1.18e-21:
		tmp = (-a * x) * t
	elif t <= -9e-177:
		tmp = (z * y) * x
	elif t <= 9e+96:
		tmp = (-j * y) * i
	else:
		tmp = (-t * x) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -5e+75)
		tmp = Float64(Float64(i * t) * b);
	elseif (t <= -1.18e-21)
		tmp = Float64(Float64(Float64(-a) * x) * t);
	elseif (t <= -9e-177)
		tmp = Float64(Float64(z * y) * x);
	elseif (t <= 9e+96)
		tmp = Float64(Float64(Float64(-j) * y) * i);
	else
		tmp = Float64(Float64(Float64(-t) * x) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (t <= -5e+75)
		tmp = (i * t) * b;
	elseif (t <= -1.18e-21)
		tmp = (-a * x) * t;
	elseif (t <= -9e-177)
		tmp = (z * y) * x;
	elseif (t <= 9e+96)
		tmp = (-j * y) * i;
	else
		tmp = (-t * x) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -5e+75], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[t, -1.18e-21], N[(N[((-a) * x), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t, -9e-177], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 9e+96], N[(N[((-j) * y), $MachinePrecision] * i), $MachinePrecision], N[(N[((-t) * x), $MachinePrecision] * a), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -5.0000000000000002e75

   1. Initial program 55.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 39.9%

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

   if -5.0000000000000002e75 < t < -1.18000000000000002e-21

   1. Initial program 84.3%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 38.9%

      \[\leadsto \left(\left(-a\right) \cdot x\right) \cdot t \]

   if -1.18000000000000002e-21 < t < -9.0000000000000007e-177

   1. Initial program 86.9%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 46.2

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

   8. Applied rewrites 46.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 38.4%

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

   if -9.0000000000000007e-177 < t < 8.99999999999999914e96

   1. Initial program 83.7%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 40.6%

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

   if 8.99999999999999914e96 < t

   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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 47.5%

      \[\leadsto \left(\left(-t\right) \cdot x\right) \cdot a \]
3. Recombined 5 regimes into one program.

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+75}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;t \leq -1.18 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(-a\right) \cdot x\right) \cdot t\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-177}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+96}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+220}:\\ \;\;\;\;\left(\left(-a\right) \cdot x\right) \cdot t\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{+68}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-129}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+203}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -7.2e+220)
   (* (* (- a) x) t)
   (if (<= x -7.5e+68)
     (* (* z y) x)
     (if (<= x 2.6e-129)
       (* (* i b) t)
       (if (<= x 2.75e+203) (* (* (- j) y) i) (* (* 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 <= -7.2e+220) {
		tmp = (-a * x) * t;
	} else if (x <= -7.5e+68) {
		tmp = (z * y) * x;
	} else if (x <= 2.6e-129) {
		tmp = (i * b) * t;
	} else if (x <= 2.75e+203) {
		tmp = (-j * y) * i;
	} 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 <= (-7.2d+220)) then
        tmp = (-a * x) * t
    else if (x <= (-7.5d+68)) then
        tmp = (z * y) * x
    else if (x <= 2.6d-129) then
        tmp = (i * b) * t
    else if (x <= 2.75d+203) then
        tmp = (-j * y) * i
    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 <= -7.2e+220) {
		tmp = (-a * x) * t;
	} else if (x <= -7.5e+68) {
		tmp = (z * y) * x;
	} else if (x <= 2.6e-129) {
		tmp = (i * b) * t;
	} else if (x <= 2.75e+203) {
		tmp = (-j * y) * i;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if x <= -7.2e+220:
		tmp = (-a * x) * t
	elif x <= -7.5e+68:
		tmp = (z * y) * x
	elif x <= 2.6e-129:
		tmp = (i * b) * t
	elif x <= 2.75e+203:
		tmp = (-j * y) * i
	else:
		tmp = (z * x) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -7.2e+220)
		tmp = Float64(Float64(Float64(-a) * x) * t);
	elseif (x <= -7.5e+68)
		tmp = Float64(Float64(z * y) * x);
	elseif (x <= 2.6e-129)
		tmp = Float64(Float64(i * b) * t);
	elseif (x <= 2.75e+203)
		tmp = Float64(Float64(Float64(-j) * y) * i);
	else
		tmp = Float64(Float64(z * 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 <= -7.2e+220)
		tmp = (-a * x) * t;
	elseif (x <= -7.5e+68)
		tmp = (z * y) * x;
	elseif (x <= 2.6e-129)
		tmp = (i * b) * t;
	elseif (x <= 2.75e+203)
		tmp = (-j * y) * i;
	else
		tmp = (z * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -7.2e+220], N[(N[((-a) * x), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[x, -7.5e+68], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 2.6e-129], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[x, 2.75e+203], N[(N[((-j) * y), $MachinePrecision] * i), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -7.20000000000000038e220

