Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 74.0% → 84.3%

Time: 21.9s

Alternatives: 28

Speedup: 0.8×

• 57.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.0% 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: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (+ (* x (- (* y z) (* t a))) (* b (- (* a i) (* z c))))
          (* j (- (* t c) (* y i))))))
   (if (<= t_1 INFINITY)
     t_1
     (fma i (fma j (- 0.0 y) (* a b)) (* z (fma c (- 0.0 b) (* x y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) + (b * ((a * i) - (z * c)))) + (j * ((t * c) - (y * i)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(i, fma(j, (0.0 - y), (a * b)), (z * fma(c, (0.0 - b), (x * y))));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 42.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 57.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right) \leq \infty:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, 0 - z, a \cdot i\right)\right)\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.25 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, b \cdot i\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, x \cdot y\right), j \cdot \mathsf{fma}\left(c, t, y \cdot \left(0 - i\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (fma y (fma j (- 0.0 i) (* x z)) (* b (fma c (- 0.0 z) (* a i))))))
   (if (<= b -1.7e+25)
     t_1
     (if (<= b 3.25e+75)
       (fma
        a
        (fma t (- 0.0 x) (* b i))
        (fma z (fma c (- 0.0 b) (* x y)) (* j (fma c t (* y (- 0.0 i))))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(y, fma(j, (0.0 - i), (x * z)), (b * fma(c, (0.0 - z), (a * i))));
	double tmp;
	if (b <= -1.7e+25) {
		tmp = t_1;
	} else if (b <= 3.25e+75) {
		tmp = fma(a, fma(t, (0.0 - x), (b * i)), fma(z, fma(c, (0.0 - b), (x * y)), (j * fma(c, t, (y * (0.0 - i))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(y, fma(j, Float64(0.0 - i), Float64(x * z)), Float64(b * fma(c, Float64(0.0 - z), Float64(a * i))))
	tmp = 0.0
	if (b <= -1.7e+25)
		tmp = t_1;
	elseif (b <= 3.25e+75)
		tmp = fma(a, fma(t, Float64(0.0 - x), Float64(b * i)), fma(z, fma(c, Float64(0.0 - b), Float64(x * y)), Float64(j * fma(c, t, Float64(y * Float64(0.0 - i))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(y * N[(j * N[(0.0 - i), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * N[(0.0 - z), $MachinePrecision] + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.7e+25], t$95$1, If[LessEqual[b, 3.25e+75], N[(a * N[(t * N[(0.0 - x), $MachinePrecision] + N[(b * i), $MachinePrecision]), $MachinePrecision] + N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(j * N[(c * t + N[(y * N[(0.0 - i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.69999999999999992e25 or 3.2499999999999999e75 < b

   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 t around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), \color{blue}{b \cdot \left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)}\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)}\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)}\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)}\right) \]

   5. Simplified 81.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, 0 - z, i \cdot a\right)\right)} \]

   if -1.69999999999999992e25 < b < 3.2499999999999999e75

   1. Initial program 77.4%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 85.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, 0 - z, a \cdot i\right)\right)\\ \mathbf{elif}\;b \leq 3.25 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, b \cdot i\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, x \cdot y\right), j \cdot \mathsf{fma}\left(c, t, y \cdot \left(0 - i\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, 0 - z, a \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{if}\;i \leq -1.1 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(i, t\_1, 0 - z \cdot \left(b \cdot c\right)\right)\\ \mathbf{elif}\;i \leq -1.02 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, 0 - z, a \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 2.9 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, 0 - z, t \cdot j\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, t\_1, z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma j (- 0.0 y) (* a b))))
   (if (<= i -1.1e+179)
     (fma i t_1 (- 0.0 (* z (* b c))))
     (if (<= i -1.02e-44)
       (fma y (fma j (- 0.0 i) (* x z)) (* b (fma c (- 0.0 z) (* a i))))
       (if (<= i 2.9e+15)
         (fma c (fma b (- 0.0 z) (* t j)) (* x (- (* y z) (* t a))))
         (fma i t_1 (* z (fma c (- 0.0 b) (* x y)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(j, (0.0 - y), (a * b));
	double tmp;
	if (i <= -1.1e+179) {
		tmp = fma(i, t_1, (0.0 - (z * (b * c))));
	} else if (i <= -1.02e-44) {
		tmp = fma(y, fma(j, (0.0 - i), (x * z)), (b * fma(c, (0.0 - z), (a * i))));
	} else if (i <= 2.9e+15) {
		tmp = fma(c, fma(b, (0.0 - z), (t * j)), (x * ((y * z) - (t * a))));
	} else {
		tmp = fma(i, t_1, (z * fma(c, (0.0 - b), (x * y))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(j, Float64(0.0 - y), Float64(a * b))
	tmp = 0.0
	if (i <= -1.1e+179)
		tmp = fma(i, t_1, Float64(0.0 - Float64(z * Float64(b * c))));
	elseif (i <= -1.02e-44)
		tmp = fma(y, fma(j, Float64(0.0 - i), Float64(x * z)), Float64(b * fma(c, Float64(0.0 - z), Float64(a * i))));
	elseif (i <= 2.9e+15)
		tmp = fma(c, fma(b, Float64(0.0 - z), Float64(t * j)), Float64(x * Float64(Float64(y * z) - Float64(t * a))));
	else
		tmp = fma(i, t_1, Float64(z * fma(c, Float64(0.0 - b), Float64(x * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.1e+179], N[(i * t$95$1 + N[(0.0 - N[(z * N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -1.02e-44], N[(y * N[(j * N[(0.0 - i), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * N[(0.0 - z), $MachinePrecision] + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.9e+15], N[(c * N[(b * N[(0.0 - z), $MachinePrecision] + N[(t * j), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * t$95$1 + N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -1.1e179

   1. Initial program 59.4%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 66.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)}\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{0 - b \cdot \left(c \cdot z\right)}\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{0 - b \cdot \left(c \cdot z\right)}\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), 0 - \color{blue}{\left(b \cdot c\right) \cdot z}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), 0 - \color{blue}{z \cdot \left(b \cdot c\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), 0 - \color{blue}{z \cdot \left(b \cdot c\right)}\right) \]

       7. *-lowering-*.f64 81.5

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), 0 - z \cdot \color{blue}{\left(b \cdot c\right)}\right) \]

   11. Simplified 81.5%

       \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{0 - z \cdot \left(b \cdot c\right)}\right) \]

   if -1.1e179 < i < -1.0199999999999999e-44

   1. Initial program 70.5%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), \color{blue}{b \cdot \left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)}\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)}\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)}\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)}\right) \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, 0 - z, i \cdot a\right)\right)} \]

   if -1.0199999999999999e-44 < i < 2.9e15

   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 i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, 0 - z, j \cdot t\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   if 2.9e15 < i

   1. Initial program 73.3%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 82.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.1 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), 0 - z \cdot \left(b \cdot c\right)\right)\\ \mathbf{elif}\;i \leq -1.02 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, 0 - z, a \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 2.9 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, 0 - z, t \cdot j\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \mathbf{if}\;i \leq -9 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 4.1 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, 0 - z, t \cdot j\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (fma i (fma j (- 0.0 y) (* a b)) (* z (fma c (- 0.0 b) (* x y))))))
   (if (<= i -9e+32)
     t_1
     (if (<= i 4.1e+16)
       (fma c (fma b (- 0.0 z) (* t j)) (* x (- (* y z) (* t a))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(i, fma(j, (0.0 - y), (a * b)), (z * fma(c, (0.0 - b), (x * y))));
	double tmp;
	if (i <= -9e+32) {
		tmp = t_1;
	} else if (i <= 4.1e+16) {
		tmp = fma(c, fma(b, (0.0 - z), (t * j)), (x * ((y * z) - (t * a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(i, fma(j, Float64(0.0 - y), Float64(a * b)), Float64(z * fma(c, Float64(0.0 - b), Float64(x * y))))
	tmp = 0.0
	if (i <= -9e+32)
		tmp = t_1;
	elseif (i <= 4.1e+16)
		tmp = fma(c, fma(b, Float64(0.0 - z), Float64(t * j)), Float64(x * Float64(Float64(y * z) - Float64(t * a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -9e+32], t$95$1, If[LessEqual[i, 4.1e+16], N[(c * N[(b * N[(0.0 - z), $MachinePrecision] + N[(t * j), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -9.0000000000000007e32 or 4.1e16 < i

   1. Initial program 69.9%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 75.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 79.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   if -9.0000000000000007e32 < i < 4.1e16

   1. Initial program 79.1%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 71.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, 0 - z, j \cdot t\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -9 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \mathbf{elif}\;i \leq 4.1 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, 0 - z, t \cdot j\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -6.1 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), 0 - z \cdot \left(b \cdot c\right)\right)\\ \mathbf{elif}\;i \leq 4 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, 0 - z, t \cdot j\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -6.1e+105)
   (fma i (fma j (- 0.0 y) (* a b)) (- 0.0 (* z (* b c))))
   (if (<= i 4e+88)
     (fma c (fma b (- 0.0 z) (* t j)) (* x (- (* y z) (* t a))))
     (* i (- (* a b) (* y j))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -6.1e+105) {
		tmp = fma(i, fma(j, (0.0 - y), (a * b)), (0.0 - (z * (b * c))));
	} else if (i <= 4e+88) {
		tmp = fma(c, fma(b, (0.0 - z), (t * j)), (x * ((y * z) - (t * a))));
	} else {
		tmp = i * ((a * b) - (y * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -6.1e+105)
		tmp = fma(i, fma(j, Float64(0.0 - y), Float64(a * b)), Float64(0.0 - Float64(z * Float64(b * c))));
	elseif (i <= 4e+88)
		tmp = fma(c, fma(b, Float64(0.0 - z), Float64(t * j)), Float64(x * Float64(Float64(y * z) - Float64(t * a))));
	else
		tmp = Float64(i * Float64(Float64(a * b) - Float64(y * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -6.1e+105], N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(0.0 - N[(z * N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 4e+88], N[(c * N[(b * N[(0.0 - z), $MachinePrecision] + N[(t * j), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(N[(a * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -6.0999999999999996e105

   1. Initial program 61.9%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 59.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 72.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)}\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{0 - b \cdot \left(c \cdot z\right)}\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{0 - b \cdot \left(c \cdot z\right)}\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), 0 - \color{blue}{\left(b \cdot c\right) \cdot z}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), 0 - \color{blue}{z \cdot \left(b \cdot c\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), 0 - \color{blue}{z \cdot \left(b \cdot c\right)}\right) \]

       7. *-lowering-*.f64 77.0

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), 0 - z \cdot \color{blue}{\left(b \cdot c\right)}\right) \]

   11. Simplified 77.0%

       \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{0 - z \cdot \left(b \cdot c\right)}\right) \]

   if -6.0999999999999996e105 < i < 3.99999999999999984e88

   1. Initial program 79.1%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 69.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, 0 - z, j \cdot t\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   if 3.99999999999999984e88 < i

   1. Initial program 73.3%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 84.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in i around inf

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

       1. *-lowering-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. --lowering--.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 75.4

