Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 72.9% → 83.4%

Time: 16.3s

Alternatives: 29

Speedup: 0.5×

• 57.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 72.9% 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: 83.4% accurate, 0.5× speedup

Math

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

FPCore

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision] - N[(N[(N[(N[(a * t), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - N[(N[(N[(i * a), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(b * z + N[(N[(N[(z * x), $MachinePrecision] * y + N[(N[((-t) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / (-c)), $MachinePrecision]), $MachinePrecision] * (-c)), $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 94.1%

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

   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 j around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in c around -inf

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

   8. Applied rewrites 59.2%

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

4. Final simplification 88.5%

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

Alternative 2: 69.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, z, i \cdot a\right)\\ \mathbf{if}\;j \leq -3.05 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, b, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;j \leq 4.4 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, b, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b + \left(c \cdot t - i \cdot y\right) \cdot j\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- c) z (* i a))))
   (if (<= j -3.05e+81)
     (fma t_1 b (* (fma (- i) y (* c t)) j))
     (if (<= j 4.4e+130)
       (fma t_1 b (* (fma (- a) t (* z y)) x))
       (+ (* (* (- z) c) b) (* (- (* c t) (* i 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(-c, z, (i * a));
	double tmp;
	if (j <= -3.05e+81) {
		tmp = fma(t_1, b, (fma(-i, y, (c * t)) * j));
	} else if (j <= 4.4e+130) {
		tmp = fma(t_1, b, (fma(-a, t, (z * y)) * x));
	} else {
		tmp = ((-z * c) * b) + (((c * t) - (i * y)) * j);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-c), z, Float64(i * a))
	tmp = 0.0
	if (j <= -3.05e+81)
		tmp = fma(t_1, b, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	elseif (j <= 4.4e+130)
		tmp = fma(t_1, b, Float64(fma(Float64(-a), t, Float64(z * y)) * x));
	else
		tmp = Float64(Float64(Float64(Float64(-z) * c) * b) + Float64(Float64(Float64(c * t) - Float64(i * y)) * j));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -3.05e+81], N[(t$95$1 * b + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 4.4e+130], N[(t$95$1 * b + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-z) * c), $MachinePrecision] * b), $MachinePrecision] + N[(N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if j < -3.05000000000000019e81

   1. Initial program 74.1%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 73.2%

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

   if -3.05000000000000019e81 < j < 4.39999999999999987e130

   1. Initial program 80.2%

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

   3. Taylor expanded in j around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 80.1%

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

   if 4.39999999999999987e130 < j

   1. Initial program 80.6%

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

   3. Taylor expanded in c around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -3.05 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot a\right), b, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;j \leq 4.4 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot a\right), b, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b + \left(c \cdot t - i \cdot y\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{if}\;j \leq -3.6 \cdot 10^{+81}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a + t\_1\\ \mathbf{elif}\;j \leq 4.4 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot a\right), b, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (- (* c t) (* i y)) j)))
   (if (<= j -3.6e+81)
     (+ (* (* i b) a) t_1)
     (if (<= j 4.4e+130)
       (fma (fma (- c) z (* i a)) b (* (fma (- a) t (* z y)) x))
       (+ (* (* (- 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 = ((c * t) - (i * y)) * j;
	double tmp;
	if (j <= -3.6e+81) {
		tmp = ((i * b) * a) + t_1;
	} else if (j <= 4.4e+130) {
		tmp = fma(fma(-c, z, (i * a)), b, (fma(-a, t, (z * y)) * x));
	} else {
		tmp = ((-z * c) * b) + t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(c * t) - Float64(i * y)) * j)
	tmp = 0.0
	if (j <= -3.6e+81)
		tmp = Float64(Float64(Float64(i * b) * a) + t_1);
	elseif (j <= 4.4e+130)
		tmp = fma(fma(Float64(-c), z, Float64(i * a)), b, Float64(fma(Float64(-a), t, Float64(z * y)) * x));
	else
		tmp = Float64(Float64(Float64(Float64(-z) * c) * b) + t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -3.6e+81], N[(N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[j, 4.4e+130], N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-z) * c), $MachinePrecision] * b), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if j < -3.60000000000000005e81

   1. Initial program 74.1%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

   if -3.60000000000000005e81 < j < 4.39999999999999987e130

   1. Initial program 80.2%

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

   3. Taylor expanded in j around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 80.1%

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

   if 4.39999999999999987e130 < j

   1. Initial program 80.6%

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

   3. Taylor expanded in c around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -3.6 \cdot 10^{+81}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{elif}\;j \leq 4.4 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot a\right), b, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b + \left(c \cdot t - i \cdot y\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-177}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot a\right), b, \left(c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;x \leq 10^{-28}:\\ \;\;\;\;\left(\left(\frac{b \cdot a}{y} - j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= x -1.55e+90)
     t_1
     (if (<= x -2.9e-177)
       (fma (fma (- c) z (* i a)) b (* (* c t) j))
       (if (<= x 1e-28) (* (* (- (/ (* b a) y) j) y) i) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -1.55e+90) {
		tmp = t_1;
	} else if (x <= -2.9e-177) {
		tmp = fma(fma(-c, z, (i * a)), b, ((c * t) * j));
	} else if (x <= 1e-28) {
		tmp = ((((b * a) / y) - j) * y) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -1.55e+90)
		tmp = t_1;
	elseif (x <= -2.9e-177)
		tmp = fma(fma(Float64(-c), z, Float64(i * a)), b, Float64(Float64(c * t) * j));
	elseif (x <= 1e-28)
		tmp = Float64(Float64(Float64(Float64(Float64(b * a) / y) - j) * y) * i);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.55e+90], t$95$1, If[LessEqual[x, -2.9e-177], N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b + N[(N[(c * t), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-28], N[(N[(N[(N[(N[(b * a), $MachinePrecision] / y), $MachinePrecision] - j), $MachinePrecision] * y), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.54999999999999994e90 or 9.99999999999999971e-29 < x

   1. Initial program 77.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 67.9

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

   5. Applied rewrites 67.9%

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

   if -1.54999999999999994e90 < x < -2.89999999999999997e-177

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 61.7%

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

   if -2.89999999999999997e-177 < x < 9.99999999999999971e-29

   1. Initial program 76.2%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 60.3%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-177}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot a\right), b, \left(c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;x \leq 10^{-28}:\\ \;\;\;\;\left(\left(\frac{b \cdot a}{y} - j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(\frac{b \cdot a}{y} - j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)))
   (if (<= z -3.8e+80)
     t_1
     (if (<= z 3.6e-293)
       (* (fma (- x) t (* i b)) a)
       (if (<= z 1.4e+80) (* (* (- (/ (* b a) y) j) y) i) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x)) * z;
	double tmp;
	if (z <= -3.8e+80) {
		tmp = t_1;
	} else if (z <= 3.6e-293) {
		tmp = fma(-x, t, (i * b)) * a;
	} else if (z <= 1.4e+80) {
		tmp = ((((b * a) / y) - j) * y) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-b), c, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -3.8e+80)
		tmp = t_1;
	elseif (z <= 3.6e-293)
		tmp = Float64(fma(Float64(-x), t, Float64(i * b)) * a);
	elseif (z <= 1.4e+80)
		tmp = Float64(Float64(Float64(Float64(Float64(b * a) / y) - j) * y) * i);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.8e+80], t$95$1, If[LessEqual[z, 3.6e-293], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 1.4e+80], N[(N[(N[(N[(N[(b * a), $MachinePrecision] / y), $MachinePrecision] - j), $MachinePrecision] * y), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.79999999999999997e80 or 1.39999999999999992e80 < z