   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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 83.8%

      \[\leadsto \left(\left(-a\right) \cdot x\right) \cdot t \]

   if -7.20000000000000038e220 < x < -7.49999999999999959e68

   1. Initial program 73.5%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 52.1

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

   8. Applied rewrites 52.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 37.5%

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

   if -7.49999999999999959e68 < x < 2.6000000000000001e-129

   1. Initial program 70.3%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 42.7

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 34.5%

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

   if 2.6000000000000001e-129 < x < 2.75000000000000014e203

   1. Initial program 84.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 35.6%

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

   if 2.75000000000000014e203 < x

   1. Initial program 72.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 51.4%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+220}:\\ \;\;\;\;\left(\left(-a\right) \cdot x\right) \cdot t\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{+68}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-129}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+203}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-218}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- j) i (* z x)) y)))
   (if (<= y -2.2e+80)
     t_1
     (if (<= y -7.8e-218)
       (* (fma (- a) x (* i b)) t)
       (if (<= y 3.8e-27) (* (fma (- z) c (* 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(-j, i, (z * x)) * y;
	double tmp;
	if (y <= -2.2e+80) {
		tmp = t_1;
	} else if (y <= -7.8e-218) {
		tmp = fma(-a, x, (i * b)) * t;
	} else if (y <= 3.8e-27) {
		tmp = fma(-z, c, (i * t)) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-j), i, Float64(z * x)) * y)
	tmp = 0.0
	if (y <= -2.2e+80)
		tmp = t_1;
	elseif (y <= -7.8e-218)
		tmp = Float64(fma(Float64(-a), x, Float64(i * b)) * t);
	elseif (y <= 3.8e-27)
		tmp = Float64(fma(Float64(-z), c, Float64(i * 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[(N[((-j) * i + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.2e+80], t$95$1, If[LessEqual[y, -7.8e-218], N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[y, 3.8e-27], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.20000000000000003e80 or 3.8e-27 < y

   1. Initial program 71.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if -2.20000000000000003e80 < y < -7.8e-218

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 57.7

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

   5. Applied rewrites 57.7%

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

   if -7.8e-218 < y < 3.8e-27

   1. Initial program 88.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + i \cdot t\right) \cdot b \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, i \cdot t\right)} \cdot b \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, i \cdot t\right) \cdot b \]

      11. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, i \cdot t\right) \cdot b \]

      12. lower-*.f64 61.8

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot t}\right) \cdot b \]

   5. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-218}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -5 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;j \leq 2.3 \cdot 10^{-290}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{elif}\;j \leq 9.5 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -5e-37)
   (* (fma (- i) y (* c a)) j)
   (if (<= j 2.3e-290)
     (* (fma (- a) x (* i b)) t)
     (if (<= j 9.5e-83)
       (* (fma (- b) c (* y x)) z)
       (* (fma (- y) j (* b t)) i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -5e-37) {
		tmp = fma(-i, y, (c * a)) * j;
	} else if (j <= 2.3e-290) {
		tmp = fma(-a, x, (i * b)) * t;
	} else if (j <= 9.5e-83) {
		tmp = fma(-b, c, (y * x)) * z;
	} else {
		tmp = fma(-y, j, (b * t)) * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -5e-37)
		tmp = Float64(fma(Float64(-i), y, Float64(c * a)) * j);
	elseif (j <= 2.3e-290)
		tmp = Float64(fma(Float64(-a), x, Float64(i * b)) * t);
	elseif (j <= 9.5e-83)
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	else
		tmp = Float64(fma(Float64(-y), j, Float64(b * t)) * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -5e-37], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[j, 2.3e-290], N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[j, 9.5e-83], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if j < -4.9999999999999997e-37