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

   11. Simplified 75.4%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6.1 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), 0 - z \cdot \left(b \cdot c\right)\right)\\ \mathbf{elif}\;i \leq 4 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, 0 - z, t \cdot j\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 30.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{+44}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-86}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-257}:\\ \;\;\;\;a \cdot \left(t \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-228}:\\ \;\;\;\;y \cdot \left(0 - i \cdot j\right)\\ \mathbf{elif}\;b \leq 26000:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+128}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z \cdot \left(b \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -8.2e+44)
   (* i (* a b))
   (if (<= b -4.5e-86)
     (* x (* y z))
     (if (<= b -3e-257)
       (* a (* t (- 0.0 x)))
       (if (<= b 1.4e-228)
         (* y (- 0.0 (* i j)))
         (if (<= b 26000.0)
           (* y (* x z))
           (if (<= b 2.7e+128) (* b (* a i)) (- 0.0 (* z (* b c))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -8.2e+44) {
		tmp = i * (a * b);
	} else if (b <= -4.5e-86) {
		tmp = x * (y * z);
	} else if (b <= -3e-257) {
		tmp = a * (t * (0.0 - x));
	} else if (b <= 1.4e-228) {
		tmp = y * (0.0 - (i * j));
	} else if (b <= 26000.0) {
		tmp = y * (x * z);
	} else if (b <= 2.7e+128) {
		tmp = b * (a * i);
	} else {
		tmp = 0.0 - (z * (b * c));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (b <= (-8.2d+44)) then
        tmp = i * (a * b)
    else if (b <= (-4.5d-86)) then
        tmp = x * (y * z)
    else if (b <= (-3d-257)) then
        tmp = a * (t * (0.0d0 - x))
    else if (b <= 1.4d-228) then
        tmp = y * (0.0d0 - (i * j))
    else if (b <= 26000.0d0) then
        tmp = y * (x * z)
    else if (b <= 2.7d+128) then
        tmp = b * (a * i)
    else
        tmp = 0.0d0 - (z * (b * c))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -8.2e+44) {
		tmp = i * (a * b);
	} else if (b <= -4.5e-86) {
		tmp = x * (y * z);
	} else if (b <= -3e-257) {
		tmp = a * (t * (0.0 - x));
	} else if (b <= 1.4e-228) {
		tmp = y * (0.0 - (i * j));
	} else if (b <= 26000.0) {
		tmp = y * (x * z);
	} else if (b <= 2.7e+128) {
		tmp = b * (a * i);
	} else {
		tmp = 0.0 - (z * (b * c));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -8.2e+44:
		tmp = i * (a * b)
	elif b <= -4.5e-86:
		tmp = x * (y * z)
	elif b <= -3e-257:
		tmp = a * (t * (0.0 - x))
	elif b <= 1.4e-228:
		tmp = y * (0.0 - (i * j))
	elif b <= 26000.0:
		tmp = y * (x * z)
	elif b <= 2.7e+128:
		tmp = b * (a * i)
	else:
		tmp = 0.0 - (z * (b * c))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -8.2e+44)
		tmp = Float64(i * Float64(a * b));
	elseif (b <= -4.5e-86)
		tmp = Float64(x * Float64(y * z));
	elseif (b <= -3e-257)
		tmp = Float64(a * Float64(t * Float64(0.0 - x)));
	elseif (b <= 1.4e-228)
		tmp = Float64(y * Float64(0.0 - Float64(i * j)));
	elseif (b <= 26000.0)
		tmp = Float64(y * Float64(x * z));
	elseif (b <= 2.7e+128)
		tmp = Float64(b * Float64(a * i));
	else
		tmp = Float64(0.0 - Float64(z * Float64(b * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -8.2e+44)
		tmp = i * (a * b);
	elseif (b <= -4.5e-86)
		tmp = x * (y * z);
	elseif (b <= -3e-257)
		tmp = a * (t * (0.0 - x));
	elseif (b <= 1.4e-228)
		tmp = y * (0.0 - (i * j));
	elseif (b <= 26000.0)
		tmp = y * (x * z);
	elseif (b <= 2.7e+128)
		tmp = b * (a * i);
	else
		tmp = 0.0 - (z * (b * c));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -8.2e+44], N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -4.5e-86], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3e-257], N[(a * N[(t * N[(0.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.4e-228], N[(y * N[(0.0 - N[(i * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 26000.0], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.7e+128], N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(z * N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 7 regimes

2. if b < -8.1999999999999993e44

   1. Initial program 72.6%

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

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 48.3

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 48.3%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 42.7

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

   8. Simplified 42.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 53.9

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

   10. Applied egg-rr 53.9%

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

   if -8.1999999999999993e44 < b < -4.4999999999999998e-86

   1. Initial program 81.5%

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

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 48.1

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

   5. Simplified 48.1%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 38.4

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

   8. Simplified 38.4%

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

   if -4.4999999999999998e-86 < b < -2.9999999999999999e-257

   1. Initial program 69.1%

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

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 47.3

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

   5. Simplified 47.3%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(0 - x\right)}\right) \]

      11. --lowering--.f64 39.0

          \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(0 - x\right)}\right) \]

   8. Simplified 39.0%

      \[\leadsto \color{blue}{a \cdot \left(t \cdot \left(0 - x\right)\right)} \]

   if -2.9999999999999999e-257 < b < 1.4000000000000001e-228

   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 j around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 57.1

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 57.1%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]

   6. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - i \cdot \left(j \cdot y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - i \cdot \left(j \cdot y\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{i \cdot \left(j \cdot y\right)} \]

      5. *-lowering-*.f64 40.7

         \[\leadsto 0 - i \cdot \color{blue}{\left(j \cdot y\right)} \]

   8. Simplified 40.7%

      \[\leadsto \color{blue}{0 - i \cdot \left(j \cdot y\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto 0 - \color{blue}{\left(i \cdot j\right) \cdot y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\left(i \cdot j\right) \cdot y} \]

      3. *-lowering-*.f64 46.9

         \[\leadsto 0 - \color{blue}{\left(i \cdot j\right)} \cdot y \]

   10. Applied egg-rr 46.9%

       \[\leadsto 0 - \color{blue}{\left(i \cdot j\right) \cdot y} \]

   if 1.4000000000000001e-228 < b < 26000

   1. Initial program 75.2%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 48.8

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

   5. Simplified 48.8%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 46.5

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

   8. Simplified 46.5%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 38.7

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

   11. Simplified 38.7%

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

   if 26000 < b < 2.70000000000000001e128

   1. Initial program 79.4%

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

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 53.0

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 53.0%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 42.7

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

   8. Simplified 42.7%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 42.7

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

   10. Applied egg-rr 42.7%

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

   if 2.70000000000000001e128 < b

   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 z around 0

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

   5. Simplified 60.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in c around inf

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \color{blue}{0 - b \cdot \left(c \cdot z\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto \color{blue}{0 - b \cdot \left(c \cdot z\right)} \]

       4. associate-*r* N/A

          \[\leadsto 0 - \color{blue}{\left(b \cdot c\right) \cdot z} \]

       5. *-commutative N/A

          \[\leadsto 0 - \color{blue}{z \cdot \left(b \cdot c\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto 0 - \color{blue}{z \cdot \left(b \cdot c\right)} \]

       7. *-lowering-*.f64 52.0

          \[\leadsto 0 - z \cdot \color{blue}{\left(b \cdot c\right)} \]

   11. Simplified 52.0%

       \[\leadsto \color{blue}{0 - z \cdot \left(b \cdot c\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{+44}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-86}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-257}:\\ \;\;\;\;a \cdot \left(t \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-228}:\\ \;\;\;\;y \cdot \left(0 - i \cdot j\right)\\ \mathbf{elif}\;b \leq 26000:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+128}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z \cdot \left(b \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 30.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+43}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-87}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;b \leq -2.05 \cdot 10^{-257}:\\ \;\;\;\;a \cdot \left(t \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-226}:\\ \;\;\;\;j \cdot \left(y \cdot \left(0 - i\right)\right)\\ \mathbf{elif}\;b \leq 450:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{+128}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z \cdot \left(b \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -6.6e+43)
   (* i (* a b))
   (if (<= b -1.26e-87)
     (* x (* y z))
     (if (<= b -2.05e-257)
       (* a (* t (- 0.0 x)))
       (if (<= b 2.3e-226)
         (* j (* y (- 0.0 i)))
         (if (<= b 450.0)
           (* y (* x z))
           (if (<= b 1.7e+128) (* b (* a i)) (- 0.0 (* z (* b c))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -6.6e+43) {
		tmp = i * (a * b);
	} else if (b <= -1.26e-87) {
		tmp = x * (y * z);
	} else if (b <= -2.05e-257) {
		tmp = a * (t * (0.0 - x));
	} else if (b <= 2.3e-226) {
		tmp = j * (y * (0.0 - i));
	} else if (b <= 450.0) {
		tmp = y * (x * z);
	} else if (b <= 1.7e+128) {
		tmp = b * (a * i);
	} else {
		tmp = 0.0 - (z * (b * c));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (b <= (-6.6d+43)) then
        tmp = i * (a * b)
    else if (b <= (-1.26d-87)) then
        tmp = x * (y * z)
    else if (b <= (-2.05d-257)) then
        tmp = a * (t * (0.0d0 - x))
    else if (b <= 2.3d-226) then
        tmp = j * (y * (0.0d0 - i))
    else if (b <= 450.0d0) then
        tmp = y * (x * z)
    else if (b <= 1.7d+128) then
        tmp = b * (a * i)
    else
        tmp = 0.0d0 - (z * (b * c))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -6.6e+43) {
		tmp = i * (a * b);
	} else if (b <= -1.26e-87) {
		tmp = x * (y * z);
	} else if (b <= -2.05e-257) {
		tmp = a * (t * (0.0 - x));
	} else if (b <= 2.3e-226) {
		tmp = j * (y * (0.0 - i));
	} else if (b <= 450.0) {
		tmp = y * (x * z);
	} else if (b <= 1.7e+128) {
		tmp = b * (a * i);
	} else {
		tmp = 0.0 - (z * (b * c));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -6.6e+43:
		tmp = i * (a * b)
	elif b <= -1.26e-87:
		tmp = x * (y * z)
	elif b <= -2.05e-257:
		tmp = a * (t * (0.0 - x))
	elif b <= 2.3e-226:
		tmp = j * (y * (0.0 - i))
	elif b <= 450.0:
		tmp = y * (x * z)
	elif b <= 1.7e+128:
		tmp = b * (a * i)
	else:
		tmp = 0.0 - (z * (b * c))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -6.6e+43)
		tmp = Float64(i * Float64(a * b));
	elseif (b <= -1.26e-87)
		tmp = Float64(x * Float64(y * z));
	elseif (b <= -2.05e-257)
		tmp = Float64(a * Float64(t * Float64(0.0 - x)));
	elseif (b <= 2.3e-226)
		tmp = Float64(j * Float64(y * Float64(0.0 - i)));
	elseif (b <= 450.0)
		tmp = Float64(y * Float64(x * z));
	elseif (b <= 1.7e+128)
		tmp = Float64(b * Float64(a * i));
	else
		tmp = Float64(0.0 - Float64(z * Float64(b * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -6.6e+43)
		tmp = i * (a * b);
	elseif (b <= -1.26e-87)
		tmp = x * (y * z);
	elseif (b <= -2.05e-257)
		tmp = a * (t * (0.0 - x));
	elseif (b <= 2.3e-226)
		tmp = j * (y * (0.0 - i));
	elseif (b <= 450.0)
		tmp = y * (x * z);
	elseif (b <= 1.7e+128)
		tmp = b * (a * i);
	else
		tmp = 0.0 - (z * (b * c));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -6.6e+43], N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.26e-87], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -2.05e-257], N[(a * N[(t * N[(0.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.3e-226], N[(j * N[(y * N[(0.0 - i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 450.0], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.7e+128], N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(z * N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 7 regimes

2. if b < -6.6000000000000003e43

   1. Initial program 72.6%

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

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 48.3

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 48.3%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 42.7

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

   8. Simplified 42.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 53.9

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

   10. Applied egg-rr 53.9%

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

   if -6.6000000000000003e43 < b < -1.26000000000000009e-87

   1. Initial program 81.5%

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

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 48.1

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

   5. Simplified 48.1%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 38.4

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

   8. Simplified 38.4%

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

   if -1.26000000000000009e-87 < b < -2.0499999999999998e-257

   1. Initial program 69.1%

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

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 47.3

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

   5. Simplified 47.3%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(0 - x\right)}\right) \]

      11. --lowering--.f64 39.0

          \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(0 - x\right)}\right) \]

   8. Simplified 39.0%

      \[\leadsto \color{blue}{a \cdot \left(t \cdot \left(0 - x\right)\right)} \]

   if -2.0499999999999998e-257 < b < 2.3e-226

   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 j around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 57.1

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 57.1%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]

   6. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      7. --lowering--.f64 46.9

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   8. Simplified 46.9%

      \[\leadsto j \cdot \color{blue}{\left(i \cdot \left(0 - y\right)\right)} \]

   if 2.3e-226 < b < 450

   1. Initial program 75.2%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 48.8

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

   5. Simplified 48.8%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 46.5

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

   8. Simplified 46.5%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 38.7

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

   11. Simplified 38.7%

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

   if 450 < b < 1.6999999999999999e128

   1. Initial program 79.4%

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

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 53.0

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 53.0%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 42.7

