Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 74.1% → 83.5%

Time: 20.0s

Alternatives: 29

Speedup: 0.5×

• 58.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot i - z \cdot c\\ t_2 := j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot t\_1\right) + t\_2 \leq \infty:\\ \;\;\;\;t\_2 - \left(\frac{b}{\frac{-1}{t\_1}} + x \cdot \left(t \cdot a - y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\mathsf{fma}\left(y, x, \frac{a \cdot \left(b \cdot i\right)}{z}\right) - b \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (- (* a i) (* z c))) (t_2 (* j (- (* t c) (* y i)))))
   (if (<= (+ (+ (* x (- (* y z) (* t a))) (* b t_1)) t_2) INFINITY)
     (- t_2 (+ (/ b (/ -1.0 t_1)) (* x (- (* t a) (* y z)))))
     (* z (- (fma y x (/ (* a (* b i)) z)) (* b c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (a * i) - (z * c);
	double t_2 = j * ((t * c) - (y * i));
	double tmp;
	if ((((x * ((y * z) - (t * a))) + (b * t_1)) + t_2) <= ((double) INFINITY)) {
		tmp = t_2 - ((b / (-1.0 / t_1)) + (x * ((t * a) - (y * z))));
	} else {
		tmp = z * (fma(y, x, ((a * (b * i)) / z)) - (b * c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(a * i) - Float64(z * c))
	t_2 = Float64(j * Float64(Float64(t * c) - Float64(y * i)))
	tmp = 0.0
	if (Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) + Float64(b * t_1)) + t_2) <= Inf)
		tmp = Float64(t_2 - Float64(Float64(b / Float64(-1.0 / t_1)) + Float64(x * Float64(Float64(t * a) - Float64(y * z)))));
	else
		tmp = Float64(z * Float64(fma(y, x, Float64(Float64(a * Float64(b * i)) / z)) - Float64(b * c)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], Infinity], N[(t$95$2 - N[(N[(b / N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(t * a), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y * x + N[(N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 88.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 88.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 88.4

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

   4. Applied rewrites 88.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 71.5%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right) \leq \infty:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) - \left(\frac{b}{\frac{-1}{a \cdot i - z \cdot c}} + x \cdot \left(t \cdot a - y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\mathsf{fma}\left(y, x, \frac{a \cdot \left(b \cdot i\right)}{z}\right) - b \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 71.5%

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

4. Final simplification 85.5%

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

Alternative 3: 59.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, -c, x \cdot y\right)\\ \mathbf{if}\;i \leq -7.4 \cdot 10^{+83}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -4 \cdot 10^{-204}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, b \cdot \left(a \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 4.8 \cdot 10^{-136}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, -t \cdot \left(x \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma b (- c) (* x y))))
   (if (<= i -7.4e+83)
     (* i (fma j (- y) (* a b)))
     (if (<= i -4e-204)
       (fma z t_1 (* b (* a i)))
       (if (<= i 4.8e-136)
         (fma c (fma j t (* z (- b))) (* x (* y z)))
         (if (<= i 2.8e+78)
           (fma z t_1 (- (* t (* x a))))
           (* i (fma b a (* y (- 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 = fma(b, -c, (x * y));
	double tmp;
	if (i <= -7.4e+83) {
		tmp = i * fma(j, -y, (a * b));
	} else if (i <= -4e-204) {
		tmp = fma(z, t_1, (b * (a * i)));
	} else if (i <= 4.8e-136) {
		tmp = fma(c, fma(j, t, (z * -b)), (x * (y * z)));
	} else if (i <= 2.8e+78) {
		tmp = fma(z, t_1, -(t * (x * a)));
	} else {
		tmp = i * fma(b, a, (y * -j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(b, Float64(-c), Float64(x * y))
	tmp = 0.0
	if (i <= -7.4e+83)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (i <= -4e-204)
		tmp = fma(z, t_1, Float64(b * Float64(a * i)));
	elseif (i <= 4.8e-136)
		tmp = fma(c, fma(j, t, Float64(z * Float64(-b))), Float64(x * Float64(y * z)));
	elseif (i <= 2.8e+78)
		tmp = fma(z, t_1, Float64(-Float64(t * Float64(x * a))));
	else
		tmp = Float64(i * fma(b, a, Float64(y * Float64(-j))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * (-c) + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -7.4e+83], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -4e-204], N[(z * t$95$1 + N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 4.8e-136], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.8e+78], N[(z * t$95$1 + (-N[(t * N[(x * a), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if i < -7.4000000000000005e83

   1. Initial program 63.0%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -7.4000000000000005e83 < i < -4e-204

   1. Initial program 78.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 78.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 78.5

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 78.5

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in j around 0

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 71.8%

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

   if -4e-204 < i < 4.7999999999999997e-136

   1. Initial program 78.4%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 78.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 69.3%

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

   if 4.7999999999999997e-136 < i < 2.8000000000000001e78

   1. Initial program 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 78.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 78.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 78.2

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

   4. Applied rewrites 78.2%

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

   5. Taylor expanded in j around 0

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

   7. Applied rewrites 78.1%

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

   8. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(b, \mathsf{neg}\left(c\right), \color{blue}{y \cdot x}\right), -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 75.7%

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

   if 2.8000000000000001e78 < i

   1. Initial program 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 65.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 65.6

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 65.6

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

   4. Applied rewrites 65.6%

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

   5. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. cancel-sign-sub N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 75.6

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

   7. Applied rewrites 75.6%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.4 \cdot 10^{+83}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -4 \cdot 10^{-204}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), b \cdot \left(a \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 4.8 \cdot 10^{-136}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), -t \cdot \left(x \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), b \cdot \left(a \cdot i\right)\right)\\ t_2 := i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{if}\;i \leq -7.4 \cdot 10^{+83}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;i \leq -1.35 \cdot 10^{-166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 9.2 \cdot 10^{-273}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(-t \cdot a\right)\right)\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma z (fma b (- c) (* x y)) (* b (* a i))))
        (t_2 (* i (fma j (- y) (* a b)))))
   (if (<= i -7.4e+83)
     t_2
     (if (<= i -1.35e-166)
       t_1
       (if (<= i 9.2e-273)
         (fma c (fma j t (* z (- b))) (* x (- (* t a))))
         (if (<= i 8.6e+154) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(z, fma(b, -c, (x * y)), (b * (a * i)));
	double t_2 = i * fma(j, -y, (a * b));
	double tmp;
	if (i <= -7.4e+83) {
		tmp = t_2;
	} else if (i <= -1.35e-166) {
		tmp = t_1;
	} else if (i <= 9.2e-273) {
		tmp = fma(c, fma(j, t, (z * -b)), (x * -(t * a)));
	} else if (i <= 8.6e+154) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(z, fma(b, Float64(-c), Float64(x * y)), Float64(b * Float64(a * i)))
	t_2 = Float64(i * fma(j, Float64(-y), Float64(a * b)))
	tmp = 0.0
	if (i <= -7.4e+83)
		tmp = t_2;
	elseif (i <= -1.35e-166)
		tmp = t_1;
	elseif (i <= 9.2e-273)
		tmp = fma(c, fma(j, t, Float64(z * Float64(-b))), Float64(x * Float64(-Float64(t * a))));
	elseif (i <= 8.6e+154)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(b * (-c) + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -7.4e+83], t$95$2, If[LessEqual[i, -1.35e-166], t$95$1, If[LessEqual[i, 9.2e-273], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision] + N[(x * (-N[(t * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 8.6e+154], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -7.4000000000000005e83 or 8.5999999999999995e154 < i

   1. Initial program 61.8%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   if -7.4000000000000005e83 < i < -1.35000000000000003e-166 or 9.19999999999999923e-273 < i < 8.5999999999999995e154

   1. Initial program 79.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 79.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 79.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 79.3