   1. Initial program 73.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -3.79999999999999997e80 < z < 3.59999999999999985e-293

   1. Initial program 85.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 3.59999999999999985e-293 < z < 1.39999999999999992e80

   1. Initial program 79.5%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 57.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(\frac{b \cdot a}{y} - j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(\frac{b}{y} \cdot a - j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)))
   (if (<= z -3.8e+80)
     t_1
     (if (<= z 3.6e-293)
       (* (fma (- x) t (* i b)) a)
       (if (<= z 1.4e+80) (* (* (- (* (/ b y) a) j) y) i) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x)) * z;
	double tmp;
	if (z <= -3.8e+80) {
		tmp = t_1;
	} else if (z <= 3.6e-293) {
		tmp = fma(-x, t, (i * b)) * a;
	} else if (z <= 1.4e+80) {
		tmp = ((((b / y) * a) - j) * y) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-b), c, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -3.8e+80)
		tmp = t_1;
	elseif (z <= 3.6e-293)
		tmp = Float64(fma(Float64(-x), t, Float64(i * b)) * a);
	elseif (z <= 1.4e+80)
		tmp = Float64(Float64(Float64(Float64(Float64(b / y) * a) - j) * y) * i);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.8e+80], t$95$1, If[LessEqual[z, 3.6e-293], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 1.4e+80], N[(N[(N[(N[(N[(b / y), $MachinePrecision] * a), $MachinePrecision] - j), $MachinePrecision] * y), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.79999999999999997e80 or 1.39999999999999992e80 < z

   1. Initial program 73.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -3.79999999999999997e80 < z < 3.59999999999999985e-293

   1. Initial program 85.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 3.59999999999999985e-293 < z < 1.39999999999999992e80

   1. Initial program 79.5%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 57.1%

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

   10. Applied rewrites 57.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(\frac{b}{y} \cdot a - j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\\ \mathbf{if}\;i \leq -1.02 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 3.9 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\left(-z\right) \cdot c, b, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- y) j (* b a)) i)))
   (if (<= i -1.02e+18)
     t_1
     (if (<= i 3.9e+118)
       (fma (* (- z) c) b (* (fma (- a) t (* z y)) x))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-y, j, (b * a)) * i;
	double tmp;
	if (i <= -1.02e+18) {
		tmp = t_1;
	} else if (i <= 3.9e+118) {
		tmp = fma((-z * c), b, (fma(-a, t, (z * y)) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-y), j, Float64(b * a)) * i)
	tmp = 0.0
	if (i <= -1.02e+18)
		tmp = t_1;
	elseif (i <= 3.9e+118)
		tmp = fma(Float64(Float64(-z) * c), b, Float64(fma(Float64(-a), t, Float64(z * y)) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[i, -1.02e+18], t$95$1, If[LessEqual[i, 3.9e+118], N[(N[((-z) * c), $MachinePrecision] * b + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.02e18 or 3.9e118 < i

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   if -1.02e18 < i < 3.9e118

   1. Initial program 79.0%

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

   3. Taylor expanded in j around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 73.5%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 68.0%

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

Alternative 8: 29.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-x\right) \cdot t\right) \cdot a\\ t_2 := \left(b \cdot a\right) \cdot i\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{+245}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq -1.5 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-34}:\\ \;\;\;\;\left(i \cdot a\right) \cdot b\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- x) t) a)) (t_2 (* (* b a) i)))
   (if (<= a -1.1e+245)
     t_2
     (if (<= a -1.5e+36)
       t_1
       (if (<= a -1e-34)
         (* (* i a) b)
         (if (<= a 1.6e-11) (* (* z y) x) (if (<= a 6.2e+121) 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 * t) * a;
	double t_2 = (b * a) * i;
	double tmp;
	if (a <= -1.1e+245) {
		tmp = t_2;
	} else if (a <= -1.5e+36) {
		tmp = t_1;
	} else if (a <= -1e-34) {
		tmp = (i * a) * b;
	} else if (a <= 1.6e-11) {
		tmp = (z * y) * x;
	} else if (a <= 6.2e+121) {
		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 * t) * a
    t_2 = (b * a) * i
    if (a <= (-1.1d+245)) then
        tmp = t_2
    else if (a <= (-1.5d+36)) then
        tmp = t_1
    else if (a <= (-1d-34)) then
        tmp = (i * a) * b
    else if (a <= 1.6d-11) then
        tmp = (z * y) * x
    else if (a <= 6.2d+121) 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 * t) * a;
	double t_2 = (b * a) * i;
	double tmp;
	if (a <= -1.1e+245) {
		tmp = t_2;
	} else if (a <= -1.5e+36) {
		tmp = t_1;
	} else if (a <= -1e-34) {
		tmp = (i * a) * b;
	} else if (a <= 1.6e-11) {
		tmp = (z * y) * x;
	} else if (a <= 6.2e+121) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-x * t) * a
	t_2 = (b * a) * i
	tmp = 0
	if a <= -1.1e+245:
		tmp = t_2
	elif a <= -1.5e+36:
		tmp = t_1
	elif a <= -1e-34:
		tmp = (i * a) * b
	elif a <= 1.6e-11:
		tmp = (z * y) * x
	elif a <= 6.2e+121:
		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) * t) * a)
	t_2 = Float64(Float64(b * a) * i)
	tmp = 0.0
	if (a <= -1.1e+245)
		tmp = t_2;
	elseif (a <= -1.5e+36)
		tmp = t_1;
	elseif (a <= -1e-34)
		tmp = Float64(Float64(i * a) * b);
	elseif (a <= 1.6e-11)
		tmp = Float64(Float64(z * y) * x);
	elseif (a <= 6.2e+121)
		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 * t) * a;
	t_2 = (b * a) * i;
	tmp = 0.0;
	if (a <= -1.1e+245)
		tmp = t_2;
	elseif (a <= -1.5e+36)
		tmp = t_1;
	elseif (a <= -1e-34)
		tmp = (i * a) * b;
	elseif (a <= 1.6e-11)
		tmp = (z * y) * x;
	elseif (a <= 6.2e+121)
		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[((-x) * t), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[a, -1.1e+245], t$95$2, If[LessEqual[a, -1.5e+36], t$95$1, If[LessEqual[a, -1e-34], N[(N[(i * a), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[a, 1.6e-11], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[a, 6.2e+121], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.1e245 or 6.20000000000000016e121 < a

   1. Initial program 72.7%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 54.4%

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

   if -1.1e245 < a < -1.5e36 or 1.59999999999999997e-11 < a < 6.20000000000000016e121

   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 j around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 45.3%