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

   8. Simplified 42.7%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 42.7

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

   10. Applied egg-rr 42.7%

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

   if 1.6999999999999999e128 < b

   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 z around 0

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

   5. Simplified 60.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in c around inf

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \color{blue}{0 - b \cdot \left(c \cdot z\right)} \]

       3. --lowering--.f64 N/A

          \[\leadsto \color{blue}{0 - b \cdot \left(c \cdot z\right)} \]

       4. associate-*r* N/A

          \[\leadsto 0 - \color{blue}{\left(b \cdot c\right) \cdot z} \]

       5. *-commutative N/A

          \[\leadsto 0 - \color{blue}{z \cdot \left(b \cdot c\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto 0 - \color{blue}{z \cdot \left(b \cdot c\right)} \]

       7. *-lowering-*.f64 52.0

          \[\leadsto 0 - z \cdot \color{blue}{\left(b \cdot c\right)} \]

   11. Simplified 52.0%

       \[\leadsto \color{blue}{0 - z \cdot \left(b \cdot c\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+43}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-87}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;b \leq -2.05 \cdot 10^{-257}:\\ \;\;\;\;a \cdot \left(t \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-226}:\\ \;\;\;\;j \cdot \left(y \cdot \left(0 - i\right)\right)\\ \mathbf{elif}\;b \leq 450:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{+128}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z \cdot \left(b \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(t \cdot c - y \cdot i\right)\\ t_2 := t\_1 + i \cdot \left(a \cdot b\right)\\ \mathbf{if}\;j \leq -7 \cdot 10^{+213}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq -2.6 \cdot 10^{-40}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;j \leq 2.2 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* t c) (* y i)))) (t_2 (+ t_1 (* i (* a b)))))
   (if (<= j -7e+213)
     t_1
     (if (<= j -2.6e-40)
       t_2
       (if (<= j 2.2e-50)
         (fma i (* a b) (* z (fma c (- 0.0 b) (* x y))))
         t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((t * c) - (y * i));
	double t_2 = t_1 + (i * (a * b));
	double tmp;
	if (j <= -7e+213) {
		tmp = t_1;
	} else if (j <= -2.6e-40) {
		tmp = t_2;
	} else if (j <= 2.2e-50) {
		tmp = fma(i, (a * b), (z * fma(c, (0.0 - b), (x * y))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(t * c) - Float64(y * i)))
	t_2 = Float64(t_1 + Float64(i * Float64(a * b)))
	tmp = 0.0
	if (j <= -7e+213)
		tmp = t_1;
	elseif (j <= -2.6e-40)
		tmp = t_2;
	elseif (j <= 2.2e-50)
		tmp = fma(i, Float64(a * b), Float64(z * fma(c, Float64(0.0 - b), Float64(x * y))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -7e+213], t$95$1, If[LessEqual[j, -2.6e-40], t$95$2, If[LessEqual[j, 2.2e-50], N[(i * N[(a * b), $MachinePrecision] + N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if j < -6.9999999999999994e213

   1. Initial program 67.8%

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

   3. Taylor expanded in j around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 88.2

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 88.2%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto j \cdot \left(c \cdot t + \color{blue}{\left(0 - y\right) \cdot i}\right) \]

      2. sub0-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 88.2

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

   7. Applied egg-rr 88.2%

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

   if -6.9999999999999994e213 < j < -2.6000000000000001e-40 or 2.1999999999999999e-50 < j

   1. Initial program 81.5%

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

   3. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 69.0

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

   5. Simplified 69.0%

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

   if -2.6000000000000001e-40 < j < 2.1999999999999999e-50

   1. Initial program 70.2%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in j around 0

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{a \cdot b}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{b \cdot a}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]

       2. *-lowering-*.f64 67.8

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{b \cdot a}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]

   11. Simplified 67.8%

       \[\leadsto \mathsf{fma}\left(i, \color{blue}{b \cdot a}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -7 \cdot 10^{+213}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{elif}\;j \leq -2.6 \cdot 10^{-40}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;j \leq 2.2 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + i \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{if}\;b \leq -4 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -1.85 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;b \leq -2.35 \cdot 10^{-254}:\\ \;\;\;\;a \cdot \left(t \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-230}:\\ \;\;\;\;y \cdot \left(0 - i \cdot j\right)\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-24}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* a i) (* z c)))))
   (if (<= b -4e+17)
     t_1
     (if (<= b -1.85e-88)
       (* x (* y z))
       (if (<= b -2.35e-254)
         (* a (* t (- 0.0 x)))
         (if (<= b 1.95e-230)
           (* y (- 0.0 (* i j)))
           (if (<= b 9.8e-24) (* 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 = b * ((a * i) - (z * c));
	double tmp;
	if (b <= -4e+17) {
		tmp = t_1;
	} else if (b <= -1.85e-88) {
		tmp = x * (y * z);
	} else if (b <= -2.35e-254) {
		tmp = a * (t * (0.0 - x));
	} else if (b <= 1.95e-230) {
		tmp = y * (0.0 - (i * j));
	} else if (b <= 9.8e-24) {
		tmp = y * (x * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = b * ((a * i) - (z * c))
    if (b <= (-4d+17)) then
        tmp = t_1
    else if (b <= (-1.85d-88)) then
        tmp = x * (y * z)
    else if (b <= (-2.35d-254)) then
        tmp = a * (t * (0.0d0 - x))
    else if (b <= 1.95d-230) then
        tmp = y * (0.0d0 - (i * j))
    else if (b <= 9.8d-24) then
        tmp = y * (x * z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * ((a * i) - (z * c));
	double tmp;
	if (b <= -4e+17) {
		tmp = t_1;
	} else if (b <= -1.85e-88) {
		tmp = x * (y * z);
	} else if (b <= -2.35e-254) {
		tmp = a * (t * (0.0 - x));
	} else if (b <= 1.95e-230) {
		tmp = y * (0.0 - (i * j));
	} else if (b <= 9.8e-24) {
		tmp = y * (x * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = b * ((a * i) - (z * c))
	tmp = 0
	if b <= -4e+17:
		tmp = t_1
	elif b <= -1.85e-88:
		tmp = x * (y * z)
	elif b <= -2.35e-254:
		tmp = a * (t * (0.0 - x))
	elif b <= 1.95e-230:
		tmp = y * (0.0 - (i * j))
	elif b <= 9.8e-24:
		tmp = y * (x * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(a * i) - Float64(z * c)))
	tmp = 0.0
	if (b <= -4e+17)
		tmp = t_1;
	elseif (b <= -1.85e-88)
		tmp = Float64(x * Float64(y * z));
	elseif (b <= -2.35e-254)
		tmp = Float64(a * Float64(t * Float64(0.0 - x)));
	elseif (b <= 1.95e-230)
		tmp = Float64(y * Float64(0.0 - Float64(i * j)));
	elseif (b <= 9.8e-24)
		tmp = Float64(y * Float64(x * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = b * ((a * i) - (z * c));
	tmp = 0.0;
	if (b <= -4e+17)
		tmp = t_1;
	elseif (b <= -1.85e-88)
		tmp = x * (y * z);
	elseif (b <= -2.35e-254)
		tmp = a * (t * (0.0 - x));
	elseif (b <= 1.95e-230)
		tmp = y * (0.0 - (i * j));
	elseif (b <= 9.8e-24)
		tmp = y * (x * z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4e+17], t$95$1, If[LessEqual[b, -1.85e-88], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -2.35e-254], N[(a * N[(t * N[(0.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-230], N[(y * N[(0.0 - N[(i * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.8e-24], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -4e17 or 9.8000000000000002e-24 < b

   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 z around 0

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

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

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

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

       3. mul-1-neg N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. distribute-lft-in N/A

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

       8. *-lowering-*.f64 N/A

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

       9. +-commutative N/A

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

       10. mul-1-neg N/A

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

       11. unsub-neg N/A

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

       12. --lowering--.f64 N/A

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 N/A

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

       15. *-commutative N/A

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

       16. *-lowering-*.f64 65.8

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

   11. Simplified 65.8%

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

   if -4e17 < b < -1.8499999999999999e-88

   1. Initial program 84.8%

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

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 47.3

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

   5. Simplified 47.3%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 39.2

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

   8. Simplified 39.2%

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

   if -1.8499999999999999e-88 < b < -2.35000000000000013e-254

   1. Initial program 69.1%

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

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 47.3

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

   5. Simplified 47.3%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(0 - x\right)}\right) \]

      11. --lowering--.f64 39.0

          \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(0 - x\right)}\right) \]

   8. Simplified 39.0%

      \[\leadsto \color{blue}{a \cdot \left(t \cdot \left(0 - x\right)\right)} \]

   if -2.35000000000000013e-254 < b < 1.9500000000000001e-230

   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 j around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 57.1

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 57.1%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]

   6. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - i \cdot \left(j \cdot y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - i \cdot \left(j \cdot y\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{i \cdot \left(j \cdot y\right)} \]

      5. *-lowering-*.f64 40.7

         \[\leadsto 0 - i \cdot \color{blue}{\left(j \cdot y\right)} \]

   8. Simplified 40.7%

      \[\leadsto \color{blue}{0 - i \cdot \left(j \cdot y\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto 0 - \color{blue}{\left(i \cdot j\right) \cdot y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\left(i \cdot j\right) \cdot y} \]

      3. *-lowering-*.f64 46.9

         \[\leadsto 0 - \color{blue}{\left(i \cdot j\right)} \cdot y \]

   10. Applied egg-rr 46.9%

       \[\leadsto 0 - \color{blue}{\left(i \cdot j\right) \cdot y} \]

   if 1.9500000000000001e-230 < b < 9.8000000000000002e-24

   1. Initial program 75.2%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 48.5

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

   5. Simplified 48.5%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 48.6

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

   8. Simplified 48.6%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 40.0

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

   11. Simplified 40.0%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+17}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{elif}\;b \leq -1.85 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;b \leq -2.35 \cdot 10^{-254}:\\ \;\;\;\;a \cdot \left(t \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-230}:\\ \;\;\;\;y \cdot \left(0 - i \cdot j\right)\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-24}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.1 \cdot 10^{+33}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -3.4 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \left(z \cdot \left(y - \frac{t \cdot a}{z}\right)\right)\\ \mathbf{elif}\;i \leq 5.2 \cdot 10^{-84}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(0 - b\right)\right)\\ \mathbf{elif}\;i \leq 2.9 \cdot 10^{+20}:\\ \;\;\;\;0 - \mathsf{fma}\left(t \cdot a, x, i \cdot \left(y \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -1.1e+33)
   (* i (fma j (- 0.0 y) (* a b)))
   (if (<= i -3.4e-99)
     (* x (* z (- y (/ (* t a) z))))
     (if (<= i 5.2e-84)
       (* c (fma j t (* z (- 0.0 b))))
       (if (<= i 2.9e+20)
         (- 0.0 (fma (* t a) x (* i (* y j))))
         (* i (- (* a b) (* y j))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -1.1e+33) {
		tmp = i * fma(j, (0.0 - y), (a * b));
	} else if (i <= -3.4e-99) {
		tmp = x * (z * (y - ((t * a) / z)));
	} else if (i <= 5.2e-84) {
		tmp = c * fma(j, t, (z * (0.0 - b)));
	} else if (i <= 2.9e+20) {
		tmp = 0.0 - fma((t * a), x, (i * (y * j)));
	} else {
		tmp = i * ((a * b) - (y * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -1.1e+33)
		tmp = Float64(i * fma(j, Float64(0.0 - y), Float64(a * b)));
	elseif (i <= -3.4e-99)
		tmp = Float64(x * Float64(z * Float64(y - Float64(Float64(t * a) / z))));
	elseif (i <= 5.2e-84)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(0.0 - b))));
	elseif (i <= 2.9e+20)
		tmp = Float64(0.0 - fma(Float64(t * a), x, Float64(i * Float64(y * j))));
	else
		tmp = Float64(i * Float64(Float64(a * b) - Float64(y * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -1.1e+33], N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -3.4e-99], N[(x * N[(z * N[(y - N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5.2e-84], N[(c * N[(j * t + N[(z * N[(0.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.9e+20], N[(0.0 - N[(N[(t * a), $MachinePrecision] * x + N[(i * N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(N[(a * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if i < -1.09999999999999997e33