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

   4. Applied rewrites 79.3%

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

   5. Taylor expanded in j around 0

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

   7. Applied rewrites 72.5%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 67.2%

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

   if -1.35000000000000003e-166 < i < 9.19999999999999923e-273

   1. Initial program 77.3%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 71.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.4 \cdot 10^{+83}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -1.35 \cdot 10^{-166}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), b \cdot \left(a \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 9.2 \cdot 10^{-273}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(-t \cdot a\right)\right)\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), b \cdot \left(a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, -c, x \cdot y\right)\\ t_2 := \mathsf{fma}\left(z, t\_1, b \cdot \left(a \cdot i\right)\right)\\ \mathbf{if}\;b \leq -29000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{-179}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_1, -t \cdot \left(x \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma b (- c) (* x y))) (t_2 (fma z t_1 (* b (* a i)))))
   (if (<= b -29000000000.0)
     t_2
     (if (<= b 2.45e-179)
       (+ (* j (- (* t c) (* y i))) (* z (* x y)))
       (if (<= b 2e-60) (fma z t_1 (- (* t (* x a)))) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(b, -c, (x * y));
	double t_2 = fma(z, t_1, (b * (a * i)));
	double tmp;
	if (b <= -29000000000.0) {
		tmp = t_2;
	} else if (b <= 2.45e-179) {
		tmp = (j * ((t * c) - (y * i))) + (z * (x * y));
	} else if (b <= 2e-60) {
		tmp = fma(z, t_1, -(t * (x * a)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(b, Float64(-c), Float64(x * y))
	t_2 = fma(z, t_1, Float64(b * Float64(a * i)))
	tmp = 0.0
	if (b <= -29000000000.0)
		tmp = t_2;
	elseif (b <= 2.45e-179)
		tmp = Float64(Float64(j * Float64(Float64(t * c) - Float64(y * i))) + Float64(z * Float64(x * y)));
	elseif (b <= 2e-60)
		tmp = fma(z, t_1, Float64(-Float64(t * Float64(x * a))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * (-c) + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * t$95$1 + N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -29000000000.0], t$95$2, If[LessEqual[b, 2.45e-179], N[(N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-60], N[(z * t$95$1 + (-N[(t * N[(x * a), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.9e10 or 1.9999999999999999e-60 < b

   1. Initial program 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 74.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 74.5

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 74.5

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

   4. Applied rewrites 74.5%

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

   5. Taylor expanded in j around 0

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

   7. Applied rewrites 71.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 77.4%

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

   if -2.9e10 < b < 2.45e-179

   1. Initial program 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 70.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 70.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.6

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

   7. Applied rewrites 67.6%

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

   if 2.45e-179 < b < 1.9999999999999999e-60

   1. Initial program 72.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 72.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 72.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in j around 0

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

   7. Applied rewrites 84.3%

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

   8. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(b, \mathsf{neg}\left(c\right), \color{blue}{y \cdot x}\right), -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -29000000000:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), b \cdot \left(a \cdot i\right)\right)\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{-179}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), -t \cdot \left(x \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), b \cdot \left(a \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+220}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma y (fma j (- i) (* x z)) (* b (fma c (- z) (* a i))))))
   (if (<= y -3.8e-71)
     t_1
     (if (<= y 1.1e+220)
       (fma z (fma b (- c) (* x y)) (* a (fma t (- x) (* b i))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(y, fma(j, -i, (x * z)), (b * fma(c, -z, (a * i))));
	double tmp;
	if (y <= -3.8e-71) {
		tmp = t_1;
	} else if (y <= 1.1e+220) {
		tmp = fma(z, fma(b, -c, (x * y)), (a * fma(t, -x, (b * i))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(y, fma(j, Float64(-i), Float64(x * z)), Float64(b * fma(c, Float64(-z), Float64(a * i))))
	tmp = 0.0
	if (y <= -3.8e-71)
		tmp = t_1;
	elseif (y <= 1.1e+220)
		tmp = fma(z, fma(b, Float64(-c), Float64(x * y)), Float64(a * fma(t, Float64(-x), Float64(b * i))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.8e-71], t$95$1, If[LessEqual[y, 1.1e+220], N[(z * N[(b * (-c) + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.79999999999999992e-71 or 1.09999999999999995e220 < y

   1. Initial program 76.0%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

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

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. mul-1-neg N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 82.4%

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

   if -3.79999999999999992e-71 < y < 1.09999999999999995e220

   1. Initial program 71.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 71.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 71.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 71.2

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

   4. Applied rewrites 71.2%

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

   5. Taylor expanded in j around 0

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

   7. Applied rewrites 76.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+220}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.1 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(j, \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right), a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 3.7 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -1.1e+57)
   (fma j (fma c t (* y (- i))) (* a (fma t (- x) (* b i))))
   (if (<= i 3.7e+36)
     (fma c (fma j t (* z (- b))) (* x (- (* y z) (* t a))))
     (fma y (fma j (- i) (* x z)) (* b (fma c (- z) (* a 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 (i <= -1.1e+57) {
		tmp = fma(j, fma(c, t, (y * -i)), (a * fma(t, -x, (b * i))));
	} else if (i <= 3.7e+36) {
		tmp = fma(c, fma(j, t, (z * -b)), (x * ((y * z) - (t * a))));
	} else {
		tmp = fma(y, fma(j, -i, (x * z)), (b * fma(c, -z, (a * i))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -1.1e+57)
		tmp = fma(j, fma(c, t, Float64(y * Float64(-i))), Float64(a * fma(t, Float64(-x), Float64(b * i))));
	elseif (i <= 3.7e+36)
		tmp = fma(c, fma(j, t, Float64(z * Float64(-b))), Float64(x * Float64(Float64(y * z) - Float64(t * a))));
	else
		tmp = fma(y, fma(j, Float64(-i), Float64(x * z)), Float64(b * fma(c, Float64(-z), Float64(a * i))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -1.1e+57], N[(j * N[(c * t + N[(y * (-i)), $MachinePrecision]), $MachinePrecision] + N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.7e+36], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.1e57

   1. Initial program 63.8%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. cancel-sign-sub N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

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

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 87.3%

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

   if -1.1e57 < i < 3.70000000000000029e36

   1. Initial program 78.5%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 74.8%

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

   if 3.70000000000000029e36 < i

   1. Initial program 66.4%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

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

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. mul-1-neg N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 78.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.1 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(j, \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right), a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 3.7 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.1 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(j, \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right), a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 1.06 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -1.1e+57)
   (fma j (fma c t (* y (- i))) (* a (fma t (- x) (* b i))))
   (if (<= i 1.06e+55)
     (fma c (fma j t (* z (- b))) (* x (- (* y z) (* t a))))
     (* i (fma b a (* y (- j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -1.1e+57) {
		tmp = fma(j, fma(c, t, (y * -i)), (a * fma(t, -x, (b * i))));
	} else if (i <= 1.06e+55) {
		tmp = fma(c, fma(j, t, (z * -b)), (x * ((y * z) - (t * a))));
	} else {
		tmp = i * fma(b, a, (y * -j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -1.1e+57)
		tmp = fma(j, fma(c, t, Float64(y * Float64(-i))), Float64(a * fma(t, Float64(-x), Float64(b * i))));
	elseif (i <= 1.06e+55)
		tmp = fma(c, fma(j, t, Float64(z * Float64(-b))), Float64(x * Float64(Float64(y * z) - Float64(t * a))));
	else
		tmp = Float64(i * fma(b, a, Float64(y * Float64(-j))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -1.1e+57], N[(j * N[(c * t + N[(y * (-i)), $MachinePrecision]), $MachinePrecision] + N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.06e+55], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.1e57

   1. Initial program 63.8%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. cancel-sign-sub N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