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

   if -1.5e36 < a < -9.99999999999999928e-35

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 43.9%

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

   10. Applied rewrites 43.9%

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

   if -9.99999999999999928e-35 < a < 1.59999999999999997e-11

   1. Initial program 86.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 35.3%

      \[\leadsto \left(z \cdot y\right) \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+245}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;a \leq -1.5 \cdot 10^{+36}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-34}:\\ \;\;\;\;\left(i \cdot a\right) \cdot b\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+121}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(\left(-y\right) \cdot j, i, \left(b \cdot a\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)))
   (if (<= z -3.8e+80)
     t_1
     (if (<= z 3.6e-293)
       (* (fma (- x) t (* i b)) a)
       (if (<= z 1.4e+80) (fma (* (- y) j) i (* (* b a) i)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x)) * z;
	double tmp;
	if (z <= -3.8e+80) {
		tmp = t_1;
	} else if (z <= 3.6e-293) {
		tmp = fma(-x, t, (i * b)) * a;
	} else if (z <= 1.4e+80) {
		tmp = fma((-y * j), i, ((b * a) * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-b), c, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -3.8e+80)
		tmp = t_1;
	elseif (z <= 3.6e-293)
		tmp = Float64(fma(Float64(-x), t, Float64(i * b)) * a);
	elseif (z <= 1.4e+80)
		tmp = fma(Float64(Float64(-y) * j), i, Float64(Float64(b * a) * i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.8e+80], t$95$1, If[LessEqual[z, 3.6e-293], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 1.4e+80], N[(N[((-y) * j), $MachinePrecision] * i + N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.79999999999999997e80 or 1.39999999999999992e80 < z

   1. Initial program 73.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -3.79999999999999997e80 < z < 3.59999999999999985e-293

   1. Initial program 85.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 3.59999999999999985e-293 < z < 1.39999999999999992e80

   1. Initial program 79.5%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 57.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(\left(-y\right) \cdot j, i, \left(b \cdot a\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot x\\ \mathbf{if}\;z \leq -7.4 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(-y\right) \cdot j\right) \cdot i\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+249}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z y) x)))
   (if (<= z -7.4e+80)
     t_1
     (if (<= z 4.8e-293)
       (* (* (- x) t) a)
       (if (<= z 1.05e+80)
         (* (* (- y) j) i)
         (if (<= z 2.5e+249) t_1 (* (* (- z) c) b)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * y) * x;
	double tmp;
	if (z <= -7.4e+80) {
		tmp = t_1;
	} else if (z <= 4.8e-293) {
		tmp = (-x * t) * a;
	} else if (z <= 1.05e+80) {
		tmp = (-y * j) * i;
	} else if (z <= 2.5e+249) {
		tmp = t_1;
	} else {
		tmp = (-z * c) * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * y) * x
    if (z <= (-7.4d+80)) then
        tmp = t_1
    else if (z <= 4.8d-293) then
        tmp = (-x * t) * a
    else if (z <= 1.05d+80) then
        tmp = (-y * j) * i
    else if (z <= 2.5d+249) then
        tmp = t_1
    else
        tmp = (-z * c) * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * y) * x;
	double tmp;
	if (z <= -7.4e+80) {
		tmp = t_1;
	} else if (z <= 4.8e-293) {
		tmp = (-x * t) * a;
	} else if (z <= 1.05e+80) {
		tmp = (-y * j) * i;
	} else if (z <= 2.5e+249) {
		tmp = t_1;
	} else {
		tmp = (-z * c) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * y) * x
	tmp = 0
	if z <= -7.4e+80:
		tmp = t_1
	elif z <= 4.8e-293:
		tmp = (-x * t) * a
	elif z <= 1.05e+80:
		tmp = (-y * j) * i
	elif z <= 2.5e+249:
		tmp = t_1
	else:
		tmp = (-z * c) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * y) * x)
	tmp = 0.0
	if (z <= -7.4e+80)
		tmp = t_1;
	elseif (z <= 4.8e-293)
		tmp = Float64(Float64(Float64(-x) * t) * a);
	elseif (z <= 1.05e+80)
		tmp = Float64(Float64(Float64(-y) * j) * i);
	elseif (z <= 2.5e+249)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-z) * c) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * y) * x;
	tmp = 0.0;
	if (z <= -7.4e+80)
		tmp = t_1;
	elseif (z <= 4.8e-293)
		tmp = (-x * t) * a;
	elseif (z <= 1.05e+80)
		tmp = (-y * j) * i;
	elseif (z <= 2.5e+249)
		tmp = t_1;
	else
		tmp = (-z * c) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -7.4e+80], t$95$1, If[LessEqual[z, 4.8e-293], N[(N[((-x) * t), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 1.05e+80], N[(N[((-y) * j), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[z, 2.5e+249], t$95$1, N[(N[((-z) * c), $MachinePrecision] * b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.39999999999999992e80 or 1.05000000000000001e80 < z < 2.4999999999999998e249

   1. Initial program 73.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 52.9%

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

   if -7.39999999999999992e80 < z < 4.7999999999999998e-293

   1. Initial program 85.5%

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

   3. Taylor expanded in j around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 34.3%

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

   if 4.7999999999999998e-293 < z < 1.05000000000000001e80

   1. Initial program 79.5%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 41.2%

      \[\leadsto \left(\left(-j\right) \cdot y\right) \cdot i \]

   if 2.4999999999999998e249 < z

   1. Initial program 73.8%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 42.0%

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

   10. Applied rewrites 61.1%

       \[\leadsto \left(c \cdot z\right) \cdot \left(-b\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+80}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(-y\right) \cdot j\right) \cdot i\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+249}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot x\\ \mathbf{if}\;z \leq -1.48 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-263}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(-y\right) \cdot i\right) \cdot j\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+249}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z y) x)))
   (if (<= z -1.48e+110)
     t_1
     (if (<= z 2.6e-263)
       (* (* b a) i)
       (if (<= z 1.05e+80)
         (* (* (- y) i) j)
         (if (<= z 2.5e+249) t_1 (* (* (- z) c) b)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * y) * x;
	double tmp;
	if (z <= -1.48e+110) {
		tmp = t_1;
	} else if (z <= 2.6e-263) {
		tmp = (b * a) * i;
	} else if (z <= 1.05e+80) {
		tmp = (-y * i) * j;
	} else if (z <= 2.5e+249) {
		tmp = t_1;
	} else {
		tmp = (-z * c) * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * y) * x
    if (z <= (-1.48d+110)) then
        tmp = t_1
    else if (z <= 2.6d-263) then
        tmp = (b * a) * i
    else if (z <= 1.05d+80) then
        tmp = (-y * i) * j
    else if (z <= 2.5d+249) then
        tmp = t_1
    else
        tmp = (-z * c) * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * y) * x;
	double tmp;
	if (z <= -1.48e+110) {
		tmp = t_1;
	} else if (z <= 2.6e-263) {
		tmp = (b * a) * i;
	} else if (z <= 1.05e+80) {
		tmp = (-y * i) * j;
	} else if (z <= 2.5e+249) {
		tmp = t_1;
	} else {
		tmp = (-z * c) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * y) * x
	tmp = 0
	if z <= -1.48e+110:
		tmp = t_1
	elif z <= 2.6e-263:
		tmp = (b * a) * i
	elif z <= 1.05e+80:
		tmp = (-y * i) * j
	elif z <= 2.5e+249:
		tmp = t_1
	else:
		tmp = (-z * c) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * y) * x)
	tmp = 0.0
	if (z <= -1.48e+110)
		tmp = t_1;
	elseif (z <= 2.6e-263)
		tmp = Float64(Float64(b * a) * i);
	elseif (z <= 1.05e+80)
		tmp = Float64(Float64(Float64(-y) * i) * j);
	elseif (z <= 2.5e+249)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-z) * c) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * y) * x;
	tmp = 0.0;
	if (z <= -1.48e+110)
		tmp = t_1;
	elseif (z <= 2.6e-263)
		tmp = (b * a) * i;
	elseif (z <= 1.05e+80)
		tmp = (-y * i) * j;
	elseif (z <= 2.5e+249)
		tmp = t_1;
	else
		tmp = (-z * c) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -1.48e+110], t$95$1, If[LessEqual[z, 2.6e-263], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[z, 1.05e+80], N[(N[((-y) * i), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, 2.5e+249], t$95$1, N[(N[((-z) * c), $MachinePrecision] * b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.48000000000000008e110 or 1.05000000000000001e80 < z < 2.4999999999999998e249