   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 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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 66.8

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   if -1.09999999999999997e33 < i < -3.40000000000000007e-99

   1. Initial program 70.0%

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

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 55.0

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

   5. Simplified 55.0%

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

   6. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 55.1

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

   8. Simplified 55.1%

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

   if -3.40000000000000007e-99 < i < 5.2e-84

   1. Initial program 82.3%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 80.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in c around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, z \cdot \color{blue}{\left(0 - b\right)}\right) \]

      11. --lowering--.f64 59.5

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, z \cdot \color{blue}{\left(0 - b\right)}\right) \]

   8. Simplified 59.5%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, z \cdot \left(0 - b\right)\right)} \]

   if 5.2e-84 < i < 2.9e20

   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 t around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto a \cdot \color{blue}{\left(0 - t \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(0 - t \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. *-lowering-*.f64 75.5

         \[\leadsto a \cdot \left(0 - \color{blue}{t \cdot x}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 75.5%

      \[\leadsto \color{blue}{a \cdot \left(0 - t \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around 0

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

      1. distribute-lft-out N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 75.9

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

   8. Simplified 75.9%

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

   if 2.9e20 < i

   1. Initial program 73.3%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 82.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in i around inf

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

       1. *-lowering-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. --lowering--.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 70.0

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

   11. Simplified 70.0%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.1 \cdot 10^{+33}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -3.4 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \left(z \cdot \left(y - \frac{t \cdot a}{z}\right)\right)\\ \mathbf{elif}\;i \leq 5.2 \cdot 10^{-84}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(0 - b\right)\right)\\ \mathbf{elif}\;i \leq 2.9 \cdot 10^{+20}:\\ \;\;\;\;0 - \mathsf{fma}\left(t \cdot a, x, i \cdot \left(y \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.2 \cdot 10^{+33}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -8.2 \cdot 10^{-103}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;i \leq 2.6 \cdot 10^{-83}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(0 - b\right)\right)\\ \mathbf{elif}\;i \leq 1.32 \cdot 10^{+16}:\\ \;\;\;\;0 - \mathsf{fma}\left(t \cdot a, x, i \cdot \left(y \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -4.2e+33)
   (* i (fma j (- 0.0 y) (* a b)))
   (if (<= i -8.2e-103)
     (* x (- (* y z) (* t a)))
     (if (<= i 2.6e-83)
       (* c (fma j t (* z (- 0.0 b))))
       (if (<= i 1.32e+16)
         (- 0.0 (fma (* t a) x (* i (* y j))))
         (* i (- (* a b) (* y j))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -4.2e+33) {
		tmp = i * fma(j, (0.0 - y), (a * b));
	} else if (i <= -8.2e-103) {
		tmp = x * ((y * z) - (t * a));
	} else if (i <= 2.6e-83) {
		tmp = c * fma(j, t, (z * (0.0 - b)));
	} else if (i <= 1.32e+16) {
		tmp = 0.0 - fma((t * a), x, (i * (y * j)));
	} else {
		tmp = i * ((a * b) - (y * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -4.2e+33)
		tmp = Float64(i * fma(j, Float64(0.0 - y), Float64(a * b)));
	elseif (i <= -8.2e-103)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	elseif (i <= 2.6e-83)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(0.0 - b))));
	elseif (i <= 1.32e+16)
		tmp = Float64(0.0 - fma(Float64(t * a), x, Float64(i * Float64(y * j))));
	else
		tmp = Float64(i * Float64(Float64(a * b) - Float64(y * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -4.2e+33], N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -8.2e-103], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.6e-83], N[(c * N[(j * t + N[(z * N[(0.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.32e+16], N[(0.0 - N[(N[(t * a), $MachinePrecision] * x + N[(i * N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(N[(a * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if i < -4.2000000000000001e33

   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 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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 66.8

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   if -4.2000000000000001e33 < i < -8.19999999999999992e-103

   1. Initial program 70.0%

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

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 55.0

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

   5. Simplified 55.0%

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

   if -8.19999999999999992e-103 < i < 2.60000000000000009e-83

   1. Initial program 82.3%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 80.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in c around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, z \cdot \color{blue}{\left(0 - b\right)}\right) \]

      11. --lowering--.f64 59.5

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, z \cdot \color{blue}{\left(0 - b\right)}\right) \]

   8. Simplified 59.5%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, z \cdot \left(0 - b\right)\right)} \]

   if 2.60000000000000009e-83 < i < 1.32e16

   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 t around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto a \cdot \color{blue}{\left(0 - t \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(0 - t \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. *-lowering-*.f64 75.5

         \[\leadsto a \cdot \left(0 - \color{blue}{t \cdot x}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 75.5%

      \[\leadsto \color{blue}{a \cdot \left(0 - t \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around 0

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

      1. distribute-lft-out N/A

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 75.9

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

   8. Simplified 75.9%

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

   if 1.32e16 < i

   1. Initial program 73.3%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 82.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in i around inf

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

       1. *-lowering-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. --lowering--.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 70.0

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

   11. Simplified 70.0%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.2 \cdot 10^{+33}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -8.2 \cdot 10^{-103}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;i \leq 2.6 \cdot 10^{-83}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(0 - b\right)\right)\\ \mathbf{elif}\;i \leq 1.32 \cdot 10^{+16}:\\ \;\;\;\;0 - \mathsf{fma}\left(t \cdot a, x, i \cdot \left(y \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -2.4 \cdot 10^{+149}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{elif}\;j \leq -1.95 \cdot 10^{+15}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{elif}\;j \leq 6.8 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(i \cdot \mathsf{fma}\left(c, \frac{t}{i}, 0 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -2.4e+149)
   (* j (- (* t c) (* y i)))
   (if (<= j -1.95e+15)
     (* i (fma j (- 0.0 y) (* a b)))
     (if (<= j 6.8e-32)
       (fma i (* a b) (* z (fma c (- 0.0 b) (* x y))))
       (* j (* i (fma c (/ t i) (- 0.0 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 (j <= -2.4e+149) {
		tmp = j * ((t * c) - (y * i));
	} else if (j <= -1.95e+15) {
		tmp = i * fma(j, (0.0 - y), (a * b));
	} else if (j <= 6.8e-32) {
		tmp = fma(i, (a * b), (z * fma(c, (0.0 - b), (x * y))));
	} else {
		tmp = j * (i * fma(c, (t / i), (0.0 - y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -2.4e+149)
		tmp = Float64(j * Float64(Float64(t * c) - Float64(y * i)));
	elseif (j <= -1.95e+15)
		tmp = Float64(i * fma(j, Float64(0.0 - y), Float64(a * b)));
	elseif (j <= 6.8e-32)
		tmp = fma(i, Float64(a * b), Float64(z * fma(c, Float64(0.0 - b), Float64(x * y))));
	else
		tmp = Float64(j * Float64(i * fma(c, Float64(t / i), Float64(0.0 - y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -2.4e+149], N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, -1.95e+15], N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 6.8e-32], N[(i * N[(a * b), $MachinePrecision] + N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(j * N[(i * N[(c * N[(t / i), $MachinePrecision] + N[(0.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if j < -2.40000000000000012e149

   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 j around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 80.3

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 80.3%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto j \cdot \left(c \cdot t + \color{blue}{\left(0 - y\right) \cdot i}\right) \]

      2. sub0-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 80.3

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

   7. Applied egg-rr 80.3%

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

   if -2.40000000000000012e149 < j < -1.95e15

   1. Initial program 90.7%

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 68.6

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 68.6%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   if -1.95e15 < j < 6.79999999999999956e-32

   1. Initial program 72.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 77.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \left(a \cdot \left(b \cdot i\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(j \cdot y\right) \cdot i}\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \cdot i + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, -1 \cdot \left(j \cdot y\right) + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, j \cdot \color{blue}{\left(-1 \cdot y\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)}, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in j around 0

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{a \cdot b}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{b \cdot a}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]

       2. *-lowering-*.f64 67.1

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{b \cdot a}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]

   11. Simplified 67.1%

       \[\leadsto \mathsf{fma}\left(i, \color{blue}{b \cdot a}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]

   if 6.79999999999999956e-32 < j

   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 j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 57.5

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 57.5%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]

   6. Taylor expanded in i around inf

      \[\leadsto j \cdot \color{blue}{\left(i \cdot \left(-1 \cdot y + \frac{c \cdot t}{i}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(\frac{c \cdot t}{i} + -1 \cdot y\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto j \cdot \left(i \cdot \left(\frac{c \cdot t}{i} + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(\frac{c \cdot t}{i} - y\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \color{blue}{\left(i \cdot \left(\frac{c \cdot t}{i} - y\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(\frac{c \cdot t}{i} + \left(\mathsf{neg}\left(y\right)\right)\right)}\right) \]

      6. associate-/l* N/A

         \[\leadsto j \cdot \left(i \cdot \left(\color{blue}{c \cdot \frac{t}{i}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \left(i \cdot \left(c \cdot \frac{t}{i} + \color{blue}{-1 \cdot y}\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\mathsf{fma}\left(c, \frac{t}{i}, -1 \cdot y\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto j \cdot \left(i \cdot \mathsf{fma}\left(c, \color{blue}{\frac{t}{i}}, -1 \cdot y\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto j \cdot \left(i \cdot \mathsf{fma}\left(c, \frac{t}{i}, \color{blue}{\mathsf{neg}\left(y\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto j \cdot \left(i \cdot \mathsf{fma}\left(c, \frac{t}{i}, \color{blue}{0 - y}\right)\right) \]

      12. --lowering--.f64 59.2

          \[\leadsto j \cdot \left(i \cdot \mathsf{fma}\left(c, \frac{t}{i}, \color{blue}{0 - y}\right)\right) \]

   8. Simplified 59.2%

      \[\leadsto j \cdot \color{blue}{\left(i \cdot \mathsf{fma}\left(c, \frac{t}{i}, 0 - y\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.4 \cdot 10^{+149}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{elif}\;j \leq -1.95 \cdot 10^{+15}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{elif}\;j \leq 6.8 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(i \cdot \mathsf{fma}\left(c, \frac{t}{i}, 0 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{if}\;j \leq -3.8 \cdot 10^{-53}:\\ \;\;\;\;t\_1 - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;j \leq 6.5 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + i \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* t c) (* y i)))))
   (if (<= j -3.8e-53)
     (- t_1 (* c (* z b)))
     (if (<= j 6.5e-53)
       (fma i (* a b) (* z (fma c (- 0.0 b) (* x y))))
       (+ t_1 (* i (* a 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 = j * ((t * c) - (y * i));
	double tmp;
	if (j <= -3.8e-53) {
		tmp = t_1 - (c * (z * b));
	} else if (j <= 6.5e-53) {
		tmp = fma(i, (a * b), (z * fma(c, (0.0 - b), (x * y))));
	} else {
		tmp = t_1 + (i * (a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(t * c) - Float64(y * i)))
	tmp = 0.0
	if (j <= -3.8e-53)
		tmp = Float64(t_1 - Float64(c * Float64(z * b)));
	elseif (j <= 6.5e-53)
		tmp = fma(i, Float64(a * b), Float64(z * fma(c, Float64(0.0 - b), Float64(x * y))));
	else
		tmp = Float64(t_1 + Float64(i * Float64(a * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -3.8e-53], N[(t$95$1 - N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 6.5e-53], N[(i * N[(a * b), $MachinePrecision] + N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if j < -3.7999999999999998e-53

   1. Initial program 80.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - b \cdot \left(c \cdot z\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(0 - b \cdot \left(c \cdot z\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(0 - \color{blue}{\left(b \cdot c\right) \cdot z}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-commutative N/A

         \[\leadsto \left(0 - \color{blue}{\left(c \cdot b\right)} \cdot z\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(0 - \color{blue}{c \cdot \left(b \cdot z\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(0 - \color{blue}{c \cdot \left(b \cdot z\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. *-lowering-*.f64 73.0

         \[\leadsto \left(0 - c \cdot \color{blue}{\left(b \cdot z\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{\left(0 - c \cdot \left(b \cdot z\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   if -3.7999999999999998e-53 < j < 6.4999999999999997e-53