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

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 87.3%

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

   if -1.1e57 < i < 1.06000000000000004e55

   1. Initial program 78.2%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 74.5%

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

   if 1.06000000000000004e55 < i

   1. Initial program 66.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 66.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 66.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. cancel-sign-sub N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 74.3

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

   7. Applied rewrites 74.3%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.1 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(j, \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right), a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 1.06 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -3.25 \cdot 10^{+66}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq 1.06 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -3.25e+66)
   (* i (fma j (- y) (* a b)))
   (if (<= i 1.06e+55)
     (fma c (fma j t (* z (- b))) (* x (- (* y z) (* t a))))
     (* i (fma b a (* y (- j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -3.25e+66) {
		tmp = i * fma(j, -y, (a * b));
	} else if (i <= 1.06e+55) {
		tmp = fma(c, fma(j, t, (z * -b)), (x * ((y * z) - (t * a))));
	} else {
		tmp = i * fma(b, a, (y * -j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -3.25e+66)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (i <= 1.06e+55)
		tmp = fma(c, fma(j, t, Float64(z * Float64(-b))), Float64(x * Float64(Float64(y * z) - Float64(t * a))));
	else
		tmp = Float64(i * fma(b, a, Float64(y * Float64(-j))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -3.25e+66], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.06e+55], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -3.2500000000000001e66

   1. Initial program 63.8%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if -3.2500000000000001e66 < i < 1.06000000000000004e55

   1. Initial program 78.2%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 74.5%

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

   if 1.06000000000000004e55 < i

   1. Initial program 66.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 66.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 66.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. cancel-sign-sub N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 74.3

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

   7. Applied rewrites 74.3%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.25 \cdot 10^{+66}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq 1.06 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -7.4 \cdot 10^{+83}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -4 \cdot 10^{-204}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), b \cdot \left(a \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 1.95 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -7.4e+83)
   (* i (fma j (- y) (* a b)))
   (if (<= i -4e-204)
     (fma z (fma b (- c) (* x y)) (* b (* a i)))
     (if (<= i 1.95e+54)
       (fma c (fma j t (* z (- b))) (* x (* y z)))
       (* i (fma b a (* y (- j))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -7.4e+83) {
		tmp = i * fma(j, -y, (a * b));
	} else if (i <= -4e-204) {
		tmp = fma(z, fma(b, -c, (x * y)), (b * (a * i)));
	} else if (i <= 1.95e+54) {
		tmp = fma(c, fma(j, t, (z * -b)), (x * (y * z)));
	} else {
		tmp = i * fma(b, a, (y * -j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -7.4e+83)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (i <= -4e-204)
		tmp = fma(z, fma(b, Float64(-c), Float64(x * y)), Float64(b * Float64(a * i)));
	elseif (i <= 1.95e+54)
		tmp = fma(c, fma(j, t, Float64(z * Float64(-b))), Float64(x * Float64(y * z)));
	else
		tmp = Float64(i * fma(b, a, Float64(y * Float64(-j))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -7.4e+83], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -4e-204], N[(z * N[(b * (-c) + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.95e+54], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -7.4000000000000005e83

   1. Initial program 63.0%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -7.4000000000000005e83 < i < -4e-204

   1. Initial program 78.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 78.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 78.5

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 78.5

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in j around 0

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 71.8%

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

   if -4e-204 < i < 1.9500000000000001e54

   1. Initial program 78.2%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 63.0%

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

   if 1.9500000000000001e54 < i

   1. Initial program 66.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 66.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 66.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. cancel-sign-sub N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 74.3

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

   7. Applied rewrites 74.3%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.4 \cdot 10^{+83}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -4 \cdot 10^{-204}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), b \cdot \left(a \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 1.95 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{if}\;b \leq -9200000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-111}:\\ \;\;\;\;z \cdot \left(x \cdot y\right) + \left(-y \cdot \left(i \cdot j\right)\right)\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-55}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+94}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma c (- z) (* a i)))))
   (if (<= b -9200000000000.0)
     t_1
     (if (<= b 1.65e-111)
       (+ (* z (* x y)) (- (* y (* i j))))
       (if (<= b 7e-55)
         (* t (fma j c (* x (- a))))
         (if (<= b 1.35e+94) (* z (fma c (- b) (* x y))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(c, -z, (a * i));
	double tmp;
	if (b <= -9200000000000.0) {
		tmp = t_1;
	} else if (b <= 1.65e-111) {
		tmp = (z * (x * y)) + -(y * (i * j));
	} else if (b <= 7e-55) {
		tmp = t * fma(j, c, (x * -a));
	} else if (b <= 1.35e+94) {
		tmp = z * fma(c, -b, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(c, Float64(-z), Float64(a * i)))
	tmp = 0.0
	if (b <= -9200000000000.0)
		tmp = t_1;
	elseif (b <= 1.65e-111)
		tmp = Float64(Float64(z * Float64(x * y)) + Float64(-Float64(y * Float64(i * j))));
	elseif (b <= 7e-55)
		tmp = Float64(t * fma(j, c, Float64(x * Float64(-a))));
	elseif (b <= 1.35e+94)
		tmp = Float64(z * fma(c, Float64(-b), Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -9200000000000.0], t$95$1, If[LessEqual[b, 1.65e-111], N[(N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision] + (-N[(y * N[(i * j), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[b, 7e-55], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+94], N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -9.2e12 or 1.3500000000000001e94 < b

   1. Initial program 74.8%

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

   3. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -9.2e12 < b < 1.65e-111

   1. Initial program 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 68.7

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 68.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 63.7

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

   7. Applied rewrites 63.7%

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

   8. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 57.6

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

   10. Applied rewrites 57.6%

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

   if 1.65e-111 < b < 7.00000000000000051e-55

   1. Initial program 91.4%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if 7.00000000000000051e-55 < b < 1.3500000000000001e94

   1. Initial program 72.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9200000000000:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-111}:\\ \;\;\;\;z \cdot \left(x \cdot y\right) + \left(-y \cdot \left(i \cdot j\right)\right)\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-55}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+94}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{if}\;b \leq -5400:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-111}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-55}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+94}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma c (- z) (* a i)))))
   (if (<= b -5400.0)
     t_1
     (if (<= b 1.95e-111)
       (* y (fma j (- i) (* x z)))
       (if (<= b 7e-55)
         (* t (fma j c (* x (- a))))
         (if (<= b 1.35e+94) (* z (fma c (- b) (* x y))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(c, -z, (a * i));
	double tmp;
	if (b <= -5400.0) {
		tmp = t_1;
	} else if (b <= 1.95e-111) {
		tmp = y * fma(j, -i, (x * z));
	} else if (b <= 7e-55) {
		tmp = t * fma(j, c, (x * -a));
	} else if (b <= 1.35e+94) {
		tmp = z * fma(c, -b, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(c, Float64(-z), Float64(a * i)))
	tmp = 0.0
	if (b <= -5400.0)
		tmp = t_1;
	elseif (b <= 1.95e-111)
		tmp = Float64(y * fma(j, Float64(-i), Float64(x * z)));
	elseif (b <= 7e-55)
		tmp = Float64(t * fma(j, c, Float64(x * Float64(-a))));
	elseif (b <= 1.35e+94)
		tmp = Float64(z * fma(c, Float64(-b), Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5400.0], t$95$1, If[LessEqual[b, 1.95e-111], N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-55], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+94], N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -5400 or 1.3500000000000001e94 < b

   1. Initial program 75.2%

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

   3. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -5400 < b < 1.9500000000000001e-111

   1. Initial program 68.0%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

   if 1.9500000000000001e-111 < b < 7.00000000000000051e-55

   1. Initial program 91.4%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if 7.00000000000000051e-55 < b < 1.3500000000000001e94