   1. Initial program 70.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 56.9%

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

   if -1.48000000000000008e110 < z < 2.6e-263

   1. Initial program 86.6%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 31.9%

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

   if 2.6e-263 < z < 1.05000000000000001e80

   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 j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 46.7

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

   5. Applied rewrites 46.7%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 36.4%

      \[\leadsto \left(\left(-y\right) \cdot i\right) \cdot j \]

   if 2.4999999999999998e249 < z

   1. Initial program 73.8%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 42.0%

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

   10. Applied rewrites 61.1%

       \[\leadsto \left(c \cdot z\right) \cdot \left(-b\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.48 \cdot 10^{+110}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-263}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(-y\right) \cdot i\right) \cdot j\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+249}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot x\\ \mathbf{if}\;z \leq -7.4 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+79}:\\ \;\;\;\;\left(\left(-j\right) \cdot i\right) \cdot y\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+249}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z y) x)))
   (if (<= z -7.4e+80)
     t_1
     (if (<= z 4.8e-293)
       (* (* (- x) t) a)
       (if (<= z 9e+79)
         (* (* (- j) i) y)
         (if (<= z 2.5e+249) t_1 (* (* (- z) c) b)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * y) * x;
	double tmp;
	if (z <= -7.4e+80) {
		tmp = t_1;
	} else if (z <= 4.8e-293) {
		tmp = (-x * t) * a;
	} else if (z <= 9e+79) {
		tmp = (-j * i) * y;
	} else if (z <= 2.5e+249) {
		tmp = t_1;
	} else {
		tmp = (-z * c) * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * y) * x
    if (z <= (-7.4d+80)) then
        tmp = t_1
    else if (z <= 4.8d-293) then
        tmp = (-x * t) * a
    else if (z <= 9d+79) then
        tmp = (-j * i) * y
    else if (z <= 2.5d+249) then
        tmp = t_1
    else
        tmp = (-z * c) * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * y) * x;
	double tmp;
	if (z <= -7.4e+80) {
		tmp = t_1;
	} else if (z <= 4.8e-293) {
		tmp = (-x * t) * a;
	} else if (z <= 9e+79) {
		tmp = (-j * i) * y;
	} else if (z <= 2.5e+249) {
		tmp = t_1;
	} else {
		tmp = (-z * c) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * y) * x
	tmp = 0
	if z <= -7.4e+80:
		tmp = t_1
	elif z <= 4.8e-293:
		tmp = (-x * t) * a
	elif z <= 9e+79:
		tmp = (-j * i) * y
	elif z <= 2.5e+249:
		tmp = t_1
	else:
		tmp = (-z * c) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * y) * x)
	tmp = 0.0
	if (z <= -7.4e+80)
		tmp = t_1;
	elseif (z <= 4.8e-293)
		tmp = Float64(Float64(Float64(-x) * t) * a);
	elseif (z <= 9e+79)
		tmp = Float64(Float64(Float64(-j) * i) * y);
	elseif (z <= 2.5e+249)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-z) * c) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * y) * x;
	tmp = 0.0;
	if (z <= -7.4e+80)
		tmp = t_1;
	elseif (z <= 4.8e-293)
		tmp = (-x * t) * a;
	elseif (z <= 9e+79)
		tmp = (-j * i) * y;
	elseif (z <= 2.5e+249)
		tmp = t_1;
	else
		tmp = (-z * c) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -7.4e+80], t$95$1, If[LessEqual[z, 4.8e-293], N[(N[((-x) * t), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 9e+79], N[(N[((-j) * i), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 2.5e+249], t$95$1, N[(N[((-z) * c), $MachinePrecision] * b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.39999999999999992e80 or 8.99999999999999987e79 < z < 2.4999999999999998e249

   1. Initial program 73.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 52.9%

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

   if -7.39999999999999992e80 < z < 4.7999999999999998e-293

   1. Initial program 85.5%

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

   3. Taylor expanded in j around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 34.3%

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

   if 4.7999999999999998e-293 < z < 8.99999999999999987e79

   1. Initial program 79.5%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 20.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 33.9%

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

   if 2.4999999999999998e249 < z

   1. Initial program 73.8%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 42.0%

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

   10. Applied rewrites 61.1%

       \[\leadsto \left(c \cdot z\right) \cdot \left(-b\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+80}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+79}:\\ \;\;\;\;\left(\left(-j\right) \cdot i\right) \cdot y\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+249}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)))
   (if (<= z -3.8e+80)
     t_1
     (if (<= z 3.6e-293)
       (* (fma (- x) t (* i b)) a)
       (if (<= z 1.4e+80) (* (fma (- y) j (* b a)) i) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x)) * z;
	double tmp;
	if (z <= -3.8e+80) {
		tmp = t_1;
	} else if (z <= 3.6e-293) {
		tmp = fma(-x, t, (i * b)) * a;
	} else if (z <= 1.4e+80) {
		tmp = fma(-y, j, (b * a)) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-b), c, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -3.8e+80)
		tmp = t_1;
	elseif (z <= 3.6e-293)
		tmp = Float64(fma(Float64(-x), t, Float64(i * b)) * a);
	elseif (z <= 1.4e+80)
		tmp = Float64(fma(Float64(-y), j, Float64(b * a)) * i);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.8e+80], t$95$1, If[LessEqual[z, 3.6e-293], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 1.4e+80], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.79999999999999997e80 or 1.39999999999999992e80 < z

   1. Initial program 73.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -3.79999999999999997e80 < z < 3.59999999999999985e-293

   1. Initial program 85.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 3.59999999999999985e-293 < z < 1.39999999999999992e80

   1. Initial program 79.5%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-257}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)))
   (if (<= z -3.8e+80)
     t_1
     (if (<= z 3.5e-257)
       (* (fma (- x) t (* i b)) a)
       (if (<= z 1.75e-32) (* (fma (- i) y (* c t)) j) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x)) * z;
	double tmp;
	if (z <= -3.8e+80) {
		tmp = t_1;
	} else if (z <= 3.5e-257) {
		tmp = fma(-x, t, (i * b)) * a;
	} else if (z <= 1.75e-32) {
		tmp = fma(-i, y, (c * t)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-b), c, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -3.8e+80)
		tmp = t_1;
	elseif (z <= 3.5e-257)
		tmp = Float64(fma(Float64(-x), t, Float64(i * b)) * a);
	elseif (z <= 1.75e-32)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.8e+80], t$95$1, If[LessEqual[z, 3.5e-257], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 1.75e-32], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.79999999999999997e80 or 1.7499999999999999e-32 < z