   1. Initial program 70.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \left(a \cdot \left(b \cdot i\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(j \cdot y\right) \cdot i}\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \cdot i + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, -1 \cdot \left(j \cdot y\right) + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, j \cdot \color{blue}{\left(-1 \cdot y\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)}, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 74.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in j around 0

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{a \cdot b}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{b \cdot a}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]

       2. *-lowering-*.f64 68.6

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{b \cdot a}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]

   11. Simplified 68.6%

       \[\leadsto \mathsf{fma}\left(i, \color{blue}{b \cdot a}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]

   if 6.4999999999999997e-53 < j

   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

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot i} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot \left(a \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(a \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-lowering-*.f64 67.2

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 67.2%

      \[\leadsto \color{blue}{i \cdot \left(b \cdot a\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -3.8 \cdot 10^{-53}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;j \leq 6.5 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + i \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{if}\;j \leq -1.55 \cdot 10^{-86}:\\ \;\;\;\;t\_1 - a \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;j \leq 7.8 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + i \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* t c) (* y i)))))
   (if (<= j -1.55e-86)
     (- t_1 (* a (* x t)))
     (if (<= j 7.8e-51)
       (fma i (* a b) (* z (fma c (- 0.0 b) (* x y))))
       (+ t_1 (* i (* a 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 = j * ((t * c) - (y * i));
	double tmp;
	if (j <= -1.55e-86) {
		tmp = t_1 - (a * (x * t));
	} else if (j <= 7.8e-51) {
		tmp = fma(i, (a * b), (z * fma(c, (0.0 - b), (x * y))));
	} else {
		tmp = t_1 + (i * (a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(t * c) - Float64(y * i)))
	tmp = 0.0
	if (j <= -1.55e-86)
		tmp = Float64(t_1 - Float64(a * Float64(x * t)));
	elseif (j <= 7.8e-51)
		tmp = fma(i, Float64(a * b), Float64(z * fma(c, Float64(0.0 - b), Float64(x * y))));
	else
		tmp = Float64(t_1 + Float64(i * Float64(a * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -1.55e-86], N[(t$95$1 - N[(a * N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 7.8e-51], N[(i * N[(a * b), $MachinePrecision] + N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if j < -1.54999999999999994e-86

   1. Initial program 79.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. neg-sub0 N/A

         \[\leadsto a \cdot \color{blue}{\left(0 - t \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(0 - t \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. *-lowering-*.f64 67.5

         \[\leadsto a \cdot \left(0 - \color{blue}{t \cdot x}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 67.5%

      \[\leadsto \color{blue}{a \cdot \left(0 - t \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   if -1.54999999999999994e-86 < j < 7.7999999999999995e-51

   1. Initial program 69.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \left(a \cdot \left(b \cdot i\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(j \cdot y\right) \cdot i}\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \cdot i + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, -1 \cdot \left(j \cdot y\right) + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, j \cdot \color{blue}{\left(-1 \cdot y\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)}, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 76.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in j around 0

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{a \cdot b}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{b \cdot a}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]

       2. *-lowering-*.f64 70.8

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{b \cdot a}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]

   11. Simplified 70.8%

       \[\leadsto \mathsf{fma}\left(i, \color{blue}{b \cdot a}, z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right) \]

   if 7.7999999999999995e-51 < j

   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

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot i} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot \left(a \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(a \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-lowering-*.f64 67.2

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 67.2%

      \[\leadsto \color{blue}{i \cdot \left(b \cdot a\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.55 \cdot 10^{-86}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) - a \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;j \leq 7.8 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + i \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.44 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-267}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \left(0 - t \cdot a\right)\\ \mathbf{elif}\;y \leq 0.29:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(y \cdot \left(0 - i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -1.44e-8)
   (* x (* y z))
   (if (<= y 1.4e-267)
     (* j (* t c))
     (if (<= y 1.06e-128)
       (* x (- 0.0 (* t a)))
       (if (<= y 0.29) (* i (* a b)) (* j (* y (- 0.0 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 (y <= -1.44e-8) {
		tmp = x * (y * z);
	} else if (y <= 1.4e-267) {
		tmp = j * (t * c);
	} else if (y <= 1.06e-128) {
		tmp = x * (0.0 - (t * a));
	} else if (y <= 0.29) {
		tmp = i * (a * b);
	} else {
		tmp = j * (y * (0.0 - i));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (y <= (-1.44d-8)) then
        tmp = x * (y * z)
    else if (y <= 1.4d-267) then
        tmp = j * (t * c)
    else if (y <= 1.06d-128) then
        tmp = x * (0.0d0 - (t * a))
    else if (y <= 0.29d0) then
        tmp = i * (a * b)
    else
        tmp = j * (y * (0.0d0 - i))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -1.44e-8) {
		tmp = x * (y * z);
	} else if (y <= 1.4e-267) {
		tmp = j * (t * c);
	} else if (y <= 1.06e-128) {
		tmp = x * (0.0 - (t * a));
	} else if (y <= 0.29) {
		tmp = i * (a * b);
	} else {
		tmp = j * (y * (0.0 - i));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if y <= -1.44e-8:
		tmp = x * (y * z)
	elif y <= 1.4e-267:
		tmp = j * (t * c)
	elif y <= 1.06e-128:
		tmp = x * (0.0 - (t * a))
	elif y <= 0.29:
		tmp = i * (a * b)
	else:
		tmp = j * (y * (0.0 - i))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -1.44e-8)
		tmp = Float64(x * Float64(y * z));
	elseif (y <= 1.4e-267)
		tmp = Float64(j * Float64(t * c));
	elseif (y <= 1.06e-128)
		tmp = Float64(x * Float64(0.0 - Float64(t * a)));
	elseif (y <= 0.29)
		tmp = Float64(i * Float64(a * b));
	else
		tmp = Float64(j * Float64(y * Float64(0.0 - i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (y <= -1.44e-8)
		tmp = x * (y * z);
	elseif (y <= 1.4e-267)
		tmp = j * (t * c);
	elseif (y <= 1.06e-128)
		tmp = x * (0.0 - (t * a));
	elseif (y <= 0.29)
		tmp = i * (a * b);
	else
		tmp = j * (y * (0.0 - i));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -1.44e-8], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e-267], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.06e-128], N[(x * N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.29], N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision], N[(j * N[(y * N[(0.0 - i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.44e-8

   1. Initial program 69.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 44.8

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 44.8%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 38.4

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   8. Simplified 38.4%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   if -1.44e-8 < y < 1.40000000000000002e-267

   1. Initial program 78.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. 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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 36.4

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 36.4%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 33.1

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]

   8. Simplified 33.1%

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]

   if 1.40000000000000002e-267 < y < 1.05999999999999995e-128

   1. Initial program 88.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 49.5

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 49.5%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot t\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot t\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{t \cdot a}\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(\mathsf{neg}\left(a\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \left(t \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(-1 \cdot a\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto x \cdot \left(t \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

      7. neg-sub0 N/A

         \[\leadsto x \cdot \left(t \cdot \color{blue}{\left(0 - a\right)}\right) \]

      8. --lowering--.f64 44.4

         \[\leadsto x \cdot \left(t \cdot \color{blue}{\left(0 - a\right)}\right) \]

   8. Simplified 44.4%

      \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(0 - a\right)\right)} \]

   if 1.05999999999999995e-128 < y < 0.28999999999999998

   1. Initial program 79.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 55.4

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 55.4%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

      2. *-lowering-*.f64 41.5

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   8. Simplified 41.5%

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot i} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot a\right)} \cdot i \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot i} \]

      5. *-lowering-*.f64 47.0

         \[\leadsto \color{blue}{\left(b \cdot a\right)} \cdot i \]

   10. Applied egg-rr 47.0%

       \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot i} \]

   if 0.28999999999999998 < y

   1. Initial program 68.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 57.8

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 57.8%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto j \cdot \color{blue}{\left(-1 \cdot \left(i \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(\mathsf{neg}\left(i \cdot y\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \color{blue}{\left(i \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \color{blue}{\left(i \cdot \left(-1 \cdot y\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      6. neg-sub0 N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      7. --lowering--.f64 47.9

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   8. Simplified 47.9%

      \[\leadsto j \cdot \color{blue}{\left(i \cdot \left(0 - y\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.44 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-267}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \left(0 - t \cdot a\right)\\ \mathbf{elif}\;y \leq 0.29:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(y \cdot \left(0 - i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{-269}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-129}:\\ \;\;\;\;a \cdot \left(t \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;y \leq 0.29:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(y \cdot \left(0 - i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -9.5e-5)
   (* x (* y z))
   (if (<= y 1.36e-269)
     (* j (* t c))
     (if (<= y 9.2e-129)
       (* a (* t (- 0.0 x)))
       (if (<= y 0.29) (* i (* a b)) (* j (* y (- 0.0 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 (y <= -9.5e-5) {
		tmp = x * (y * z);
	} else if (y <= 1.36e-269) {
		tmp = j * (t * c);
	} else if (y <= 9.2e-129) {
		tmp = a * (t * (0.0 - x));
	} else if (y <= 0.29) {
		tmp = i * (a * b);
	} else {
		tmp = j * (y * (0.0 - i));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (y <= (-9.5d-5)) then
        tmp = x * (y * z)
    else if (y <= 1.36d-269) then
        tmp = j * (t * c)
    else if (y <= 9.2d-129) then
        tmp = a * (t * (0.0d0 - x))
    else if (y <= 0.29d0) then
        tmp = i * (a * b)
    else
        tmp = j * (y * (0.0d0 - i))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -9.5e-5) {
		tmp = x * (y * z);
	} else if (y <= 1.36e-269) {
		tmp = j * (t * c);
	} else if (y <= 9.2e-129) {
		tmp = a * (t * (0.0 - x));
	} else if (y <= 0.29) {
		tmp = i * (a * b);
	} else {
		tmp = j * (y * (0.0 - i));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if y <= -9.5e-5:
		tmp = x * (y * z)
	elif y <= 1.36e-269:
		tmp = j * (t * c)
	elif y <= 9.2e-129:
		tmp = a * (t * (0.0 - x))
	elif y <= 0.29:
		tmp = i * (a * b)
	else:
		tmp = j * (y * (0.0 - i))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -9.5e-5)
		tmp = Float64(x * Float64(y * z));
	elseif (y <= 1.36e-269)
		tmp = Float64(j * Float64(t * c));
	elseif (y <= 9.2e-129)
		tmp = Float64(a * Float64(t * Float64(0.0 - x)));
	elseif (y <= 0.29)
		tmp = Float64(i * Float64(a * b));
	else
		tmp = Float64(j * Float64(y * Float64(0.0 - i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (y <= -9.5e-5)
		tmp = x * (y * z);
	elseif (y <= 1.36e-269)
		tmp = j * (t * c);
	elseif (y <= 9.2e-129)
		tmp = a * (t * (0.0 - x));
	elseif (y <= 0.29)
		tmp = i * (a * b);
	else
		tmp = j * (y * (0.0 - i));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -9.5e-5], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.36e-269], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.2e-129], N[(a * N[(t * N[(0.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.29], N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision], N[(j * N[(y * N[(0.0 - i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -9.5000000000000005e-5

   1. Initial program 69.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 44.8

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 44.8%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 38.4

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   8. Simplified 38.4%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   if -9.5000000000000005e-5 < y < 1.36e-269

   1. Initial program 78.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. 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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 36.4

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 36.4%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 33.1

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]

   8. Simplified 33.1%

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]

   if 1.36e-269 < y < 9.1999999999999998e-129

   1. Initial program 88.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 49.5

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 49.5%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \color{blue}{\left(t \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(t \cdot \left(-1 \cdot x\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. neg-sub0 N/A

          \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(0 - x\right)}\right) \]

      11. --lowering--.f64 44.3

          \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(0 - x\right)}\right) \]

   8. Simplified 44.3%

      \[\leadsto \color{blue}{a \cdot \left(t \cdot \left(0 - x\right)\right)} \]

   if 9.1999999999999998e-129 < y < 0.28999999999999998

   1. Initial program 79.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 55.4

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 55.4%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