   1. Initial program 72.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5400:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-111}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-55}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+94}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{if}\;i \leq -2.2 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -7.5 \cdot 10^{-271}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;i \leq 7.6 \cdot 10^{-107}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{+53}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- y) (* a b)))))
   (if (<= i -2.2e+29)
     t_1
     (if (<= i -7.5e-271)
       (* t (fma j c (* x (- a))))
       (if (<= i 7.6e-107)
         (* c (fma j t (* z (- b))))
         (if (<= i 8.2e+53) (* x (- (* y z) (* t a))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * fma(j, -y, (a * b));
	double tmp;
	if (i <= -2.2e+29) {
		tmp = t_1;
	} else if (i <= -7.5e-271) {
		tmp = t * fma(j, c, (x * -a));
	} else if (i <= 7.6e-107) {
		tmp = c * fma(j, t, (z * -b));
	} else if (i <= 8.2e+53) {
		tmp = x * ((y * z) - (t * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * fma(j, Float64(-y), Float64(a * b)))
	tmp = 0.0
	if (i <= -2.2e+29)
		tmp = t_1;
	elseif (i <= -7.5e-271)
		tmp = Float64(t * fma(j, c, Float64(x * Float64(-a))));
	elseif (i <= 7.6e-107)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-b))));
	elseif (i <= 8.2e+53)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2.2e+29], t$95$1, If[LessEqual[i, -7.5e-271], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7.6e-107], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 8.2e+53], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -2.2000000000000001e29 or 8.20000000000000037e53 < i

   1. Initial program 65.0%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if -2.2000000000000001e29 < i < -7.50000000000000031e-271

   1. Initial program 78.5%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 51.2

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

   5. Applied rewrites 51.2%

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

   if -7.50000000000000031e-271 < i < 7.6000000000000004e-107

   1. Initial program 81.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(-1 \cdot z\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      10. lower-neg.f64 55.7

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right) \]

   5. Applied rewrites 55.7%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)} \]

   if 7.6000000000000004e-107 < i < 8.20000000000000037e53

   1. Initial program 76.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. lower-*.f64 56.2

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Applied rewrites 56.2%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{+29}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -7.5 \cdot 10^{-271}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;i \leq 7.6 \cdot 10^{-107}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{+53}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{if}\;i \leq -2.2 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -7.5 \cdot 10^{-271}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{-86}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;i \leq 7.2 \cdot 10^{+74}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- y) (* a b)))))
   (if (<= i -2.2e+29)
     t_1
     (if (<= i -7.5e-271)
       (* t (fma j c (* x (- a))))
       (if (<= i 3.1e-86)
         (* c (fma j t (* z (- b))))
         (if (<= i 7.2e+74) (* a (fma t (- x) (* b i))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * fma(j, -y, (a * b));
	double tmp;
	if (i <= -2.2e+29) {
		tmp = t_1;
	} else if (i <= -7.5e-271) {
		tmp = t * fma(j, c, (x * -a));
	} else if (i <= 3.1e-86) {
		tmp = c * fma(j, t, (z * -b));
	} else if (i <= 7.2e+74) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * fma(j, Float64(-y), Float64(a * b)))
	tmp = 0.0
	if (i <= -2.2e+29)
		tmp = t_1;
	elseif (i <= -7.5e-271)
		tmp = Float64(t * fma(j, c, Float64(x * Float64(-a))));
	elseif (i <= 3.1e-86)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-b))));
	elseif (i <= 7.2e+74)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2.2e+29], t$95$1, If[LessEqual[i, -7.5e-271], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.1e-86], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7.2e+74], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -2.2000000000000001e29 or 7.19999999999999975e74 < i

   1. Initial program 63.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      11. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \mathsf{neg}\left(y\right), \color{blue}{b \cdot a}\right) \]

      12. lower-*.f64 76.8

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot a}\right) \]

   5. Applied rewrites 76.8%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot a\right)} \]

   if -2.2000000000000001e29 < i < -7.50000000000000031e-271

   1. Initial program 78.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(a \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto t \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(a \cdot x\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto t \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(a \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto t \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(a \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto t \cdot \mathsf{fma}\left(j, c, \color{blue}{a \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto t \cdot \mathsf{fma}\left(j, c, a \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto t \cdot \mathsf{fma}\left(j, c, \color{blue}{a \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto t \cdot \mathsf{fma}\left(j, c, a \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. lower-neg.f64 51.2

          \[\leadsto t \cdot \mathsf{fma}\left(j, c, a \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Applied rewrites 51.2%

      \[\leadsto \color{blue}{t \cdot \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right)} \]

   if -7.50000000000000031e-271 < i < 3.09999999999999989e-86

   1. Initial program 81.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(-1 \cdot z\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      10. lower-neg.f64 53.5

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right) \]

   5. Applied rewrites 53.5%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)} \]

   if 3.09999999999999989e-86 < i < 7.19999999999999975e74

   1. Initial program 78.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 56.7

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 56.7%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{+29}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -7.5 \cdot 10^{-271}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{-86}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;i \leq 7.2 \cdot 10^{+74}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{if}\;c \leq -2.1 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -3.2 \cdot 10^{+75}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;c \leq 190:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{elif}\;c \leq 5.7 \cdot 10^{+187}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (fma j t (* z (- b))))))
   (if (<= c -2.1e+102)
     t_1
     (if (<= c -3.2e+75)
       (* z (* x y))
       (if (<= c 190.0)
         (* a (fma t (- x) (* b i)))
         (if (<= c 5.7e+187) t_1 (* b (fma c (- z) (* a i)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * fma(j, t, (z * -b));
	double tmp;
	if (c <= -2.1e+102) {
		tmp = t_1;
	} else if (c <= -3.2e+75) {
		tmp = z * (x * y);
	} else if (c <= 190.0) {
		tmp = a * fma(t, -x, (b * i));
	} else if (c <= 5.7e+187) {
		tmp = t_1;
	} else {
		tmp = b * fma(c, -z, (a * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * fma(j, t, Float64(z * Float64(-b))))
	tmp = 0.0
	if (c <= -2.1e+102)
		tmp = t_1;
	elseif (c <= -3.2e+75)
		tmp = Float64(z * Float64(x * y));
	elseif (c <= 190.0)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	elseif (c <= 5.7e+187)
		tmp = t_1;
	else
		tmp = Float64(b * fma(c, Float64(-z), Float64(a * i)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.1e+102], t$95$1, If[LessEqual[c, -3.2e+75], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 190.0], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5.7e+187], t$95$1, N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.10000000000000001e102 or 190 < c < 5.7000000000000004e187

   1. Initial program 58.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(-1 \cdot z\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      10. lower-neg.f64 64.6

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right) \]

   5. Applied rewrites 64.6%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)} \]

   if -2.10000000000000001e102 < c < -3.19999999999999985e75

   1. Initial program 99.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.4%

      \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 94.2%

       \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   if -3.19999999999999985e75 < c < 190

   1. Initial program 80.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 54.3

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 54.3%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   if 5.7000000000000004e187 < c

   1. Initial program 59.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \]

      3. remove-double-neg N/A

         \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \]

      9. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \color{blue}{a \cdot i}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot z, a \cdot i\right)} \]

      15. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      16. lower-neg.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(z\right), \color{blue}{i \cdot a}\right) \]

      18. lower-*.f64 80.7

          \[\leadsto b \cdot \mathsf{fma}\left(c, -z, \color{blue}{i \cdot a}\right) \]

   5. Applied rewrites 80.7%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+102}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq -3.2 \cdot 10^{+75}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;c \leq 190:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{elif}\;c \leq 5.7 \cdot 10^{+187}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -3 \cdot 10^{+66}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq 1.95 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -3e+66)
   (* i (fma j (- y) (* a b)))
   (if (<= i 1.95e+54)
     (fma c (fma j t (* z (- b))) (* x (* y z)))
     (* i (fma b a (* y (- j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -3e+66) {
		tmp = i * fma(j, -y, (a * b));
	} else if (i <= 1.95e+54) {
		tmp = fma(c, fma(j, t, (z * -b)), (x * (y * z)));
	} else {
		tmp = i * fma(b, a, (y * -j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -3e+66)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (i <= 1.95e+54)
		tmp = fma(c, fma(j, t, Float64(z * Float64(-b))), Float64(x * Float64(y * z)));
	else
		tmp = Float64(i * fma(b, a, Float64(y * Float64(-j))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -3e+66], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.95e+54], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -3.00000000000000002e66