   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 t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(i\right), y, a \cdot c\right)}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

      14. distribute-rgt-out-- N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)}\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\mathsf{fma}\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right), t, z \cdot \left(x \cdot y - b \cdot c\right)\right)}\right) \]

   5. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-a, x, i \cdot b\right), t, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j \]

      3. mul-1-neg N/A

         \[\leadsto \left(a \cdot c + \color{blue}{\left(-1 \cdot i\right)} \cdot y\right) \cdot j \]

      4. associate-*r* N/A

         \[\leadsto \left(a \cdot c + \color{blue}{-1 \cdot \left(i \cdot y\right)}\right) \cdot j \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot y\right) + a \cdot c\right)} \cdot j \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot y\right) + a \cdot c\right) \cdot j} \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right) \cdot y} + a \cdot c\right) \cdot j \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)} \cdot j \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right) \cdot j \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right) \cdot j \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j \]

      12. lower-*.f64 60.3

          \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j \]

   8. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j} \]

   if -4.9999999999999997e-37 < j < 2.3000000000000001e-290

   1. Initial program 71.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 61.0

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 61.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   if 2.3000000000000001e-290 < j < 9.50000000000000051e-83

   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 z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b\right)\right) \cdot c\right)} \cdot z \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)}\right) \cdot z \]

      5. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)} + x \cdot y\right) \cdot z \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot c} + x \cdot y\right) \cdot z \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(b\right), c, x \cdot y\right)} \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 59.6

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]

   if 9.50000000000000051e-83 < j

   1. Initial program 76.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) \cdot i} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \cdot i \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \cdot i \]

      7. *-lft-identity N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{b \cdot t}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, b \cdot t\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, b \cdot t\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, b \cdot t\right) \cdot i \]

      11. lower-*.f64 60.8

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot t}\right) \cdot i \]

   5. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -5 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;j \leq 2.3 \cdot 10^{-290}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{elif}\;j \leq 9.5 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+134} \lor \neg \left(z \leq 2.2 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -3.5e+134) (not (<= z 2.2e-17)))
   (* (fma (- b) c (* y x)) z)
   (* (fma (- a) x (* i b)) t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((z <= -3.5e+134) || !(z <= 2.2e-17)) {
		tmp = fma(-b, c, (y * x)) * z;
	} else {
		tmp = fma(-a, x, (i * b)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -3.5e+134) || !(z <= 2.2e-17))
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	else
		tmp = Float64(fma(Float64(-a), x, Float64(i * b)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -3.5e+134], N[Not[LessEqual[z, 2.2e-17]], $MachinePrecision]], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.50000000000000003e134 or 2.2e-17 < z

   1. Initial program 70.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b\right)\right) \cdot c\right)} \cdot z \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)}\right) \cdot z \]

      5. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)} + x \cdot y\right) \cdot z \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot c} + x \cdot y\right) \cdot z \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(b\right), c, x \cdot y\right)} \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 67.1

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]

   if -3.50000000000000003e134 < z < 2.2e-17

   1. Initial program 81.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 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. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 52.8