      2. *-lowering-*.f64 41.5

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   8. Simplified 41.5%

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot i} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot a\right)} \cdot i \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot i} \]

      5. *-lowering-*.f64 47.0

         \[\leadsto \color{blue}{\left(b \cdot a\right)} \cdot i \]

   10. Applied egg-rr 47.0%

       \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot i} \]

   if 0.28999999999999998 < y

   1. Initial program 68.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 57.8

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 57.8%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto j \cdot \color{blue}{\left(-1 \cdot \left(i \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(\mathsf{neg}\left(i \cdot y\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \color{blue}{\left(i \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \color{blue}{\left(i \cdot \left(-1 \cdot y\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      6. neg-sub0 N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      7. --lowering--.f64 47.9

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   8. Simplified 47.9%

      \[\leadsto j \cdot \color{blue}{\left(i \cdot \left(0 - y\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{-269}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-129}:\\ \;\;\;\;a \cdot \left(t \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;y \leq 0.29:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(y \cdot \left(0 - i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.9 \cdot 10^{+38}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -2.7 \cdot 10^{-98}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{-12}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(0 - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -1.9e+38)
   (* i (fma j (- 0.0 y) (* a b)))
   (if (<= i -2.7e-98)
     (* x (- (* y z) (* t a)))
     (if (<= i 3.8e-12)
       (* c (fma j t (* z (- 0.0 b))))
       (* i (- (* a b) (* y j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -1.9e+38) {
		tmp = i * fma(j, (0.0 - y), (a * b));
	} else if (i <= -2.7e-98) {
		tmp = x * ((y * z) - (t * a));
	} else if (i <= 3.8e-12) {
		tmp = c * fma(j, t, (z * (0.0 - b)));
	} else {
		tmp = i * ((a * b) - (y * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -1.9e+38)
		tmp = Float64(i * fma(j, Float64(0.0 - y), Float64(a * b)));
	elseif (i <= -2.7e-98)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	elseif (i <= 3.8e-12)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(0.0 - b))));
	else
		tmp = Float64(i * Float64(Float64(a * b) - Float64(y * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -1.9e+38], N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -2.7e-98], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.8e-12], N[(c * N[(j * t + N[(z * N[(0.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(N[(a * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -1.8999999999999999e38

   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 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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 66.8

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   if -1.8999999999999999e38 < i < -2.6999999999999999e-98

   1. Initial program 70.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 55.0

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 55.0%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   if -2.6999999999999999e-98 < i < 3.79999999999999996e-12

   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 z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

      2. +-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{z \cdot \left(\mathsf{neg}\left(b\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, z \cdot \color{blue}{\left(-1 \cdot b\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{z \cdot \left(-1 \cdot b\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, z \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

      10. neg-sub0 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, z \cdot \color{blue}{\left(0 - b\right)}\right) \]

      11. --lowering--.f64 56.6

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, z \cdot \color{blue}{\left(0 - b\right)}\right) \]

   8. Simplified 56.6%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, z \cdot \left(0 - b\right)\right)} \]

   if 3.79999999999999996e-12 < i

   1. Initial program 71.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 80.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \left(a \cdot \left(b \cdot i\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(j \cdot y\right) \cdot i}\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \cdot i + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, -1 \cdot \left(j \cdot y\right) + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, j \cdot \color{blue}{\left(-1 \cdot y\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)}, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 79.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

       2. +-commutative N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b + -1 \cdot \left(j \cdot y\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto i \cdot \left(a \cdot b + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right) \]

       4. unsub-neg N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b - j \cdot y\right)} \]

       5. --lowering--.f64 N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b - j \cdot y\right)} \]

       6. *-commutative N/A

          \[\leadsto i \cdot \left(\color{blue}{b \cdot a} - j \cdot y\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto i \cdot \left(\color{blue}{b \cdot a} - j \cdot y\right) \]

       8. *-commutative N/A

          \[\leadsto i \cdot \left(b \cdot a - \color{blue}{y \cdot j}\right) \]

       9. *-lowering-*.f64 67.4

          \[\leadsto i \cdot \left(b \cdot a - \color{blue}{y \cdot j}\right) \]

   11. Simplified 67.4%

       \[\leadsto \color{blue}{i \cdot \left(b \cdot a - y \cdot j\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.9 \cdot 10^{+38}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -2.7 \cdot 10^{-98}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{-12}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(0 - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(a \cdot b - y \cdot j\right)\\ \mathbf{if}\;i \leq -6.4 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -1.85 \cdot 10^{-101}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-12}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(0 - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (- (* a b) (* y j)))))
   (if (<= i -6.4e+32)
     t_1
     (if (<= i -1.85e-101)
       (* x (- (* y z) (* t a)))
       (if (<= i 8.6e-12) (* c (fma j t (* z (- 0.0 b)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * ((a * b) - (y * j));
	double tmp;
	if (i <= -6.4e+32) {
		tmp = t_1;
	} else if (i <= -1.85e-101) {
		tmp = x * ((y * z) - (t * a));
	} else if (i <= 8.6e-12) {
		tmp = c * fma(j, t, (z * (0.0 - b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * Float64(Float64(a * b) - Float64(y * j)))
	tmp = 0.0
	if (i <= -6.4e+32)
		tmp = t_1;
	elseif (i <= -1.85e-101)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	elseif (i <= 8.6e-12)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(0.0 - b))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(N[(a * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -6.4e+32], t$95$1, If[LessEqual[i, -1.85e-101], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 8.6e-12], N[(c * N[(j * t + N[(z * N[(0.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -6.3999999999999998e32 or 8.59999999999999971e-12 < i

   1. Initial program 69.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 74.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \left(a \cdot \left(b \cdot i\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(j \cdot y\right) \cdot i}\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \cdot i + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, -1 \cdot \left(j \cdot y\right) + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, j \cdot \color{blue}{\left(-1 \cdot y\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)}, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 77.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

       2. +-commutative N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b + -1 \cdot \left(j \cdot y\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto i \cdot \left(a \cdot b + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right) \]

       4. unsub-neg N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b - j \cdot y\right)} \]

       5. --lowering--.f64 N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b - j \cdot y\right)} \]

       6. *-commutative N/A

          \[\leadsto i \cdot \left(\color{blue}{b \cdot a} - j \cdot y\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto i \cdot \left(\color{blue}{b \cdot a} - j \cdot y\right) \]

       8. *-commutative N/A

          \[\leadsto i \cdot \left(b \cdot a - \color{blue}{y \cdot j}\right) \]

       9. *-lowering-*.f64 66.3

          \[\leadsto i \cdot \left(b \cdot a - \color{blue}{y \cdot j}\right) \]

   11. Simplified 66.3%

       \[\leadsto \color{blue}{i \cdot \left(b \cdot a - y \cdot j\right)} \]

   if -6.3999999999999998e32 < i < -1.85000000000000002e-101

   1. Initial program 70.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 55.0

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 55.0%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   if -1.85000000000000002e-101 < i < 8.59999999999999971e-12

   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 z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

      2. +-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + -1 \cdot \left(b \cdot z\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{z \cdot \left(\mathsf{neg}\left(b\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, z \cdot \color{blue}{\left(-1 \cdot b\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{z \cdot \left(-1 \cdot b\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, z \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

      10. neg-sub0 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, z \cdot \color{blue}{\left(0 - b\right)}\right) \]

      11. --lowering--.f64 56.6

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, z \cdot \color{blue}{\left(0 - b\right)}\right) \]

   8. Simplified 56.6%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, z \cdot \left(0 - b\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6.4 \cdot 10^{+32}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \mathbf{elif}\;i \leq -1.85 \cdot 10^{-101}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-12}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(0 - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(a \cdot b - y \cdot j\right)\\ \mathbf{if}\;i \leq -1.65 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -1.15 \cdot 10^{-103}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;i \leq 3.7 \cdot 10^{-13}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, 0 - z, t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (- (* a b) (* y j)))))
   (if (<= i -1.65e+35)
     t_1
     (if (<= i -1.15e-103)
       (* x (- (* y z) (* t a)))
       (if (<= i 3.7e-13) (* c (fma b (- 0.0 z) (* 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 = i * ((a * b) - (y * j));
	double tmp;
	if (i <= -1.65e+35) {
		tmp = t_1;
	} else if (i <= -1.15e-103) {
		tmp = x * ((y * z) - (t * a));
	} else if (i <= 3.7e-13) {
		tmp = c * fma(b, (0.0 - z), (t * j));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * Float64(Float64(a * b) - Float64(y * j)))
	tmp = 0.0
	if (i <= -1.65e+35)
		tmp = t_1;
	elseif (i <= -1.15e-103)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	elseif (i <= 3.7e-13)
		tmp = Float64(c * fma(b, Float64(0.0 - z), Float64(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[(i * N[(N[(a * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.65e+35], t$95$1, If[LessEqual[i, -1.15e-103], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.7e-13], N[(c * N[(b * N[(0.0 - z), $MachinePrecision] + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.6500000000000001e35 or 3.69999999999999989e-13 < i

   1. Initial program 69.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 74.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \left(a \cdot \left(b \cdot i\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(j \cdot y\right) \cdot i}\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \cdot i + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, -1 \cdot \left(j \cdot y\right) + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, j \cdot \color{blue}{\left(-1 \cdot y\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)}, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 77.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

       2. +-commutative N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b + -1 \cdot \left(j \cdot y\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto i \cdot \left(a \cdot b + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right) \]

       4. unsub-neg N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b - j \cdot y\right)} \]

       5. --lowering--.f64 N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b - j \cdot y\right)} \]

       6. *-commutative N/A

          \[\leadsto i \cdot \left(\color{blue}{b \cdot a} - j \cdot y\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto i \cdot \left(\color{blue}{b \cdot a} - j \cdot y\right) \]

       8. *-commutative N/A

          \[\leadsto i \cdot \left(b \cdot a - \color{blue}{y \cdot j}\right) \]

       9. *-lowering-*.f64 66.3

          \[\leadsto i \cdot \left(b \cdot a - \color{blue}{y \cdot j}\right) \]

   11. Simplified 66.3%

       \[\leadsto \color{blue}{i \cdot \left(b \cdot a - y \cdot j\right)} \]

   if -1.6500000000000001e35 < i < -1.15e-103

   1. Initial program 70.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 55.0

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 55.0%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   if -1.15e-103 < i < 3.69999999999999989e-13

   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 c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} + j \cdot t\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + j \cdot t\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)} + j \cdot t\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, j \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{0 - z}, j \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{0 - z}, j \cdot t\right) \]

      12. *-lowering-*.f64 55.6

          \[\leadsto c \cdot \mathsf{fma}\left(b, 0 - z, \color{blue}{j \cdot t}\right) \]

   5. Simplified 55.6%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(b, 0 - z, j \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.65 \cdot 10^{+35}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \mathbf{elif}\;i \leq -1.15 \cdot 10^{-103}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;i \leq 3.7 \cdot 10^{-13}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, 0 - z, t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 31.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+43}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-87}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-267}:\\ \;\;\;\;a \cdot \left(t \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;b \leq 125000:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -6e+43)
   (* i (* a b))
   (if (<= b -1.35e-87)
     (* x (* y z))
     (if (<= b 1.95e-267)
       (* a (* t (- 0.0 x)))
       (if (<= b 125000.0) (* y (* x z)) (* a (* b 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 <= -6e+43) {
		tmp = i * (a * b);
	} else if (b <= -1.35e-87) {
		tmp = x * (y * z);
	} else if (b <= 1.95e-267) {
		tmp = a * (t * (0.0 - x));
	} else if (b <= 125000.0) {
		tmp = y * (x * z);
	} else {
		tmp = a * (b * i);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (b <= (-6d+43)) then
        tmp = i * (a * b)
    else if (b <= (-1.35d-87)) then
        tmp = x * (y * z)
    else if (b <= 1.95d-267) then
        tmp = a * (t * (0.0d0 - x))
    else if (b <= 125000.0d0) then
        tmp = y * (x * z)
    else
        tmp = a * (b * i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -6e+43) {
		tmp = i * (a * b);
	} else if (b <= -1.35e-87) {
		tmp = x * (y * z);
	} else if (b <= 1.95e-267) {
		tmp = a * (t * (0.0 - x));
	} else if (b <= 125000.0) {
		tmp = y * (x * z);
	} else {
		tmp = a * (b * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -6e+43:
		tmp = i * (a * b)
	elif b <= -1.35e-87:
		tmp = x * (y * z)
	elif b <= 1.95e-267:
		tmp = a * (t * (0.0 - x))
	elif b <= 125000.0:
		tmp = y * (x * z)
	else:
		tmp = a * (b * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -6e+43)
		tmp = Float64(i * Float64(a * b));
	elseif (b <= -1.35e-87)
		tmp = Float64(x * Float64(y * z));
	elseif (b <= 1.95e-267)
		tmp = Float64(a * Float64(t * Float64(0.0 - x)));
	elseif (b <= 125000.0)
		tmp = Float64(y * Float64(x * z));
	else
		tmp = Float64(a * Float64(b * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -6e+43)
		tmp = i * (a * b);
	elseif (b <= -1.35e-87)
		tmp = x * (y * z);
	elseif (b <= 1.95e-267)
		tmp = a * (t * (0.0 - x));
	elseif (b <= 125000.0)
		tmp = y * (x * z);
	else
		tmp = a * (b * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -6e+43], N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.35e-87], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-267], N[(a * N[(t * N[(0.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 125000.0], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -6.00000000000000033e43