   1. Initial program 63.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      11. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \mathsf{neg}\left(y\right), \color{blue}{b \cdot a}\right) \]

      12. lower-*.f64 82.0

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot a}\right) \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot a\right)} \]

   if -3.00000000000000002e66 < i < 1.9500000000000001e54

   1. Initial program 78.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right), x \cdot \left(y \cdot z\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 60.5%

      \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right), x \cdot \left(y \cdot z\right)\right) \]

   if 1.9500000000000001e54 < i

   1. Initial program 66.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{1}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 66.8

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z} - i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{z \cdot c} - i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      13. lower-*.f64 66.8

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{z \cdot c} - i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{z \cdot c - \color{blue}{i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{z \cdot c - \color{blue}{a \cdot i}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      16. lower-*.f64 66.8

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{z \cdot c - \color{blue}{a \cdot i}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   4. Applied rewrites 66.8%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{1}{z \cdot c - a \cdot i}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(-1 \cdot a\right) \cdot b}\right) \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot b\right) \]

      4. cancel-sign-sub N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

      5. +-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(a \cdot b + -1 \cdot \left(j \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto i \cdot \left(\color{blue}{b \cdot a} + -1 \cdot \left(j \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(b, a, -1 \cdot \left(j \cdot y\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(b, a, \color{blue}{\mathsf{neg}\left(j \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto i \cdot \mathsf{fma}\left(b, a, \mathsf{neg}\left(\color{blue}{y \cdot j}\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto i \cdot \mathsf{fma}\left(b, a, \color{blue}{y \cdot \left(\mathsf{neg}\left(j\right)\right)}\right) \]

      11. mul-1-neg N/A

          \[\leadsto i \cdot \mathsf{fma}\left(b, a, y \cdot \color{blue}{\left(-1 \cdot j\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(b, a, \color{blue}{y \cdot \left(-1 \cdot j\right)}\right) \]

      13. mul-1-neg N/A

          \[\leadsto i \cdot \mathsf{fma}\left(b, a, y \cdot \color{blue}{\left(\mathsf{neg}\left(j\right)\right)}\right) \]

      14. lower-neg.f64 74.3

          \[\leadsto i \cdot \mathsf{fma}\left(b, a, y \cdot \color{blue}{\left(-j\right)}\right) \]

   7. Applied rewrites 74.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3 \cdot 10^{+66}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq 1.95 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 51.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{if}\;b \leq -5400:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-128}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\\ \mathbf{elif}\;b \leq 1.52 \cdot 10^{+94}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma c (- z) (* a i)))))
   (if (<= b -5400.0)
     t_1
     (if (<= b 1.1e-128)
       (* y (fma j (- i) (* x z)))
       (if (<= b 1.52e+94) (* x (- (* y z) (* t a))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(c, -z, (a * i));
	double tmp;
	if (b <= -5400.0) {
		tmp = t_1;
	} else if (b <= 1.1e-128) {
		tmp = y * fma(j, -i, (x * z));
	} else if (b <= 1.52e+94) {
		tmp = x * ((y * z) - (t * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(c, Float64(-z), Float64(a * i)))
	tmp = 0.0
	if (b <= -5400.0)
		tmp = t_1;
	elseif (b <= 1.1e-128)
		tmp = Float64(y * fma(j, Float64(-i), Float64(x * z)));
	elseif (b <= 1.52e+94)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5400.0], t$95$1, If[LessEqual[b, 1.1e-128], N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.52e+94], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5400 or 1.5199999999999999e94 < b

   1. Initial program 75.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \]

      3. remove-double-neg N/A

         \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \]

      9. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \color{blue}{a \cdot i}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot z, a \cdot i\right)} \]

      15. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      16. lower-neg.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(z\right), \color{blue}{i \cdot a}\right) \]

      18. lower-*.f64 71.7

          \[\leadsto b \cdot \mathsf{fma}\left(c, -z, \color{blue}{i \cdot a}\right) \]

   5. Applied rewrites 71.7%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)} \]

   if -5400 < b < 1.10000000000000005e-128

   1. Initial program 68.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(-1 \cdot i\right) \cdot j} + x \cdot z\right) \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{j \cdot \left(-1 \cdot i\right)} + x \cdot z\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot i, x \cdot z\right)} \]

      5. neg-mul-1 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(i\right)}, x \cdot z\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(i\right)}, x \cdot z\right) \]

      7. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \mathsf{neg}\left(i\right), \color{blue}{z \cdot x}\right) \]

      8. lower-*.f64 55.9

         \[\leadsto y \cdot \mathsf{fma}\left(j, -i, \color{blue}{z \cdot x}\right) \]

   5. Applied rewrites 55.9%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(j, -i, z \cdot x\right)} \]

   if 1.10000000000000005e-128 < b < 1.5199999999999999e94

   1. Initial program 78.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. lower-*.f64 60.3

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Applied rewrites 60.3%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5400:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-128}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\\ \mathbf{elif}\;b \leq 1.52 \cdot 10^{+94}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{if}\;i \leq -2.2 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{-86}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;i \leq 7.2 \cdot 10^{+74}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- y) (* a b)))))
   (if (<= i -2.2e+29)
     t_1
     (if (<= i 3.1e-86)
       (* c (fma j t (* z (- b))))
       (if (<= i 7.2e+74) (* a (fma t (- x) (* b i))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * fma(j, -y, (a * b));
	double tmp;
	if (i <= -2.2e+29) {
		tmp = t_1;
	} else if (i <= 3.1e-86) {
		tmp = c * fma(j, t, (z * -b));
	} else if (i <= 7.2e+74) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * fma(j, Float64(-y), Float64(a * b)))
	tmp = 0.0
	if (i <= -2.2e+29)
		tmp = t_1;
	elseif (i <= 3.1e-86)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-b))));
	elseif (i <= 7.2e+74)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2.2e+29], t$95$1, If[LessEqual[i, 3.1e-86], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7.2e+74], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.2000000000000001e29 or 7.19999999999999975e74 < i

   1. Initial program 63.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      11. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \mathsf{neg}\left(y\right), \color{blue}{b \cdot a}\right) \]

      12. lower-*.f64 76.8

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot a}\right) \]

   5. Applied rewrites 76.8%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot a\right)} \]

   if -2.2000000000000001e29 < i < 3.09999999999999989e-86

   1. Initial program 79.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(-1 \cdot z\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      10. lower-neg.f64 47.3

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right) \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)} \]

   if 3.09999999999999989e-86 < i < 7.19999999999999975e74

   1. Initial program 78.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 56.7

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 56.7%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{+29}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{-86}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;i \leq 7.2 \cdot 10^{+74}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{if}\;a \leq -5 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-175}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;a \leq 1.08 \cdot 10^{-97}:\\ \;\;\;\;z \cdot \left(c \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (fma t (- x) (* b i)))))
   (if (<= a -5e+25)
     t_1
     (if (<= a 1.4e-175)
       (* z (* x y))
       (if (<= a 1.08e-97) (* z (* c (- b))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * fma(t, -x, (b * i));
	double tmp;
	if (a <= -5e+25) {
		tmp = t_1;
	} else if (a <= 1.4e-175) {
		tmp = z * (x * y);
	} else if (a <= 1.08e-97) {
		tmp = z * (c * -b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * fma(t, Float64(-x), Float64(b * i)))
	tmp = 0.0
	if (a <= -5e+25)
		tmp = t_1;
	elseif (a <= 1.4e-175)
		tmp = Float64(z * Float64(x * y));
	elseif (a <= 1.08e-97)
		tmp = Float64(z * Float64(c * Float64(-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[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5e+25], t$95$1, If[LessEqual[a, 1.4e-175], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.08e-97], N[(z * N[(c * (-b)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.00000000000000024e25 or 1.0799999999999999e-97 < a