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+134} \lor \neg \left(z \leq 2.2 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -3 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;j \leq 1.1 \cdot 10^{-291}:\\ \;\;\;\;\left(\left(-a\right) \cdot x\right) \cdot t\\ \mathbf{elif}\;j \leq 10^{-27}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -3e-38)
   (* (* (- i) j) y)
   (if (<= j 1.1e-291)
     (* (* (- a) x) t)
     (if (<= j 1e-27) (* (* z x) y) (* (* (- j) y) i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -3e-38) {
		tmp = (-i * j) * y;
	} else if (j <= 1.1e-291) {
		tmp = (-a * x) * t;
	} else if (j <= 1e-27) {
		tmp = (z * x) * y;
	} else {
		tmp = (-j * y) * i;
	}
	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 (j <= (-3d-38)) then
        tmp = (-i * j) * y
    else if (j <= 1.1d-291) then
        tmp = (-a * x) * t
    else if (j <= 1d-27) then
        tmp = (z * x) * y
    else
        tmp = (-j * y) * i
    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 (j <= -3e-38) {
		tmp = (-i * j) * y;
	} else if (j <= 1.1e-291) {
		tmp = (-a * x) * t;
	} else if (j <= 1e-27) {
		tmp = (z * x) * y;
	} else {
		tmp = (-j * y) * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if j <= -3e-38:
		tmp = (-i * j) * y
	elif j <= 1.1e-291:
		tmp = (-a * x) * t
	elif j <= 1e-27:
		tmp = (z * x) * y
	else:
		tmp = (-j * y) * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -3e-38)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	elseif (j <= 1.1e-291)
		tmp = Float64(Float64(Float64(-a) * x) * t);
	elseif (j <= 1e-27)
		tmp = Float64(Float64(z * x) * y);
	else
		tmp = Float64(Float64(Float64(-j) * y) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (j <= -3e-38)
		tmp = (-i * j) * y;
	elseif (j <= 1.1e-291)
		tmp = (-a * x) * t;
	elseif (j <= 1e-27)
		tmp = (z * x) * y;
	else
		tmp = (-j * y) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -3e-38], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[j, 1.1e-291], N[(N[((-a) * x), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[j, 1e-27], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], N[(N[((-j) * y), $MachinePrecision] * i), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if j < -2.99999999999999989e-38

   1. Initial program 79.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) + x \cdot z\right) \cdot y \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot i} + x \cdot z\right) \cdot y \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(j\right), i, x \cdot z\right)} \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 48.5

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(i \cdot j\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 40.3%

      \[\leadsto \left(\left(-i\right) \cdot j\right) \cdot y \]

   if -2.99999999999999989e-38 < j < 1.10000000000000001e-291

   1. Initial program 70.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 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. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 59.9

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot \left(a \cdot x\right)\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 34.5%

      \[\leadsto \left(\left(-a\right) \cdot x\right) \cdot t \]

   if 1.10000000000000001e-291 < j < 1e-27

   1. Initial program 80.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) + x \cdot z\right) \cdot y \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot i} + x \cdot z\right) \cdot y \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(j\right), i, x \cdot z\right)} \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 42.1

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 33.0%

      \[\leadsto \left(z \cdot x\right) \cdot y \]

   if 1e-27 < j

   1. Initial program 77.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) - b \cdot \left(c \cdot z - i \cdot t\right)} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right) + j \cdot \left(a \cdot c - i \cdot y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(b\right), c \cdot z - i \cdot t, j \cdot \left(a \cdot c - i \cdot y\right)\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c \cdot z - i \cdot t, j \cdot \left(a \cdot c - i \cdot y\right)\right) \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(-b, \color{blue}{c \cdot z + \left(\mathsf{neg}\left(i\right)\right) \cdot t}, j \cdot \left(a \cdot c - i \cdot y\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-b, \color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot t + c \cdot z}, j \cdot \left(a \cdot c - i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-b, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(i\right), t, c \cdot z\right)}, j \cdot \left(a \cdot c - i \cdot y\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(\color{blue}{-i}, t, c \cdot z\right), j \cdot \left(a \cdot c - i \cdot y\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, \color{blue}{c \cdot z}\right), j \cdot \left(a \cdot c - i \cdot y\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j}\right) \]

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \color{blue}{\left(a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \color{blue}{\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c\right)} \cdot j\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(i\right), y, a \cdot c\right)} \cdot j\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right) \cdot j\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j\right) \]

      17. lower-*.f64 74.6

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j\right) \]