   1. Initial program 74.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if -3.79999999999999997e80 < z < 3.50000000000000029e-257

   1. Initial program 85.0%

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

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 3.50000000000000029e-257 < z < 1.7499999999999999e-32

   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 j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-257}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- j) i (* z x)) y)))
   (if (<= y -7.5e-82)
     t_1
     (if (<= y -1.8e-140)
       (* (fma (- x) a (* j c)) t)
       (if (<= y 1.36e+63) (* (fma (- c) z (* i a)) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-j, i, (z * x)) * y;
	double tmp;
	if (y <= -7.5e-82) {
		tmp = t_1;
	} else if (y <= -1.8e-140) {
		tmp = fma(-x, a, (j * c)) * t;
	} else if (y <= 1.36e+63) {
		tmp = fma(-c, z, (i * a)) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-j), i, Float64(z * x)) * y)
	tmp = 0.0
	if (y <= -7.5e-82)
		tmp = t_1;
	elseif (y <= -1.8e-140)
		tmp = Float64(fma(Float64(-x), a, Float64(j * c)) * t);
	elseif (y <= 1.36e+63)
		tmp = Float64(fma(Float64(-c), z, Float64(i * a)) * b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-j) * i + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -7.5e-82], t$95$1, If[LessEqual[y, -1.8e-140], N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[y, 1.36e+63], N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.4999999999999997e-82 or 1.36000000000000006e63 < y

   1. Initial program 73.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 65.8

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

   5. Applied rewrites 65.8%

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

   if -7.4999999999999997e-82 < y < -1.8e-140

   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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   if -1.8e-140 < y < 1.36000000000000006e63

   1. Initial program 87.4%

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- j) i (* z x)) y)))
   (if (<= y -2.3e-17)
     t_1
     (if (<= y -8.2e-141)
       (* (fma (- a) t (* z y)) x)
       (if (<= y 1.36e+63) (* (fma (- c) z (* i a)) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-j, i, (z * x)) * y;
	double tmp;
	if (y <= -2.3e-17) {
		tmp = t_1;
	} else if (y <= -8.2e-141) {
		tmp = fma(-a, t, (z * y)) * x;
	} else if (y <= 1.36e+63) {
		tmp = fma(-c, z, (i * a)) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-j), i, Float64(z * x)) * y)
	tmp = 0.0
	if (y <= -2.3e-17)
		tmp = t_1;
	elseif (y <= -8.2e-141)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	elseif (y <= 1.36e+63)
		tmp = Float64(fma(Float64(-c), z, Float64(i * a)) * b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-j) * i + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.3e-17], t$95$1, If[LessEqual[y, -8.2e-141], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 1.36e+63], N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.30000000000000009e-17 or 1.36000000000000006e63 < y

   1. Initial program 71.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(j \cdot i\right)} + x \cdot z\right) \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right) \cdot i} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 67.2

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   if -2.30000000000000009e-17 < y < -8.20000000000000005e-141

   1. Initial program 80.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 58.8

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   if -8.20000000000000005e-141 < y < 1.36000000000000006e63

   1. Initial program 87.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot i - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      13. neg-mul-1 N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      14. remove-double-neg N/A

          \[\leadsto \left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}\right) \cdot b \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)} \cdot b \]

      16. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right) \cdot b \]

      17. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right) \cdot b \]

      18. lower-*.f64 52.1

          \[\leadsto \mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right) \cdot b \]

   5. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, a \cdot i\right) \cdot b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 51.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -5.3 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= x -5.3e-43)
     t_1
     (if (<= x 1.4e-160)
       (* (fma (- c) z (* i a)) b)
       (if (<= x 2e-15) (* (fma (- i) y (* c t)) j) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -5.3e-43) {
		tmp = t_1;
	} else if (x <= 1.4e-160) {
		tmp = fma(-c, z, (i * a)) * b;
	} else if (x <= 2e-15) {
		tmp = fma(-i, y, (c * t)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -5.3e-43)
		tmp = t_1;
	elseif (x <= 1.4e-160)
		tmp = Float64(fma(Float64(-c), z, Float64(i * a)) * b);
	elseif (x <= 2e-15)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -5.3e-43], t$95$1, If[LessEqual[x, 1.4e-160], N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[x, 2e-15], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.3000000000000003e-43 or 2.0000000000000002e-15 < x

   1. Initial program 79.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 62.4

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 62.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   if -5.3000000000000003e-43 < x < 1.40000000000000008e-160

   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 b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot i - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      13. neg-mul-1 N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      14. remove-double-neg N/A

          \[\leadsto \left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}\right) \cdot b \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)} \cdot b \]

      16. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right) \cdot b \]

      17. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right) \cdot b \]

      18. lower-*.f64 53.5

          \[\leadsto \mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right) \cdot b \]

   5. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, a \cdot i\right) \cdot b} \]

   if 1.40000000000000008e-160 < x < 2.0000000000000002e-15

   1. Initial program 84.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right)} \cdot j \]

      5. neg-mul-1 N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + c \cdot t\right) \cdot j \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, y, c \cdot t\right)} \cdot j \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, c \cdot t\right) \cdot j \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, y, c \cdot t\right) \cdot j \]

      9. lower-*.f64 55.2

         \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot t}\right) \cdot j \]

   5. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 30.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot x\\ \mathbf{if}\;z \leq -7.4 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+79}:\\ \;\;\;\;\left(\left(-j\right) \cdot i\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z y) x)))
   (if (<= z -7.4e+80)
     t_1
     (if (<= z 4.8e-293)
       (* (* (- x) t) a)
       (if (<= z 9e+79) (* (* (- j) i) 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 = (z * y) * x;
	double tmp;
	if (z <= -7.4e+80) {
		tmp = t_1;
	} else if (z <= 4.8e-293) {
		tmp = (-x * t) * a;
	} else if (z <= 9e+79) {
		tmp = (-j * i) * 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 = (z * y) * x
    if (z <= (-7.4d+80)) then
        tmp = t_1
    else if (z <= 4.8d-293) then
        tmp = (-x * t) * a
    else if (z <= 9d+79) then
        tmp = (-j * i) * 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 = (z * y) * x;
	double tmp;
	if (z <= -7.4e+80) {
		tmp = t_1;
	} else if (z <= 4.8e-293) {
		tmp = (-x * t) * a;
	} else if (z <= 9e+79) {
		tmp = (-j * i) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * y) * x
	tmp = 0
	if z <= -7.4e+80:
		tmp = t_1
	elif z <= 4.8e-293:
		tmp = (-x * t) * a
	elif z <= 9e+79:
		tmp = (-j * i) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * y) * x)
	tmp = 0.0
	if (z <= -7.4e+80)
		tmp = t_1;
	elseif (z <= 4.8e-293)
		tmp = Float64(Float64(Float64(-x) * t) * a);
	elseif (z <= 9e+79)
		tmp = Float64(Float64(Float64(-j) * i) * 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 = (z * y) * x;
	tmp = 0.0;
	if (z <= -7.4e+80)
		tmp = t_1;
	elseif (z <= 4.8e-293)
		tmp = (-x * t) * a;
	elseif (z <= 9e+79)
		tmp = (-j * i) * 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[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -7.4e+80], t$95$1, If[LessEqual[z, 4.8e-293], N[(N[((-x) * t), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 9e+79], N[(N[((-j) * i), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.39999999999999992e80 or 8.99999999999999987e79 < z