   1. Initial program 69.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 64.0

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 64.0%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   if -5.00000000000000024e25 < a < 1.4e-175

   1. Initial program 79.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 21.2%

      \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.9%

       \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   if 1.4e-175 < a < 1.0799999999999999e-97

   1. Initial program 67.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 61.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.0%

      \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 56.1%

       \[\leadsto z \cdot \color{blue}{\left(c \cdot \left(-b\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+25}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-175}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;a \leq 1.08 \cdot 10^{-97}:\\ \;\;\;\;z \cdot \left(c \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 30.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+26}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-175}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \left(c \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -3.8e+26)
   (* a (* b i))
   (if (<= a 1.4e-175)
     (* z (* x y))
     (if (<= a 5e-6) (* z (* c (- b))) (* b (* a 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 (a <= -3.8e+26) {
		tmp = a * (b * i);
	} else if (a <= 1.4e-175) {
		tmp = z * (x * y);
	} else if (a <= 5e-6) {
		tmp = z * (c * -b);
	} else {
		tmp = b * (a * 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 (a <= (-3.8d+26)) then
        tmp = a * (b * i)
    else if (a <= 1.4d-175) then
        tmp = z * (x * y)
    else if (a <= 5d-6) then
        tmp = z * (c * -b)
    else
        tmp = b * (a * 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 (a <= -3.8e+26) {
		tmp = a * (b * i);
	} else if (a <= 1.4e-175) {
		tmp = z * (x * y);
	} else if (a <= 5e-6) {
		tmp = z * (c * -b);
	} else {
		tmp = b * (a * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if a <= -3.8e+26:
		tmp = a * (b * i)
	elif a <= 1.4e-175:
		tmp = z * (x * y)
	elif a <= 5e-6:
		tmp = z * (c * -b)
	else:
		tmp = b * (a * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -3.8e+26)
		tmp = Float64(a * Float64(b * i));
	elseif (a <= 1.4e-175)
		tmp = Float64(z * Float64(x * y));
	elseif (a <= 5e-6)
		tmp = Float64(z * Float64(c * Float64(-b)));
	else
		tmp = Float64(b * Float64(a * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (a <= -3.8e+26)
		tmp = a * (b * i);
	elseif (a <= 1.4e-175)
		tmp = z * (x * y);
	elseif (a <= 5e-6)
		tmp = z * (c * -b);
	else
		tmp = b * (a * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -3.8e+26], N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.4e-175], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5e-6], N[(z * N[(c * (-b)), $MachinePrecision]), $MachinePrecision], N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -3.8000000000000002e26

   1. Initial program 67.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 69.5

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto a \cdot \left(b \cdot \color{blue}{i}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 50.2%

      \[\leadsto a \cdot \left(b \cdot \color{blue}{i}\right) \]

   if -3.8000000000000002e26 < a < 1.4e-175

   1. Initial program 79.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 21.2%

      \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.9%

       \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   if 1.4e-175 < a < 5.00000000000000041e-6

   1. Initial program 79.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 20.5%

      \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 46.8%

       \[\leadsto z \cdot \color{blue}{\left(c \cdot \left(-b\right)\right)} \]

   if 5.00000000000000041e-6 < a

   1. Initial program 65.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \]

      3. remove-double-neg N/A

         \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \]

      9. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \color{blue}{a \cdot i}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot z, a \cdot i\right)} \]

      15. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      16. lower-neg.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(z\right), \color{blue}{i \cdot a}\right) \]

      18. lower-*.f64 51.9

          \[\leadsto b \cdot \mathsf{fma}\left(c, -z, \color{blue}{i \cdot a}\right) \]

   5. Applied rewrites 51.9%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto b \cdot \left(a \cdot \color{blue}{i}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 42.7%

      \[\leadsto b \cdot \left(a \cdot \color{blue}{i}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+26}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-175}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \left(c \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 30.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+26}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{-175}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;-b \cdot \left(z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -3.8e+26)
   (* a (* b i))
   (if (<= a 2.35e-175)
     (* z (* x y))
     (if (<= a 3.5e-5) (- (* b (* z c))) (* b (* a 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 (a <= -3.8e+26) {
		tmp = a * (b * i);
	} else if (a <= 2.35e-175) {
		tmp = z * (x * y);
	} else if (a <= 3.5e-5) {
		tmp = -(b * (z * c));
	} else {
		tmp = b * (a * 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 (a <= (-3.8d+26)) then
        tmp = a * (b * i)
    else if (a <= 2.35d-175) then
        tmp = z * (x * y)
    else if (a <= 3.5d-5) then
        tmp = -(b * (z * c))
    else
        tmp = b * (a * 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 (a <= -3.8e+26) {
		tmp = a * (b * i);
	} else if (a <= 2.35e-175) {
		tmp = z * (x * y);
	} else if (a <= 3.5e-5) {
		tmp = -(b * (z * c));
	} else {
		tmp = b * (a * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if a <= -3.8e+26:
		tmp = a * (b * i)
	elif a <= 2.35e-175:
		tmp = z * (x * y)
	elif a <= 3.5e-5:
		tmp = -(b * (z * c))
	else:
		tmp = b * (a * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -3.8e+26)
		tmp = Float64(a * Float64(b * i));
	elseif (a <= 2.35e-175)
		tmp = Float64(z * Float64(x * y));
	elseif (a <= 3.5e-5)
		tmp = Float64(-Float64(b * Float64(z * c)));
	else
		tmp = Float64(b * Float64(a * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (a <= -3.8e+26)
		tmp = a * (b * i);
	elseif (a <= 2.35e-175)
		tmp = z * (x * y);
	elseif (a <= 3.5e-5)
		tmp = -(b * (z * c));
	else
		tmp = b * (a * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -3.8e+26], N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.35e-175], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.5e-5], (-N[(b * N[(z * c), $MachinePrecision]), $MachinePrecision]), N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -3.8000000000000002e26

   1. Initial program 67.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 69.5

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto a \cdot \left(b \cdot \color{blue}{i}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 50.2%

      \[\leadsto a \cdot \left(b \cdot \color{blue}{i}\right) \]

   if -3.8000000000000002e26 < a < 2.34999999999999999e-175

   1. Initial program 79.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 21.2%

      \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.9%

       \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   if 2.34999999999999999e-175 < a < 3.4999999999999997e-5

   1. Initial program 79.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.8%

      \[\leadsto -b \cdot \left(c \cdot z\right) \]

   if 3.4999999999999997e-5 < a

   1. Initial program 65.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \]

      3. remove-double-neg N/A

         \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \]

      9. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \color{blue}{a \cdot i}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot z, a \cdot i\right)} \]

      15. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      16. lower-neg.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(z\right), \color{blue}{i \cdot a}\right) \]

      18. lower-*.f64 51.9

          \[\leadsto b \cdot \mathsf{fma}\left(c, -z, \color{blue}{i \cdot a}\right) \]

   5. Applied rewrites 51.9%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto b \cdot \left(a \cdot \color{blue}{i}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 42.7%

      \[\leadsto b \cdot \left(a \cdot \color{blue}{i}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+26}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{-175}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;-b \cdot \left(z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 51.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.1 \cdot 10^{+57}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq 2.15 \cdot 10^{+54}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, y, c \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -1.1e+57)
   (* i (fma j (- y) (* a b)))
   (if (<= i 2.15e+54)
     (* z (fma x y (* c (- b))))
     (* i (fma b a (* y (- j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -1.1e+57) {
		tmp = i * fma(j, -y, (a * b));
	} else if (i <= 2.15e+54) {
		tmp = z * fma(x, y, (c * -b));
	} else {
		tmp = i * fma(b, a, (y * -j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -1.1e+57)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (i <= 2.15e+54)
		tmp = Float64(z * fma(x, y, Float64(c * Float64(-b))));
	else
		tmp = Float64(i * fma(b, a, Float64(y * Float64(-j))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -1.1e+57], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.15e+54], N[(z * N[(x * y + N[(c * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.1e57