   5. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\left(i \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.0%

      \[\leadsto \left(\left(-j\right) \cdot y\right) \cdot \color{blue}{i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -3 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;j \leq 1.1 \cdot 10^{-291}:\\ \;\;\;\;\left(\left(-a\right) \cdot x\right) \cdot t\\ \mathbf{elif}\;j \leq 10^{-27}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 29.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{+99}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;t \leq -9.6 \cdot 10^{-170}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+98}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -2.2e+99)
   (* (* i t) b)
   (if (<= t -9.6e-170)
     (* (* z x) y)
     (if (<= t 1.7e+98) (* (* (- j) y) i) (* (* i b) t)))))

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 <= -2.2e+99) {
		tmp = (i * t) * b;
	} else if (t <= -9.6e-170) {
		tmp = (z * x) * y;
	} else if (t <= 1.7e+98) {
		tmp = (-j * y) * i;
	} else {
		tmp = (i * b) * t;
	}
	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 <= (-2.2d+99)) then
        tmp = (i * t) * b
    else if (t <= (-9.6d-170)) then
        tmp = (z * x) * y
    else if (t <= 1.7d+98) then
        tmp = (-j * y) * i
    else
        tmp = (i * b) * t
    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 <= -2.2e+99) {
		tmp = (i * t) * b;
	} else if (t <= -9.6e-170) {
		tmp = (z * x) * y;
	} else if (t <= 1.7e+98) {
		tmp = (-j * y) * i;
	} else {
		tmp = (i * b) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if t <= -2.2e+99:
		tmp = (i * t) * b
	elif t <= -9.6e-170:
		tmp = (z * x) * y
	elif t <= 1.7e+98:
		tmp = (-j * y) * i
	else:
		tmp = (i * b) * t
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -2.2e+99)
		tmp = Float64(Float64(i * t) * b);
	elseif (t <= -9.6e-170)
		tmp = Float64(Float64(z * x) * y);
	elseif (t <= 1.7e+98)
		tmp = Float64(Float64(Float64(-j) * y) * i);
	else
		tmp = Float64(Float64(i * b) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (t <= -2.2e+99)
		tmp = (i * t) * b;
	elseif (t <= -9.6e-170)
		tmp = (z * x) * y;
	elseif (t <= 1.7e+98)
		tmp = (-j * y) * i;
	else
		tmp = (i * b) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -2.2e+99], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[t, -9.6e-170], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t, 1.7e+98], N[(N[((-j) * y), $MachinePrecision] * i), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.19999999999999978e99

   1. Initial program 60.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot t - c \cdot z\right) \cdot b} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(i \cdot t - c \cdot z\right) \cdot b} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right)} \cdot b \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot z + i \cdot t\right)} \cdot b \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)} + i \cdot t\right) \cdot b \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + i \cdot t\right) \cdot b \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + i \cdot t\right) \cdot b \]

      8. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + i \cdot t\right) \cdot b \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, i \cdot t\right)} \cdot b \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, i \cdot t\right) \cdot b \]

      11. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, i \cdot t\right) \cdot b \]

      12. lower-*.f64 54.5

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot t}\right) \cdot b \]

   5. Applied rewrites 54.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(i \cdot t\right) \cdot b \]
   7. Step-by-step derivation

   8. Applied rewrites 41.1%

      \[\leadsto \left(i \cdot t\right) \cdot b \]

   if -2.19999999999999978e99 < t < -9.5999999999999998e-170

   1. Initial program 79.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) + x \cdot z\right) \cdot y \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot i} + x \cdot z\right) \cdot y \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(j\right), i, x \cdot z\right)} \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 40.4

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 40.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 29.0%

      \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -9.5999999999999998e-170 < t < 1.69999999999999986e98

   1. Initial program 83.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) - b \cdot \left(c \cdot z - i \cdot t\right)} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right) + j \cdot \left(a \cdot c - i \cdot y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(b\right), c \cdot z - i \cdot t, j \cdot \left(a \cdot c - i \cdot y\right)\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c \cdot z - i \cdot t, j \cdot \left(a \cdot c - i \cdot y\right)\right) \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(-b, \color{blue}{c \cdot z + \left(\mathsf{neg}\left(i\right)\right) \cdot t}, j \cdot \left(a \cdot c - i \cdot y\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-b, \color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot t + c \cdot z}, j \cdot \left(a \cdot c - i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-b, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(i\right), t, c \cdot z\right)}, j \cdot \left(a \cdot c - i \cdot y\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(\color{blue}{-i}, t, c \cdot z\right), j \cdot \left(a \cdot c - i \cdot y\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, \color{blue}{c \cdot z}\right), j \cdot \left(a \cdot c - i \cdot y\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j}\right) \]