   1. Initial program 73.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 57.4

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 49.1%

      \[\leadsto \left(z \cdot y\right) \cdot x \]

   if -7.39999999999999992e80 < z < 4.7999999999999998e-293

   1. Initial program 85.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around 0

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right) \cdot b}\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b} + x \cdot \left(y \cdot z - a \cdot t\right) \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b + x \cdot \left(y \cdot z - a \cdot t\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(c \cdot z - a \cdot i\right), b, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)}, b, x \cdot \left(y \cdot z - a \cdot t\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right), b, x \cdot \left(y \cdot z - a \cdot t\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}, b, x \cdot \left(y \cdot z - a \cdot t\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, x \cdot \left(y \cdot z - a \cdot t\right)\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, x \cdot \left(y \cdot z - a \cdot t\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}, b, x \cdot \left(y \cdot z - a \cdot t\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)}, b, x \cdot \left(y \cdot z - a \cdot t\right)\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right), b, x \cdot \left(y \cdot z - a \cdot t\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right), b, x \cdot \left(y \cdot z - a \cdot t\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right), b, x \cdot \left(y \cdot z - a \cdot t\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x}\right) \]

   5. Applied rewrites 74.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto -1 \cdot \color{blue}{\left(a \cdot \left(t \cdot x\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.3%

      \[\leadsto \left(\left(-t\right) \cdot x\right) \cdot \color{blue}{a} \]

   if 4.7999999999999998e-293 < z < 8.99999999999999987e79

   1. Initial program 79.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(c \cdot z - a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 20.2%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]

   9. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\left(i \cdot \left(j \cdot y\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 33.9%

       \[\leadsto \left(-y\right) \cdot \color{blue}{\left(j \cdot i\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+80}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+79}:\\ \;\;\;\;\left(\left(-j\right) \cdot i\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 51.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -5.3 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= x -5.3e-43)
     t_1
     (if (<= x 1.8e-42) (* (fma (- c) z (* i a)) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -5.3e-43) {
		tmp = t_1;
	} else if (x <= 1.8e-42) {
		tmp = fma(-c, z, (i * a)) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -5.3e-43)
		tmp = t_1;
	elseif (x <= 1.8e-42)
		tmp = Float64(fma(Float64(-c), z, Float64(i * a)) * b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -5.3e-43], t$95$1, If[LessEqual[x, 1.8e-42], N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.3000000000000003e-43 or 1.8000000000000001e-42 < x

   1. Initial program 79.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. 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. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 60.5

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   if -5.3000000000000003e-43 < x < 1.8000000000000001e-42

   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 b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot i - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      13. neg-mul-1 N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      14. remove-double-neg N/A

          \[\leadsto \left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}\right) \cdot b \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)} \cdot b \]

      16. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right) \cdot b \]

      17. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right) \cdot b \]

      18. lower-*.f64 51.2

          \[\leadsto \mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right) \cdot b \]

   5. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, a \cdot i\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ \mathbf{if}\;c \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 8.4 \cdot 10^{+182}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) z (* j t)) c)))
   (if (<= c -5.5e-9)
     t_1
     (if (<= c 8.4e+182) (* (fma (- a) t (* z y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, z, (j * t)) * c;
	double tmp;
	if (c <= -5.5e-9) {
		tmp = t_1;
	} else if (c <= 8.4e+182) {
		tmp = fma(-a, t, (z * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-b), z, Float64(j * t)) * c)
	tmp = 0.0
	if (c <= -5.5e-9)
		tmp = t_1;
	elseif (c <= 8.4e+182)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * z + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[c, -5.5e-9], t$95$1, If[LessEqual[c, 8.4e+182], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.4999999999999996e-9 or 8.3999999999999996e182 < c

   1. Initial program 70.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot t - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(j \cdot t - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \cdot c \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot z} + j \cdot t\right) \cdot c \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, z, j \cdot t\right)} \cdot c \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, z, j \cdot t\right) \cdot c \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, z, j \cdot t\right) \cdot c \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, z, \color{blue}{t \cdot j}\right) \cdot c \]

      11. lower-*.f64 64.7

          \[\leadsto \mathsf{fma}\left(-b, z, \color{blue}{t \cdot j}\right) \cdot c \]

   5. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, z, t \cdot j\right) \cdot c} \]

   if -5.4999999999999996e-9 < c < 8.3999999999999996e182

   1. Initial program 83.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 49.2

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ \mathbf{elif}\;c \leq 8.4 \cdot 10^{+182}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 45.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;t \leq -4.1 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= t -4.1e+29)
     t_1
     (if (<= t 3.5e-117) (* (fma (- b) c (* y x)) z) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (t <= -4.1e+29) {
		tmp = t_1;
	} else if (t <= 3.5e-117) {
		tmp = fma(-b, c, (y * x)) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	tmp = 0.0
	if (t <= -4.1e+29)
		tmp = t_1;
	elseif (t <= 3.5e-117)
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -4.1e+29], t$95$1, If[LessEqual[t, 3.5e-117], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.1000000000000003e29 or 3.4999999999999998e-117 < t

   1. Initial program 72.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 55.1

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 55.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   if -4.1000000000000003e29 < t < 3.4999999999999998e-117

   1. Initial program 87.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      10. lower-*.f64 52.1

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{x \cdot y}\right) \cdot z \]

   5. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, x \cdot y\right) \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 29.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-56}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+87}:\\ \;\;\;\;\left(c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -3.4e-56)
   (* (* z y) x)
   (if (<= y 3.5e-119)
     (* (* b a) i)
     (if (<= y 3.5e+87) (* (* c t) j) (* (* z x) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -3.4e-56) {
		tmp = (z * y) * x;
	} else if (y <= 3.5e-119) {
		tmp = (b * a) * i;
	} else if (y <= 3.5e+87) {
		tmp = (c * t) * j;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (y <= (-3.4d-56)) then
        tmp = (z * y) * x
    else if (y <= 3.5d-119) then
        tmp = (b * a) * i
    else if (y <= 3.5d+87) then
        tmp = (c * t) * j
    else
        tmp = (z * x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -3.4e-56) {
		tmp = (z * y) * x;
	} else if (y <= 3.5e-119) {
		tmp = (b * a) * i;
	} else if (y <= 3.5e+87) {
		tmp = (c * t) * j;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if y <= -3.4e-56:
		tmp = (z * y) * x
	elif y <= 3.5e-119:
		tmp = (b * a) * i
	elif y <= 3.5e+87:
		tmp = (c * t) * j
	else:
		tmp = (z * x) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -3.4e-56)
		tmp = Float64(Float64(z * y) * x);
	elseif (y <= 3.5e-119)
		tmp = Float64(Float64(b * a) * i);
	elseif (y <= 3.5e+87)
		tmp = Float64(Float64(c * t) * j);
	else
		tmp = Float64(Float64(z * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (y <= -3.4e-56)
		tmp = (z * y) * x;
	elseif (y <= 3.5e-119)
		tmp = (b * a) * i;
	elseif (y <= 3.5e+87)
		tmp = (c * t) * j;
	else
		tmp = (z * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -3.4e-56], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 3.5e-119], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[y, 3.5e+87], N[(N[(c * t), $MachinePrecision] * j), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.39999999999999982e-56