   1. Initial program 63.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      11. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \mathsf{neg}\left(y\right), \color{blue}{b \cdot a}\right) \]

      12. lower-*.f64 82.0

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot a}\right) \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot a\right)} \]

   if -1.1e57 < i < 2.14999999999999988e54

   1. Initial program 78.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\left(-1 \cdot \frac{a \cdot \left(t \cdot x\right)}{z} + \left(x \cdot y + \frac{j \cdot \left(c \cdot t - i \cdot y\right)}{z}\right)\right) - \left(-1 \cdot \frac{a \cdot \left(b \cdot i\right)}{z} + b \cdot c\right)\right)} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{z \cdot \left(\mathsf{fma}\left(y, x, \frac{\mathsf{fma}\left(j, \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right), a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)\right)}{z}\right) - c \cdot b\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(x, y, \mathsf{neg}\left(b \cdot c\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto z \cdot \mathsf{fma}\left(x, y, \color{blue}{b \cdot \left(\mathsf{neg}\left(c\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto z \cdot \mathsf{fma}\left(x, y, b \cdot \color{blue}{\left(-1 \cdot c\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(x, y, \color{blue}{b \cdot \left(-1 \cdot c\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto z \cdot \mathsf{fma}\left(x, y, b \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)}\right) \]

      8. lower-neg.f64 53.3

         \[\leadsto z \cdot \mathsf{fma}\left(x, y, b \cdot \color{blue}{\left(-c\right)}\right) \]

   8. Applied rewrites 53.3%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)} \]

   if 2.14999999999999988e54 < i

   1. Initial program 66.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{1}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 66.8

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z} - i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{z \cdot c} - i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      13. lower-*.f64 66.8

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{z \cdot c} - i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{z \cdot c - \color{blue}{i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{z \cdot c - \color{blue}{a \cdot i}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      16. lower-*.f64 66.8

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{z \cdot c - \color{blue}{a \cdot i}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   4. Applied rewrites 66.8%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{1}{z \cdot c - a \cdot i}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(-1 \cdot a\right) \cdot b}\right) \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot b\right) \]

      4. cancel-sign-sub N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

      5. +-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(a \cdot b + -1 \cdot \left(j \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto i \cdot \left(\color{blue}{b \cdot a} + -1 \cdot \left(j \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(b, a, -1 \cdot \left(j \cdot y\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(b, a, \color{blue}{\mathsf{neg}\left(j \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto i \cdot \mathsf{fma}\left(b, a, \mathsf{neg}\left(\color{blue}{y \cdot j}\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto i \cdot \mathsf{fma}\left(b, a, \color{blue}{y \cdot \left(\mathsf{neg}\left(j\right)\right)}\right) \]

      11. mul-1-neg N/A

          \[\leadsto i \cdot \mathsf{fma}\left(b, a, y \cdot \color{blue}{\left(-1 \cdot j\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(b, a, \color{blue}{y \cdot \left(-1 \cdot j\right)}\right) \]

      13. mul-1-neg N/A

          \[\leadsto i \cdot \mathsf{fma}\left(b, a, y \cdot \color{blue}{\left(\mathsf{neg}\left(j\right)\right)}\right) \]

      14. lower-neg.f64 74.3

          \[\leadsto i \cdot \mathsf{fma}\left(b, a, y \cdot \color{blue}{\left(-j\right)}\right) \]

   7. Applied rewrites 74.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.1 \cdot 10^{+57}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq 2.15 \cdot 10^{+54}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, y, c \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 51.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{if}\;i \leq -1.1 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 2.15 \cdot 10^{+54}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, y, c \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- y) (* a b)))))
   (if (<= i -1.1e+57)
     t_1
     (if (<= i 2.15e+54) (* z (fma x y (* c (- 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 = i * fma(j, -y, (a * b));
	double tmp;
	if (i <= -1.1e+57) {
		tmp = t_1;
	} else if (i <= 2.15e+54) {
		tmp = z * fma(x, y, (c * -b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * fma(j, Float64(-y), Float64(a * b)))
	tmp = 0.0
	if (i <= -1.1e+57)
		tmp = t_1;
	elseif (i <= 2.15e+54)
		tmp = Float64(z * fma(x, y, Float64(c * Float64(-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[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.1e+57], t$95$1, If[LessEqual[i, 2.15e+54], N[(z * N[(x * y + N[(c * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.1e57 or 2.14999999999999988e54 < i

   1. Initial program 65.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      11. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \mathsf{neg}\left(y\right), \color{blue}{b \cdot a}\right) \]

      12. lower-*.f64 77.7

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot a}\right) \]

   5. Applied rewrites 77.7%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot a\right)} \]

   if -1.1e57 < i < 2.14999999999999988e54

   1. Initial program 78.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\left(-1 \cdot \frac{a \cdot \left(t \cdot x\right)}{z} + \left(x \cdot y + \frac{j \cdot \left(c \cdot t - i \cdot y\right)}{z}\right)\right) - \left(-1 \cdot \frac{a \cdot \left(b \cdot i\right)}{z} + b \cdot c\right)\right)} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{z \cdot \left(\mathsf{fma}\left(y, x, \frac{\mathsf{fma}\left(j, \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right), a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)\right)}{z}\right) - c \cdot b\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(x, y, \mathsf{neg}\left(b \cdot c\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto z \cdot \mathsf{fma}\left(x, y, \color{blue}{b \cdot \left(\mathsf{neg}\left(c\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto z \cdot \mathsf{fma}\left(x, y, b \cdot \color{blue}{\left(-1 \cdot c\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(x, y, \color{blue}{b \cdot \left(-1 \cdot c\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto z \cdot \mathsf{fma}\left(x, y, b \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)}\right) \]

      8. lower-neg.f64 53.3

         \[\leadsto z \cdot \mathsf{fma}\left(x, y, b \cdot \color{blue}{\left(-c\right)}\right) \]

   8. Applied rewrites 53.3%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.1 \cdot 10^{+57}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq 2.15 \cdot 10^{+54}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, y, c \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 46.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{if}\;b \leq -3800:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{+24}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma c (- z) (* a i)))))
   (if (<= b -3800.0)
     t_1
     (if (<= b 1.08e+24) (* a (fma t (- x) (* b i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(c, -z, (a * i));
	double tmp;
	if (b <= -3800.0) {
		tmp = t_1;
	} else if (b <= 1.08e+24) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(c, Float64(-z), Float64(a * i)))
	tmp = 0.0
	if (b <= -3800.0)
		tmp = t_1;
	elseif (b <= 1.08e+24)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3800.0], t$95$1, If[LessEqual[b, 1.08e+24], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -3800 or 1.0799999999999999e24 < b

   1. Initial program 75.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \]

      3. remove-double-neg N/A

         \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \]

      9. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \color{blue}{a \cdot i}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot z, a \cdot i\right)} \]

      15. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      16. lower-neg.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(z\right), \color{blue}{i \cdot a}\right) \]