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \color{blue}{\left(a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \color{blue}{\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c\right)} \cdot j\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(i\right), y, a \cdot c\right)} \cdot j\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right) \cdot j\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j\right) \]

      17. lower-*.f64 75.6

          \[\leadsto \mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j\right) \]

   5. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\left(i \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.9%

      \[\leadsto \left(\left(-j\right) \cdot y\right) \cdot \color{blue}{i} \]

   if 1.69999999999999986e98 < t

   1. Initial program 71.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 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. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 76.7

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 41.5%

      \[\leadsto \left(i \cdot b\right) \cdot t \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{+99}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;t \leq -9.6 \cdot 10^{-170}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+98}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+59}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;b \leq 78000000000:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -6e+59)
   (* (* i b) t)
   (if (<= b 78000000000.0) (* (* (- i) j) y) (* (* (- c) z) b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -6e+59) {
		tmp = (i * b) * t;
	} else if (b <= 78000000000.0) {
		tmp = (-i * j) * y;
	} else {
		tmp = (-c * z) * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (b <= (-6d+59)) then
        tmp = (i * b) * t
    else if (b <= 78000000000.0d0) then
        tmp = (-i * j) * y
    else
        tmp = (-c * z) * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -6e+59) {
		tmp = (i * b) * t;
	} else if (b <= 78000000000.0) {
		tmp = (-i * j) * y;
	} else {
		tmp = (-c * z) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -6e+59:
		tmp = (i * b) * t
	elif b <= 78000000000.0:
		tmp = (-i * j) * y
	else:
		tmp = (-c * z) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -6e+59)
		tmp = Float64(Float64(i * b) * t);
	elseif (b <= 78000000000.0)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	else
		tmp = Float64(Float64(Float64(-c) * z) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -6e+59)
		tmp = (i * b) * t;
	elseif (b <= 78000000000.0)
		tmp = (-i * j) * y;
	else
		tmp = (-c * z) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -6e+59], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[b, 78000000000.0], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.0000000000000001e59

   1. Initial program 77.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 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. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 60.6

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 48.2%

      \[\leadsto \left(i \cdot b\right) \cdot t \]

   if -6.0000000000000001e59 < b < 7.8e10

   1. Initial program 76.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) + x \cdot z\right) \cdot y \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot i} + x \cdot z\right) \cdot y \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(j\right), i, x \cdot z\right)} \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 52.1

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(i \cdot j\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 32.8%

      \[\leadsto \left(\left(-i\right) \cdot j\right) \cdot y \]

   if 7.8e10 < b

   1. Initial program 78.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot t - c \cdot z\right) \cdot b} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(i \cdot t - c \cdot z\right) \cdot b} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right)} \cdot b \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot z + i \cdot t\right)} \cdot b \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)} + i \cdot t\right) \cdot b \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + i \cdot t\right) \cdot b \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + i \cdot t\right) \cdot b \]

      8. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + i \cdot t\right) \cdot b \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, i \cdot t\right)} \cdot b \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, i \cdot t\right) \cdot b \]

      11. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, i \cdot t\right) \cdot b \]

      12. lower-*.f64 63.2

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot t}\right) \cdot b \]

   5. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot \left(c \cdot z\right)\right) \cdot b \]
   7. Step-by-step derivation