   1. Initial program 77.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 51.2

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 41.8%

      \[\leadsto \left(z \cdot y\right) \cdot x \]

   if -3.39999999999999982e-56 < y < 3.5e-119

   1. Initial program 86.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 36.9

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 36.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in b around inf

      \[\leadsto \left(a \cdot b\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 31.1%

      \[\leadsto \left(b \cdot a\right) \cdot i \]

   if 3.5e-119 < y < 3.49999999999999986e87

   1. Initial program 81.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right)} \cdot j \]

      5. neg-mul-1 N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + c \cdot t\right) \cdot j \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, y, c \cdot t\right)} \cdot j \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, c \cdot t\right) \cdot j \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, y, c \cdot t\right) \cdot j \]

      9. lower-*.f64 42.4

         \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot t}\right) \cdot j \]

   5. Applied rewrites 42.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]

   6. Taylor expanded in c around inf

      \[\leadsto \left(c \cdot t\right) \cdot j \]
   7. Step-by-step derivation

   8. Applied rewrites 32.2%

      \[\leadsto \left(c \cdot t\right) \cdot j \]

   if 3.49999999999999986e87 < y

   1. Initial program 66.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(j \cdot i\right)} + x \cdot z\right) \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right) \cdot i} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 78.5

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 50.3%

      \[\leadsto \left(z \cdot x\right) \cdot y \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 23: 29.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-57}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-119}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+87}:\\ \;\;\;\;\left(c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -3.7e-57)
   (* (* z y) x)
   (if (<= y 2.35e-119)
     (* (* i b) a)
     (if (<= y 3.5e+87) (* (* c t) j) (* (* z x) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -3.7e-57) {
		tmp = (z * y) * x;
	} else if (y <= 2.35e-119) {
		tmp = (i * b) * a;
	} else if (y <= 3.5e+87) {
		tmp = (c * t) * j;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (y <= (-3.7d-57)) then
        tmp = (z * y) * x
    else if (y <= 2.35d-119) then
        tmp = (i * b) * a
    else if (y <= 3.5d+87) then
        tmp = (c * t) * j
    else
        tmp = (z * x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -3.7e-57) {
		tmp = (z * y) * x;
	} else if (y <= 2.35e-119) {
		tmp = (i * b) * a;
	} else if (y <= 3.5e+87) {
		tmp = (c * t) * j;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if y <= -3.7e-57:
		tmp = (z * y) * x
	elif y <= 2.35e-119:
		tmp = (i * b) * a
	elif y <= 3.5e+87:
		tmp = (c * t) * j
	else:
		tmp = (z * x) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -3.7e-57)
		tmp = Float64(Float64(z * y) * x);
	elseif (y <= 2.35e-119)
		tmp = Float64(Float64(i * b) * a);
	elseif (y <= 3.5e+87)
		tmp = Float64(Float64(c * t) * j);
	else
		tmp = Float64(Float64(z * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (y <= -3.7e-57)
		tmp = (z * y) * x;
	elseif (y <= 2.35e-119)
		tmp = (i * b) * a;
	elseif (y <= 3.5e+87)
		tmp = (c * t) * j;
	else
		tmp = (z * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -3.7e-57], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 2.35e-119], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[y, 3.5e+87], N[(N[(c * t), $MachinePrecision] * j), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.7e-57

   1. Initial program 77.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 51.2

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 41.8%

      \[\leadsto \left(z \cdot y\right) \cdot x \]

   if -3.7e-57 < y < 2.35000000000000001e-119

   1. Initial program 86.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(c \cdot z - a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.9%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]

   if 2.35000000000000001e-119 < y < 3.49999999999999986e87

   1. Initial program 81.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right)} \cdot j \]

      5. neg-mul-1 N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + c \cdot t\right) \cdot j \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, y, c \cdot t\right)} \cdot j \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, c \cdot t\right) \cdot j \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, y, c \cdot t\right) \cdot j \]

      9. lower-*.f64 42.4

         \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot t}\right) \cdot j \]

   5. Applied rewrites 42.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]

   6. Taylor expanded in c around inf

      \[\leadsto \left(c \cdot t\right) \cdot j \]
   7. Step-by-step derivation

   8. Applied rewrites 32.2%

      \[\leadsto \left(c \cdot t\right) \cdot j \]

   if 3.49999999999999986e87 < y

   1. Initial program 66.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(j \cdot i\right)} + x \cdot z\right) \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right) \cdot i} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 78.5

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 50.3%

      \[\leadsto \left(z \cdot x\right) \cdot y \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-57}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-119}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+87}:\\ \;\;\;\;\left(c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 29.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot x\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{-57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-119}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+87}:\\ \;\;\;\;\left(c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z y) x)))
   (if (<= y -3.7e-57)
     t_1
     (if (<= y 2.35e-119)
       (* (* i b) a)
       (if (<= y 2.9e+87) (* (* c t) j) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * y) * x;
	double tmp;
	if (y <= -3.7e-57) {
		tmp = t_1;
	} else if (y <= 2.35e-119) {
		tmp = (i * b) * a;
	} else if (y <= 2.9e+87) {
		tmp = (c * t) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * y) * x
    if (y <= (-3.7d-57)) then
        tmp = t_1
    else if (y <= 2.35d-119) then
        tmp = (i * b) * a
    else if (y <= 2.9d+87) then
        tmp = (c * t) * j
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * y) * x;
	double tmp;
	if (y <= -3.7e-57) {
		tmp = t_1;
	} else if (y <= 2.35e-119) {
		tmp = (i * b) * a;
	} else if (y <= 2.9e+87) {
		tmp = (c * t) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * y) * x
	tmp = 0
	if y <= -3.7e-57:
		tmp = t_1
	elif y <= 2.35e-119:
		tmp = (i * b) * a
	elif y <= 2.9e+87:
		tmp = (c * t) * j
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * y) * x)
	tmp = 0.0
	if (y <= -3.7e-57)
		tmp = t_1;
	elseif (y <= 2.35e-119)
		tmp = Float64(Float64(i * b) * a);
	elseif (y <= 2.9e+87)
		tmp = Float64(Float64(c * t) * j);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * y) * x;
	tmp = 0.0;
	if (y <= -3.7e-57)
		tmp = t_1;
	elseif (y <= 2.35e-119)
		tmp = (i * b) * a;
	elseif (y <= 2.9e+87)
		tmp = (c * t) * j;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -3.7e-57], t$95$1, If[LessEqual[y, 2.35e-119], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[y, 2.9e+87], N[(N[(c * t), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.7e-57 or 2.8999999999999998e87 < y

   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 x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 52.5