      18. lower-*.f64 70.0

          \[\leadsto b \cdot \mathsf{fma}\left(c, -z, \color{blue}{i \cdot a}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)} \]

   if -3800 < b < 1.0799999999999999e24

   1. Initial program 70.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 39.8

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 39.8%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3800:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{+24}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 29.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (* x y))))
   (if (<= x -2.65e+94) t_1 (if (<= x 2.8e-19) (* a (* b i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (x <= -2.65e+94) {
		tmp = t_1;
	} else if (x <= 2.8e-19) {
		tmp = a * (b * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (x * y)
    if (x <= (-2.65d+94)) then
        tmp = t_1
    else if (x <= 2.8d-19) then
        tmp = a * (b * i)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (x <= -2.65e+94) {
		tmp = t_1;
	} else if (x <= 2.8e-19) {
		tmp = a * (b * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = z * (x * y)
	tmp = 0
	if x <= -2.65e+94:
		tmp = t_1
	elif x <= 2.8e-19:
		tmp = a * (b * i)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * Float64(x * y))
	tmp = 0.0
	if (x <= -2.65e+94)
		tmp = t_1;
	elseif (x <= 2.8e-19)
		tmp = Float64(a * Float64(b * i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = z * (x * y);
	tmp = 0.0;
	if (x <= -2.65e+94)
		tmp = t_1;
	elseif (x <= 2.8e-19)
		tmp = a * (b * i);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.65e+94], t$95$1, If[LessEqual[x, 2.8e-19], N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.65000000000000002e94 or 2.80000000000000003e-19 < x

   1. Initial program 74.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 71.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.1%

      \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.8%

       \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   if -2.65000000000000002e94 < x < 2.80000000000000003e-19

   1. Initial program 72.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 45.4

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 45.4%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto a \cdot \left(b \cdot \color{blue}{i}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 36.1%

      \[\leadsto a \cdot \left(b \cdot \color{blue}{i}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+94}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 28.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -2.2 \cdot 10^{+156}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;j \leq 3.6 \cdot 10^{+189}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -2.2e+156)
   (* c (* t j))
   (if (<= j 3.6e+189) (* z (* x y)) (* j (* t c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -2.2e+156) {
		tmp = c * (t * j);
	} else if (j <= 3.6e+189) {
		tmp = z * (x * y);
	} else {
		tmp = j * (t * c);
	}
	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 <= (-2.2d+156)) then
        tmp = c * (t * j)
    else if (j <= 3.6d+189) then
        tmp = z * (x * y)
    else
        tmp = j * (t * c)
    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 <= -2.2e+156) {
		tmp = c * (t * j);
	} else if (j <= 3.6e+189) {
		tmp = z * (x * y);
	} else {
		tmp = j * (t * c);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if j <= -2.2e+156:
		tmp = c * (t * j)
	elif j <= 3.6e+189:
		tmp = z * (x * y)
	else:
		tmp = j * (t * c)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -2.2e+156)
		tmp = Float64(c * Float64(t * j));
	elseif (j <= 3.6e+189)
		tmp = Float64(z * Float64(x * y));
	else
		tmp = Float64(j * Float64(t * c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (j <= -2.2e+156)
		tmp = c * (t * j);
	elseif (j <= 3.6e+189)
		tmp = z * (x * y);
	else
		tmp = j * (t * c);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -2.2e+156], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 3.6e+189], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if j < -2.20000000000000004e156

   1. Initial program 75.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.4%

      \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   if -2.20000000000000004e156 < j < 3.60000000000000008e189

   1. Initial program 72.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 9.6%

      \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.0%

       \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   if 3.60000000000000008e189 < j

   1. Initial program 71.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.3%

      \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.3%

       \[\leadsto \left(c \cdot t\right) \cdot j \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.2 \cdot 10^{+156}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;j \leq 3.6 \cdot 10^{+189}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 28.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(t \cdot j\right)\\ \mathbf{if}\;j \leq -2.2 \cdot 10^{+156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 4 \cdot 10^{+189}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* t j))))
   (if (<= j -2.2e+156) t_1 (if (<= j 4e+189) (* z (* x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2.2e+156) {
		tmp = t_1;
	} else if (j <= 4e+189) {
		tmp = z * (x * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (t * j)
    if (j <= (-2.2d+156)) then
        tmp = t_1
    else if (j <= 4d+189) then
        tmp = z * (x * y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2.2e+156) {
		tmp = t_1;
	} else if (j <= 4e+189) {
		tmp = z * (x * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (t * j)
	tmp = 0
	if j <= -2.2e+156:
		tmp = t_1
	elif j <= 4e+189:
		tmp = z * (x * y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(t * j))
	tmp = 0.0
	if (j <= -2.2e+156)
		tmp = t_1;
	elseif (j <= 4e+189)
		tmp = Float64(z * Float64(x * y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (t * j);
	tmp = 0.0;
	if (j <= -2.2e+156)
		tmp = t_1;
	elseif (j <= 4e+189)
		tmp = z * (x * y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -2.2e+156], t$95$1, If[LessEqual[j, 4e+189], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -2.20000000000000004e156 or 4.0000000000000001e189 < j

   1. Initial program 74.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 54.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.7%

      \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   if -2.20000000000000004e156 < j < 4.0000000000000001e189

   1. Initial program 72.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 9.6%

      \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.0%

       \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.2 \cdot 10^{+156}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;j \leq 4 \cdot 10^{+189}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 27.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(t \cdot j\right)\\ \mathbf{if}\;j \leq -2.2 \cdot 10^{+156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 2.2 \cdot 10^{+190}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* t j))))
   (if (<= j -2.2e+156) t_1 (if (<= j 2.2e+190) (* x (* y z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2.2e+156) {
		tmp = t_1;
	} else if (j <= 2.2e+190) {
		tmp = x * (y * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (t * j)
    if (j <= (-2.2d+156)) then
        tmp = t_1
    else if (j <= 2.2d+190) then
        tmp = x * (y * z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2.2e+156) {
		tmp = t_1;
	} else if (j <= 2.2e+190) {
		tmp = x * (y * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (t * j)
	tmp = 0
	if j <= -2.2e+156:
		tmp = t_1
	elif j <= 2.2e+190:
		tmp = x * (y * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(t * j))
	tmp = 0.0
	if (j <= -2.2e+156)
		tmp = t_1;
	elseif (j <= 2.2e+190)
		tmp = Float64(x * Float64(y * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (t * j);
	tmp = 0.0;
	if (j <= -2.2e+156)
		tmp = t_1;
	elseif (j <= 2.2e+190)
		tmp = x * (y * z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -2.2e+156], t$95$1, If[LessEqual[j, 2.2e+190], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -2.20000000000000004e156 or 2.2e190 < j

   1. Initial program 74.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 54.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.7%

      \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   if -2.20000000000000004e156 < j < 2.2e190

   1. Initial program 72.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

      7. associate-*l* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      11. sub-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 27.5%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 22.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(t \cdot j\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* c (* t j)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return c * (t * j);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = c * (t * j)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return c * (t * j);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return c * (t * j)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(c * Float64(t * j))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = c * (t * j);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in i around 0

   \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t\right)\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

   4. mul-1-neg N/A

      \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \left(b \cdot \color{blue}{\left(z \cdot c\right)}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + -1 \cdot \color{blue}{\left(\left(b \cdot z\right) \cdot c\right)}\right) \]

   7. associate-*l* N/A

      \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(c \cdot \left(j \cdot t\right) + \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

   8. *-commutative N/A

      \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(j \cdot t\right) \cdot c} + \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c\right) \]

   9. distribute-rgt-in N/A

      \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{c \cdot \left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \left(j \cdot t + \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

   11. sub-neg N/A

       \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + c \cdot \color{blue}{\left(j \cdot t - b \cdot z\right)} \]

   12. +-commutative N/A

       \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(c, j \cdot t - b \cdot z, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

5. Applied rewrites 60.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

6. Taylor expanded in j around inf

   \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
7. Step-by-step derivation

8. Applied rewrites 16.8%

   \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]
9. Add Preprocessing

Developer Target 1: 69.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1015122364899489/125000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -942510763643697/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -238547917063487/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 10535888557455487/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))