   8. Applied rewrites 43.1%

      \[\leadsto \left(\left(-c\right) \cdot z\right) \cdot b \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+59}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;b \leq 78000000000:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 30.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+17} \lor \neg \left(z \leq 2.2 \cdot 10^{-17}\right):\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -2.5e+17) (not (<= z 2.2e-17))) (* (* z x) y) (* (* i b) t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((z <= -2.5e+17) || !(z <= 2.2e-17)) {
		tmp = (z * x) * y;
	} else {
		tmp = (i * b) * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if ((z <= (-2.5d+17)) .or. (.not. (z <= 2.2d-17))) then
        tmp = (z * x) * y
    else
        tmp = (i * b) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((z <= -2.5e+17) || !(z <= 2.2e-17)) {
		tmp = (z * x) * y;
	} else {
		tmp = (i * b) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (z <= -2.5e+17) or not (z <= 2.2e-17):
		tmp = (z * x) * y
	else:
		tmp = (i * b) * t
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -2.5e+17) || !(z <= 2.2e-17))
		tmp = Float64(Float64(z * x) * y);
	else
		tmp = Float64(Float64(i * b) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((z <= -2.5e+17) || ~((z <= 2.2e-17)))
		tmp = (z * x) * y;
	else
		tmp = (i * b) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -2.5e+17], N[Not[LessEqual[z, 2.2e-17]], $MachinePrecision]], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5e17 or 2.2e-17 < z

   1. Initial program 72.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) + x \cdot z\right) \cdot y \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot i} + x \cdot z\right) \cdot y \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(j\right), i, x \cdot z\right)} \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 51.0

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 39.2%

      \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -2.5e17 < z < 2.2e-17

   1. Initial program 81.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 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. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 52.1

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 30.3%

      \[\leadsto \left(i \cdot b\right) \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+17} \lor \neg \left(z \leq 2.2 \cdot 10^{-17}\right):\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 22.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(i \cdot b\right) \cdot t \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* i b) t))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (i * b) * t;
}

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 = (i * b) * t
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 (i * b) * t;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (i * b) * t

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(i * b) * t)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (i * b) * t;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision]

Derivation

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 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. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

   4. metadata-eval N/A

      \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

   5. *-lft-identity N/A

      \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

   6. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

   7. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

   9. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   11. lower-*.f64 41.2

       \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

5. Applied rewrites 41.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(b \cdot i\right) \cdot t \]
7. Step-by-step derivation

8. Applied rewrites 23.5%

   \[\leadsto \left(i \cdot b\right) \cdot t \]

9. Final simplification 23.5%

   \[\leadsto \left(i \cdot b\right) \cdot t \]
10. Add Preprocessing

Developer Target 1: 58.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_2 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + t\_1\\ \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* c a) (* y i))))
        (t_2
         (+
          (-
           (* x (- (* y z) (* t a)))
           (/
            (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0)))
            (+ (* c z) (* t i))))
          t_1)))
   (if (< x -1.469694296777705e-64)
     t_2
     (if (< x 3.2113527362226803e-147)
       (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) t_1))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (pow((c * z), 2.0) - pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = j * ((c * a) - (y * i))
    t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ** 2.0d0) - ((t * i) ** 2.0d0))) / ((c * z) + (t * i)))) + t_1
    if (x < (-1.469694296777705d-64)) then
        tmp = t_2
    else if (x < 3.2113527362226803d-147) then
        tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (Math.pow((c * z), 2.0) - Math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((c * a) - (y * i))
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (math.pow((c * z), 2.0) - math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1
	tmp = 0
	if x < -1.469694296777705e-64:
		tmp = t_2
	elif x < 3.2113527362226803e-147:
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_2 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(b * Float64((Float64(c * z) ^ 2.0) - (Float64(t * i) ^ 2.0))) / Float64(Float64(c * z) + Float64(t * i)))) + t_1)
	tmp = 0.0
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = Float64(Float64(Float64(Float64(b * i) - Float64(x * a)) * t) - Float64(Float64(z * Float64(c * b)) - t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * ((c * a) - (y * i));
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ^ 2.0) - ((t * i) ^ 2.0))) / ((c * z) + (t * i)))) + t_1;
	tmp = 0.0;
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[Power[N[(c * z), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(t * i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Less[x, -1.469694296777705e-64], t$95$2, If[Less[x, 3.2113527362226803e-147], N[(N[(N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] - N[(N[(z * N[(c * b), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024338 
(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)))))