   1. Initial program 72.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 48.3

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 48.3%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

      2. *-lowering-*.f64 42.7

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   8. Simplified 42.7%

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot i} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot a\right)} \cdot i \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot i} \]

      5. *-lowering-*.f64 53.9

         \[\leadsto \color{blue}{\left(b \cdot a\right)} \cdot i \]

   10. Applied egg-rr 53.9%

       \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot i} \]

   if -6.00000000000000033e43 < b < -1.34999999999999992e-87

   1. Initial program 81.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 48.1

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 48.1%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 38.4

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   8. Simplified 38.4%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   if -1.34999999999999992e-87 < b < 1.94999999999999988e-267

   1. Initial program 70.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 40.5

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 40.5%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \color{blue}{\left(t \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(t \cdot \left(-1 \cdot x\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. neg-sub0 N/A

          \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(0 - x\right)}\right) \]

      11. --lowering--.f64 34.5

          \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(0 - x\right)}\right) \]

   8. Simplified 34.5%

      \[\leadsto \color{blue}{a \cdot \left(t \cdot \left(0 - x\right)\right)} \]

   if 1.94999999999999988e-267 < b < 125000

   1. Initial program 76.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 45.4

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 45.4%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \frac{a \cdot \left(t \cdot x\right)}{y} + x \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \frac{a \cdot \left(t \cdot x\right)}{y} + x \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z + -1 \cdot \frac{a \cdot \left(t \cdot x\right)}{y}\right)} \]

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(x \cdot z + \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot \left(t \cdot x\right)}{y}\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z - \frac{a \cdot \left(t \cdot x\right)}{y}\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z - \frac{a \cdot \left(t \cdot x\right)}{y}\right)} \]

      6. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{z \cdot x} - \frac{a \cdot \left(t \cdot x\right)}{y}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \left(\color{blue}{z \cdot x} - \frac{a \cdot \left(t \cdot x\right)}{y}\right) \]

      8. associate-/l* N/A

         \[\leadsto y \cdot \left(z \cdot x - \color{blue}{a \cdot \frac{t \cdot x}{y}}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \left(z \cdot x - \color{blue}{a \cdot \frac{t \cdot x}{y}}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto y \cdot \left(z \cdot x - a \cdot \color{blue}{\frac{t \cdot x}{y}}\right) \]

      11. *-lowering-*.f64 43.4

          \[\leadsto y \cdot \left(z \cdot x - a \cdot \frac{\color{blue}{t \cdot x}}{y}\right) \]

   8. Simplified 43.4%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot x - a \cdot \frac{t \cdot x}{y}\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto y \cdot \color{blue}{\left(x \cdot z\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} \]

       2. *-lowering-*.f64 36.8

          \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} \]

   11. Simplified 36.8%

       \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} \]

   if 125000 < b

   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 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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 49.5

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 49.5%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

      2. *-lowering-*.f64 38.3

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   8. Simplified 38.3%

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+43}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-87}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-267}:\\ \;\;\;\;a \cdot \left(t \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;b \leq 125000:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 52.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+21}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (fma c (- 0.0 b) (* x y)))))
   (if (<= z -6.8e+104)
     t_1
     (if (<= z 1.85e+21) (* i (- (* a b) (* y 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 * fma(c, (0.0 - b), (x * y));
	double tmp;
	if (z <= -6.8e+104) {
		tmp = t_1;
	} else if (z <= 1.85e+21) {
		tmp = i * ((a * b) - (y * j));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * fma(c, Float64(0.0 - b), Float64(x * y)))
	tmp = 0.0
	if (z <= -6.8e+104)
		tmp = t_1;
	elseif (z <= 1.85e+21)
		tmp = Float64(i * Float64(Float64(a * b) - Float64(y * 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[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.8e+104], t$95$1, If[LessEqual[z, 1.85e+21], N[(i * N[(N[(a * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.7999999999999994e104 or 1.85e21 < z

   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 z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. neg-sub0 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{0 - b}, x \cdot y\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{0 - b}, x \cdot y\right) \]

      11. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

      12. *-lowering-*.f64 64.2

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 64.2%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, y \cdot x\right) \]

      2. neg-lowering-neg.f64 64.2

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{-b}, y \cdot x\right) \]

   7. Applied egg-rr 64.2%

      \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{-b}, y \cdot x\right) \]

   if -6.7999999999999994e104 < z < 1.85e21

   1. Initial program 75.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 75.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \left(a \cdot \left(b \cdot i\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(j \cdot y\right) \cdot i}\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \cdot i + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, -1 \cdot \left(j \cdot y\right) + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, j \cdot \color{blue}{\left(-1 \cdot y\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)}, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 56.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

       2. +-commutative N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b + -1 \cdot \left(j \cdot y\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto i \cdot \left(a \cdot b + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right) \]

       4. unsub-neg N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b - j \cdot y\right)} \]

       5. --lowering--.f64 N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b - j \cdot y\right)} \]

       6. *-commutative N/A

          \[\leadsto i \cdot \left(\color{blue}{b \cdot a} - j \cdot y\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto i \cdot \left(\color{blue}{b \cdot a} - j \cdot y\right) \]

       8. *-commutative N/A

          \[\leadsto i \cdot \left(b \cdot a - \color{blue}{y \cdot j}\right) \]

       9. *-lowering-*.f64 51.9

          \[\leadsto i \cdot \left(b \cdot a - \color{blue}{y \cdot j}\right) \]

   11. Simplified 51.9%

       \[\leadsto \color{blue}{i \cdot \left(b \cdot a - y \cdot j\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+104}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+21}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 52.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{if}\;b \leq -1.16 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* a i) (* z c)))))
   (if (<= b -1.16e+20) t_1 (if (<= b 3.1e-6) (* x (- (* y z) (* t a))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * ((a * i) - (z * c));
	double tmp;
	if (b <= -1.16e+20) {
		tmp = t_1;
	} else if (b <= 3.1e-6) {
		tmp = x * ((y * z) - (t * 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 = b * ((a * i) - (z * c))
    if (b <= (-1.16d+20)) then
        tmp = t_1
    else if (b <= 3.1d-6) then
        tmp = x * ((y * z) - (t * 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 = b * ((a * i) - (z * c));
	double tmp;
	if (b <= -1.16e+20) {
		tmp = t_1;
	} else if (b <= 3.1e-6) {
		tmp = x * ((y * z) - (t * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = b * ((a * i) - (z * c))
	tmp = 0
	if b <= -1.16e+20:
		tmp = t_1
	elif b <= 3.1e-6:
		tmp = x * ((y * z) - (t * a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(a * i) - Float64(z * c)))
	tmp = 0.0
	if (b <= -1.16e+20)
		tmp = t_1;
	elseif (b <= 3.1e-6)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * 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 = b * ((a * i) - (z * c));
	tmp = 0.0;
	if (b <= -1.16e+20)
		tmp = t_1;
	elseif (b <= 3.1e-6)
		tmp = x * ((y * z) - (t * 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[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.16e+20], t$95$1, If[LessEqual[b, 3.1e-6], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.16e20 or 3.1e-6 < b

   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 z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 70.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \left(a \cdot \left(b \cdot i\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(j \cdot y\right) \cdot i}\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \cdot i + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, -1 \cdot \left(j \cdot y\right) + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, j \cdot \color{blue}{\left(-1 \cdot y\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)}, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 71.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right) + a \cdot \left(b \cdot i\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{b \cdot \left(\mathsf{neg}\left(c \cdot z\right)\right)} + a \cdot \left(b \cdot i\right) \]

       3. mul-1-neg N/A

          \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z\right)\right)} + a \cdot \left(b \cdot i\right) \]

       4. *-commutative N/A

          \[\leadsto b \cdot \left(-1 \cdot \left(c \cdot z\right)\right) + \color{blue}{\left(b \cdot i\right) \cdot a} \]

       5. associate-*r* N/A

          \[\leadsto b \cdot \left(-1 \cdot \left(c \cdot z\right)\right) + \color{blue}{b \cdot \left(i \cdot a\right)} \]

       6. *-commutative N/A

          \[\leadsto b \cdot \left(-1 \cdot \left(c \cdot z\right)\right) + b \cdot \color{blue}{\left(a \cdot i\right)} \]

       7. distribute-lft-in N/A

          \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]

       9. +-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(a \cdot i + -1 \cdot \left(c \cdot z\right)\right)} \]

       10. mul-1-neg N/A

           \[\leadsto b \cdot \left(a \cdot i + \color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)}\right) \]

       11. unsub-neg N/A

           \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

       12. --lowering--.f64 N/A

           \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

       13. *-commutative N/A

           \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

       15. *-commutative N/A

           \[\leadsto b \cdot \left(i \cdot a - \color{blue}{z \cdot c}\right) \]

       16. *-lowering-*.f64 67.1

           \[\leadsto b \cdot \left(i \cdot a - \color{blue}{z \cdot c}\right) \]

   11. Simplified 67.1%

       \[\leadsto \color{blue}{b \cdot \left(i \cdot a - z \cdot c\right)} \]

   if -1.16e20 < b < 3.1e-6

   1. Initial program 75.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 44.2

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 44.2%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.16 \cdot 10^{+20}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 46.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+22}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* a i) (* z c)))))
   (if (<= z -8.2e+104)
     t_1
     (if (<= z 1.2e+22) (* i (- (* a b) (* y 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 = b * ((a * i) - (z * c));
	double tmp;
	if (z <= -8.2e+104) {
		tmp = t_1;
	} else if (z <= 1.2e+22) {
		tmp = i * ((a * b) - (y * 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 = b * ((a * i) - (z * c))
    if (z <= (-8.2d+104)) then
        tmp = t_1
    else if (z <= 1.2d+22) then
        tmp = i * ((a * b) - (y * 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 = b * ((a * i) - (z * c));
	double tmp;
	if (z <= -8.2e+104) {
		tmp = t_1;
	} else if (z <= 1.2e+22) {
		tmp = i * ((a * b) - (y * j));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = b * ((a * i) - (z * c))
	tmp = 0
	if z <= -8.2e+104:
		tmp = t_1
	elif z <= 1.2e+22:
		tmp = i * ((a * b) - (y * j))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(a * i) - Float64(z * c)))
	tmp = 0.0
	if (z <= -8.2e+104)
		tmp = t_1;
	elseif (z <= 1.2e+22)
		tmp = Float64(i * Float64(Float64(a * b) - Float64(y * 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 = b * ((a * i) - (z * c));
	tmp = 0.0;
	if (z <= -8.2e+104)
		tmp = t_1;
	elseif (z <= 1.2e+22)
		tmp = i * ((a * b) - (y * 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[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.2e+104], t$95$1, If[LessEqual[z, 1.2e+22], N[(i * N[(N[(a * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.1999999999999997e104 or 1.2e22 < z