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 44.3%

      \[\leadsto \left(z \cdot y\right) \cdot x \]

   if -3.7e-57 < y < 2.35000000000000001e-119

   1. Initial program 86.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(c \cdot z - a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.9%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]

   if 2.35000000000000001e-119 < y < 2.8999999999999998e87

   1. Initial program 81.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right)} \cdot j \]

      5. neg-mul-1 N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + c \cdot t\right) \cdot j \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, y, c \cdot t\right)} \cdot j \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, c \cdot t\right) \cdot j \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, y, c \cdot t\right) \cdot j \]

      9. lower-*.f64 42.4

         \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot t}\right) \cdot j \]

   5. Applied rewrites 42.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]

   6. Taylor expanded in c around inf

      \[\leadsto \left(c \cdot t\right) \cdot j \]
   7. Step-by-step derivation

   8. Applied rewrites 32.2%

      \[\leadsto \left(c \cdot t\right) \cdot j \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-57}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-119}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+87}:\\ \;\;\;\;\left(c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 41.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b 4.7e+127) (* (fma (- a) t (* z y)) x) (* (* 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 (b <= 4.7e+127) {
		tmp = fma(-a, t, (z * y)) * x;
	} else {
		tmp = (b * a) * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= 4.7e+127)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	else
		tmp = Float64(Float64(b * a) * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, 4.7e+127], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.70000000000000035e127

   1. Initial program 78.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 47.5

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   if 4.70000000000000035e127 < b

   1. Initial program 84.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 67.7

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in b around inf

      \[\leadsto \left(a \cdot b\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 57.5%

      \[\leadsto \left(b \cdot a\right) \cdot i \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 29.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot x\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{-57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-65}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z y) x)))
   (if (<= y -3.7e-57) t_1 (if (<= y 1.7e-65) (* (* i b) a) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * y) * x;
	double tmp;
	if (y <= -3.7e-57) {
		tmp = t_1;
	} else if (y <= 1.7e-65) {
		tmp = (i * b) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * y) * x
    if (y <= (-3.7d-57)) then
        tmp = t_1
    else if (y <= 1.7d-65) then
        tmp = (i * b) * a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * y) * x;
	double tmp;
	if (y <= -3.7e-57) {
		tmp = t_1;
	} else if (y <= 1.7e-65) {
		tmp = (i * b) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * y) * x
	tmp = 0
	if y <= -3.7e-57:
		tmp = t_1
	elif y <= 1.7e-65:
		tmp = (i * b) * a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * y) * x)
	tmp = 0.0
	if (y <= -3.7e-57)
		tmp = t_1;
	elseif (y <= 1.7e-65)
		tmp = Float64(Float64(i * b) * a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * y) * x;
	tmp = 0.0;
	if (y <= -3.7e-57)
		tmp = t_1;
	elseif (y <= 1.7e-65)
		tmp = (i * b) * a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -3.7e-57], t$95$1, If[LessEqual[y, 1.7e-65], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7e-57 or 1.69999999999999993e-65 < y

   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. 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. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 50.9

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 38.9%

      \[\leadsto \left(z \cdot y\right) \cdot x \]

   if -3.7e-57 < y < 1.69999999999999993e-65

   1. Initial program 85.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(c \cdot z - a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

   5. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.9%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-57}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-65}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 28.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\left(i \cdot a\right) \cdot b\\ \mathbf{elif}\;i \leq 2.4 \cdot 10^{-31}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -2e-76)
   (* (* i a) b)
   (if (<= i 2.4e-31) (* (* j t) c) (* (* i b) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -2e-76) {
		tmp = (i * a) * b;
	} else if (i <= 2.4e-31) {
		tmp = (j * t) * c;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (i <= (-2d-76)) then
        tmp = (i * a) * b
    else if (i <= 2.4d-31) then
        tmp = (j * t) * c
    else
        tmp = (i * b) * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -2e-76) {
		tmp = (i * a) * b;
	} else if (i <= 2.4e-31) {
		tmp = (j * t) * c;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if i <= -2e-76:
		tmp = (i * a) * b
	elif i <= 2.4e-31:
		tmp = (j * t) * c
	else:
		tmp = (i * b) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -2e-76)
		tmp = Float64(Float64(i * a) * b);
	elseif (i <= 2.4e-31)
		tmp = Float64(Float64(j * t) * c);
	else
		tmp = Float64(Float64(i * b) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (i <= -2e-76)
		tmp = (i * a) * b;
	elseif (i <= 2.4e-31)
		tmp = (j * t) * c;
	else
		tmp = (i * b) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -2e-76], N[(N[(i * a), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[i, 2.4e-31], N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.99999999999999985e-76

   1. Initial program 81.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(c \cdot z - a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

   5. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 32.3%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]
   9. Step-by-step derivation

   10. Applied rewrites 35.8%

       \[\leadsto \left(a \cdot i\right) \cdot b \]

   if -1.99999999999999985e-76 < i < 2.4e-31

   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 x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(c \cdot z - a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.7%

      \[\leadsto \left(t \cdot j\right) \cdot \color{blue}{c} \]

   if 2.4e-31 < i

   1. Initial program 76.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(c \cdot z - a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

   5. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\left(i \cdot a\right) \cdot b\\ \mathbf{elif}\;i \leq 2.4 \cdot 10^{-31}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 21.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-108}:\\ \;\;\;\;\left(i \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t 2.7e-108) (* (* i a) b) (* (* i b) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= 2.7e-108) {
		tmp = (i * a) * b;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (t <= 2.7d-108) then
        tmp = (i * a) * b
    else
        tmp = (i * b) * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= 2.7e-108) {
		tmp = (i * a) * b;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if t <= 2.7e-108:
		tmp = (i * a) * b
	else:
		tmp = (i * b) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= 2.7e-108)
		tmp = Float64(Float64(i * a) * b);
	else
		tmp = Float64(Float64(i * b) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (t <= 2.7e-108)
		tmp = (i * a) * b;
	else
		tmp = (i * b) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, 2.7e-108], N[(N[(i * a), $MachinePrecision] * b), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.70000000000000005e-108

   1. Initial program 80.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(c \cdot z - a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

   5. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 20.3%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]
   9. Step-by-step derivation

   10. Applied rewrites 23.1%

       \[\leadsto \left(a \cdot i\right) \cdot b \]

   if 2.70000000000000005e-108 < t

   1. Initial program 75.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(c \cdot z - a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

   5. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 18.4%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-108}:\\ \;\;\;\;\left(i \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 21.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(i \cdot a\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* i a) b))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (i * a) * b;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = (i * a) * b
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (i * a) * b;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (i * a) * b

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(i * a) * b)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (i * a) * b;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(i * a), $MachinePrecision] * b), $MachinePrecision]

Derivation

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 x around 0

   \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(c \cdot z - a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

   9. distribute-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

   11. neg-mul-1 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

   12. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)}, b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

   14. neg-mul-1 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

   15. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-c}, z, a \cdot i\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right), b, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j}\right) \]

5. Applied rewrites 57.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-c, z, a \cdot i\right), b, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

6. Taylor expanded in a around inf

   \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
7. Step-by-step derivation

8. Applied rewrites 19.6%

   \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]
9. Step-by-step derivation

10. Applied rewrites 19.6%

    \[\leadsto \left(a \cdot i\right) \cdot b \]

11. Final simplification 19.6%

    \[\leadsto \left(i \cdot a\right) \cdot b \]
12. Add Preprocessing

Developer Target 1: 68.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2024277 
(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)))))