   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 z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 81.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \left(a \cdot \left(b \cdot i\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(j \cdot y\right) \cdot i}\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \cdot i + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, -1 \cdot \left(j \cdot y\right) + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, j \cdot \color{blue}{\left(-1 \cdot y\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)}, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 73.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right) + a \cdot \left(b \cdot i\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{b \cdot \left(\mathsf{neg}\left(c \cdot z\right)\right)} + a \cdot \left(b \cdot i\right) \]

       3. mul-1-neg N/A

          \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z\right)\right)} + a \cdot \left(b \cdot i\right) \]

       4. *-commutative N/A

          \[\leadsto b \cdot \left(-1 \cdot \left(c \cdot z\right)\right) + \color{blue}{\left(b \cdot i\right) \cdot a} \]

       5. associate-*r* N/A

          \[\leadsto b \cdot \left(-1 \cdot \left(c \cdot z\right)\right) + \color{blue}{b \cdot \left(i \cdot a\right)} \]

       6. *-commutative N/A

          \[\leadsto b \cdot \left(-1 \cdot \left(c \cdot z\right)\right) + b \cdot \color{blue}{\left(a \cdot i\right)} \]

       7. distribute-lft-in N/A

          \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]

       9. +-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(a \cdot i + -1 \cdot \left(c \cdot z\right)\right)} \]

       10. mul-1-neg N/A

           \[\leadsto b \cdot \left(a \cdot i + \color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)}\right) \]

       11. unsub-neg N/A

           \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

       12. --lowering--.f64 N/A

           \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

       13. *-commutative N/A

           \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

       15. *-commutative N/A

           \[\leadsto b \cdot \left(i \cdot a - \color{blue}{z \cdot c}\right) \]

       16. *-lowering-*.f64 49.0

           \[\leadsto b \cdot \left(i \cdot a - \color{blue}{z \cdot c}\right) \]

   11. Simplified 49.0%

       \[\leadsto \color{blue}{b \cdot \left(i \cdot a - z \cdot c\right)} \]

   if -8.1999999999999997e104 < z < 1.2e22

   1. Initial program 76.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \left(a \cdot \left(b \cdot i\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(j \cdot y\right) \cdot i}\right)\right) + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \cdot i + a \cdot \left(b \cdot i\right)\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} + z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, -1 \cdot \left(j \cdot y\right) + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, j \cdot \color{blue}{\left(-1 \cdot y\right)} + a \cdot b, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)}, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right), z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 57.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)\right)} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

       2. +-commutative N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b + -1 \cdot \left(j \cdot y\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto i \cdot \left(a \cdot b + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right) \]

       4. unsub-neg N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b - j \cdot y\right)} \]

       5. --lowering--.f64 N/A

          \[\leadsto i \cdot \color{blue}{\left(a \cdot b - j \cdot y\right)} \]

       6. *-commutative N/A

          \[\leadsto i \cdot \left(\color{blue}{b \cdot a} - j \cdot y\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto i \cdot \left(\color{blue}{b \cdot a} - j \cdot y\right) \]

       8. *-commutative N/A

          \[\leadsto i \cdot \left(b \cdot a - \color{blue}{y \cdot j}\right) \]

       9. *-lowering-*.f64 51.6

          \[\leadsto i \cdot \left(b \cdot a - \color{blue}{y \cdot j}\right) \]

   11. Simplified 51.6%

       \[\leadsto \color{blue}{i \cdot \left(b \cdot a - y \cdot j\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+104}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+22}:\\ \;\;\;\;i \cdot \left(a \cdot b - y \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 30.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot i\right)\\ \mathbf{if}\;i \leq -1.65 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -1.5 \cdot 10^{-135}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;i \leq 9 \cdot 10^{-13}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (* b i))))
   (if (<= i -1.65e+41)
     t_1
     (if (<= i -1.5e-135) (* x (* y z)) (if (<= i 9e-13) (* j (* t c)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (b * i);
	double tmp;
	if (i <= -1.65e+41) {
		tmp = t_1;
	} else if (i <= -1.5e-135) {
		tmp = x * (y * z);
	} else if (i <= 9e-13) {
		tmp = j * (t * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * (b * i)
    if (i <= (-1.65d+41)) then
        tmp = t_1
    else if (i <= (-1.5d-135)) then
        tmp = x * (y * z)
    else if (i <= 9d-13) then
        tmp = j * (t * c)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (b * i);
	double tmp;
	if (i <= -1.65e+41) {
		tmp = t_1;
	} else if (i <= -1.5e-135) {
		tmp = x * (y * z);
	} else if (i <= 9e-13) {
		tmp = j * (t * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = a * (b * i)
	tmp = 0
	if i <= -1.65e+41:
		tmp = t_1
	elif i <= -1.5e-135:
		tmp = x * (y * z)
	elif i <= 9e-13:
		tmp = j * (t * c)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * Float64(b * i))
	tmp = 0.0
	if (i <= -1.65e+41)
		tmp = t_1;
	elseif (i <= -1.5e-135)
		tmp = Float64(x * Float64(y * z));
	elseif (i <= 9e-13)
		tmp = Float64(j * Float64(t * c));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = a * (b * i);
	tmp = 0.0;
	if (i <= -1.65e+41)
		tmp = t_1;
	elseif (i <= -1.5e-135)
		tmp = x * (y * z);
	elseif (i <= 9e-13)
		tmp = j * (t * c);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.65e+41], t$95$1, If[LessEqual[i, -1.5e-135], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 9e-13], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.65e41 or 9e-13 < i

   1. Initial program 68.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 49.8

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 49.8%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

      2. *-lowering-*.f64 37.7

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   8. Simplified 37.7%

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   if -1.65e41 < i < -1.50000000000000006e-135

   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 x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 49.5

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 49.5%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 35.4

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   8. Simplified 35.4%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   if -1.50000000000000006e-135 < i < 9e-13

   1. Initial program 82.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 39.3

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 39.3%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 33.0

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]

   8. Simplified 33.0%

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.65 \cdot 10^{+41}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;i \leq -1.5 \cdot 10^{-135}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;i \leq 9 \cdot 10^{-13}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 30.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+43}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;b \leq 180000:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -7.8e+43)
   (* i (* a b))
   (if (<= b 180000.0) (* x (* y z)) (* a (* b 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 <= -7.8e+43) {
		tmp = i * (a * b);
	} else if (b <= 180000.0) {
		tmp = x * (y * z);
	} else {
		tmp = a * (b * i);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (b <= (-7.8d+43)) then
        tmp = i * (a * b)
    else if (b <= 180000.0d0) then
        tmp = x * (y * z)
    else
        tmp = a * (b * i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -7.8e+43) {
		tmp = i * (a * b);
	} else if (b <= 180000.0) {
		tmp = x * (y * z);
	} else {
		tmp = a * (b * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -7.8e+43:
		tmp = i * (a * b)
	elif b <= 180000.0:
		tmp = x * (y * z)
	else:
		tmp = a * (b * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -7.8e+43)
		tmp = Float64(i * Float64(a * b));
	elseif (b <= 180000.0)
		tmp = Float64(x * Float64(y * z));
	else
		tmp = Float64(a * Float64(b * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -7.8e+43)
		tmp = i * (a * b);
	elseif (b <= 180000.0)
		tmp = x * (y * z);
	else
		tmp = a * (b * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -7.8e+43], N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 180000.0], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.8000000000000001e43

   1. Initial program 72.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 48.3

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 48.3%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

      2. *-lowering-*.f64 42.7

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   8. Simplified 42.7%

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot i} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot a\right)} \cdot i \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot i} \]

      5. *-lowering-*.f64 53.9

         \[\leadsto \color{blue}{\left(b \cdot a\right)} \cdot i \]

   10. Applied egg-rr 53.9%

       \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot i} \]

   if -7.8000000000000001e43 < b < 1.8e5

   1. Initial program 75.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 43.8

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 43.8%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 27.2

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   8. Simplified 27.2%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   if 1.8e5 < b

   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 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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 49.5

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 49.5%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

      2. *-lowering-*.f64 38.3

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   8. Simplified 38.3%

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+43}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;b \leq 180000:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 31.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot i\right)\\ \mathbf{if}\;i \leq -0.0078:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 8.5 \cdot 10^{-12}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (* b i))))
   (if (<= i -0.0078) t_1 (if (<= i 8.5e-12) (* j (* t c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (b * i);
	double tmp;
	if (i <= -0.0078) {
		tmp = t_1;
	} else if (i <= 8.5e-12) {
		tmp = j * (t * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * (b * i)
    if (i <= (-0.0078d0)) then
        tmp = t_1
    else if (i <= 8.5d-12) then
        tmp = j * (t * c)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (b * i);
	double tmp;
	if (i <= -0.0078) {
		tmp = t_1;
	} else if (i <= 8.5e-12) {
		tmp = j * (t * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = a * (b * i)
	tmp = 0
	if i <= -0.0078:
		tmp = t_1
	elif i <= 8.5e-12:
		tmp = j * (t * c)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * Float64(b * i))
	tmp = 0.0
	if (i <= -0.0078)
		tmp = t_1;
	elseif (i <= 8.5e-12)
		tmp = Float64(j * Float64(t * c));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = a * (b * i);
	tmp = 0.0;
	if (i <= -0.0078)
		tmp = t_1;
	elseif (i <= 8.5e-12)
		tmp = j * (t * c);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -0.0078], t$95$1, If[LessEqual[i, 8.5e-12], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -0.0077999999999999996 or 8.4999999999999997e-12 < i

   1. Initial program 69.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 47.6

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 47.6%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

      2. *-lowering-*.f64 35.2

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   8. Simplified 35.2%

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   if -0.0077999999999999996 < i < 8.4999999999999997e-12

   1. Initial program 80.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 38.8

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 38.8%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 29.9

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]

   8. Simplified 29.9%

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.0078:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;i \leq 8.5 \cdot 10^{-12}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 30.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot i\right)\\ \mathbf{if}\;i \leq -2.1 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 4.4 \cdot 10^{-13}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (* b i))))
   (if (<= i -2.1e+23) t_1 (if (<= i 4.4e-13) (* 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 = a * (b * i);
	double tmp;
	if (i <= -2.1e+23) {
		tmp = t_1;
	} else if (i <= 4.4e-13) {
		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 = a * (b * i)
    if (i <= (-2.1d+23)) then
        tmp = t_1
    else if (i <= 4.4d-13) 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 = a * (b * i);
	double tmp;
	if (i <= -2.1e+23) {
		tmp = t_1;
	} else if (i <= 4.4e-13) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = a * (b * i)
	tmp = 0
	if i <= -2.1e+23:
		tmp = t_1
	elif i <= 4.4e-13:
		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(a * Float64(b * i))
	tmp = 0.0
	if (i <= -2.1e+23)
		tmp = t_1;
	elseif (i <= 4.4e-13)
		tmp = Float64(c * Float64(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 = a * (b * i);
	tmp = 0.0;
	if (i <= -2.1e+23)
		tmp = t_1;
	elseif (i <= 4.4e-13)
		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[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2.1e+23], t$95$1, If[LessEqual[i, 4.4e-13], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -2.1000000000000001e23 or 4.39999999999999993e-13 < i

   1. Initial program 69.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 49.1

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 49.1%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

      2. *-lowering-*.f64 36.8

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   8. Simplified 36.8%

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   if -2.1000000000000001e23 < i < 4.39999999999999993e-13

   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 inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 36.9

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 36.9%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

      2. *-lowering-*.f64 28.4

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

   8. Simplified 28.4%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.1 \cdot 10^{+23}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;i \leq 4.4 \cdot 10^{-13}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 22.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(b \cdot i\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* a (* b i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (b * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = a * (b * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (b * i);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return a * (b * i)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(a * Float64(b * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = a * (b * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]

Derivation

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 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. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

   3. mul-1-neg N/A

      \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

   7. remove-double-neg N/A

      \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

   9. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

   10. neg-sub0 N/A

       \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

   12. *-commutative N/A

       \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   13. *-lowering-*.f64 37.8

       \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

5. Simplified 37.8%

   \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

6. Taylor expanded in t around 0

   \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   2. *-lowering-*.f64 22.5

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

8. Simplified 22.5%

   \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

9. Final simplification 22.5%

   \[\leadsto a \cdot \left(b \cdot i\right) \]
10. Add Preprocessing

Developer Target 1: 68.9% 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 2024196 
(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)))))