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

Percentage Accurate: 73.4% → 81.9%

Time: 22.8s

Alternatives: 26

Speedup: 0.8×

• 58.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 81.4%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 91.1%

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

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

   1. Initial program 93.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

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

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

      14. *-commutative N/A

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

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

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

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

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

      17. mul-1-neg N/A

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

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

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

      19. mul-1-neg N/A

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

   5. Simplified 55.1%

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

4. Final simplification 85.7%

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

Alternative 2: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, y, \mathsf{fma}\left(y, 0 - i \cdot j, 0\right)\right) - z \cdot \left(b \cdot c\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, y \cdot \left(0 - i\right)\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, x \cdot y\right), t \cdot \mathsf{fma}\left(a, 0 - x, b \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, 0 - z, t \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -8.2e+179)
   (- (fma (* x z) y (fma y (- 0.0 (* i j)) 0.0)) (* z (* b c)))
   (if (<= y 6.8e+123)
     (fma
      j
      (fma a c (* y (- 0.0 i)))
      (fma z (fma c (- 0.0 b) (* x y)) (* t (fma a (- 0.0 x) (* b i)))))
     (fma y (fma j (- 0.0 i) (* x z)) (* b (fma c (- 0.0 z) (* t i)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -8.2e+179) {
		tmp = fma((x * z), y, fma(y, (0.0 - (i * j)), 0.0)) - (z * (b * c));
	} else if (y <= 6.8e+123) {
		tmp = fma(j, fma(a, c, (y * (0.0 - i))), fma(z, fma(c, (0.0 - b), (x * y)), (t * fma(a, (0.0 - x), (b * i)))));
	} else {
		tmp = fma(y, fma(j, (0.0 - i), (x * z)), (b * fma(c, (0.0 - z), (t * i))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -8.2e+179)
		tmp = Float64(fma(Float64(x * z), y, fma(y, Float64(0.0 - Float64(i * j)), 0.0)) - Float64(z * Float64(b * c)));
	elseif (y <= 6.8e+123)
		tmp = fma(j, fma(a, c, Float64(y * Float64(0.0 - i))), fma(z, fma(c, Float64(0.0 - b), Float64(x * y)), Float64(t * fma(a, Float64(0.0 - x), Float64(b * i)))));
	else
		tmp = fma(y, fma(j, Float64(0.0 - i), Float64(x * z)), Float64(b * fma(c, Float64(0.0 - z), Float64(t * i))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -8.2e+179], N[(N[(N[(x * z), $MachinePrecision] * y + N[(y * N[(0.0 - N[(i * j), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] - N[(z * N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e+123], N[(j * N[(a * c + N[(y * N[(0.0 - i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(t * N[(a * N[(0.0 - x), $MachinePrecision] + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(j * N[(0.0 - i), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * N[(0.0 - z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.20000000000000021e179

   1. Initial program 45.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

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

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

      14. *-commutative N/A

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

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

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

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

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

      17. mul-1-neg N/A

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

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

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

      19. mul-1-neg N/A

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

   5. Simplified 74.6%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. neg-sub0 N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

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

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

      19. *-lowering-*.f64 81.2

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

   8. Simplified 81.2%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub0-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. +-lft-identity N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

      16. sub0-neg N/A

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

      17. associate-*r* N/A

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

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

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

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

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

      20. --lowering--.f64 81.3

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

   10. Applied egg-rr 81.3%

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

   if -8.20000000000000021e179 < y < 6.80000000000000002e123

   1. Initial program 80.2%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 85.5%

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

   if 6.80000000000000002e123 < y

   1. Initial program 61.3%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

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

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

      14. *-commutative N/A

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

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

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

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

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

      17. mul-1-neg N/A

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

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

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

      19. mul-1-neg N/A

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

   5. Simplified 82.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, y, \mathsf{fma}\left(y, 0 - i \cdot j, 0\right)\right) - z \cdot \left(b \cdot c\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, y \cdot \left(0 - i\right)\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, x \cdot y\right), t \cdot \mathsf{fma}\left(a, 0 - x, b \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, 0 - z, t \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+84}:\\ \;\;\;\;t \cdot \left(b \cdot \mathsf{fma}\left(0 - a, \frac{x}{b}, i\right)\right)\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{+25}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-57}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \mathbf{elif}\;t \leq -1.38 \cdot 10^{-105}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, 0 - b \cdot c\right)\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, y \cdot \left(0 - i\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, 0 - x, b \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -1.4e+84)
   (* t (* b (fma (- 0.0 a) (/ x b) i)))
   (if (<= t -5.8e+25)
     (* i (fma j (- 0.0 y) (* t b)))
     (if (<= t -6e-57)
       (* a (fma x (- 0.0 t) (* c j)))
       (if (<= t -1.38e-105)
         (* z (fma y x (- 0.0 (* b c))))
         (if (<= t 1.18e-14)
           (fma j (fma a c (* y (- 0.0 i))) (* x (* y z)))
           (* t (fma a (- 0.0 x) (* 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 (t <= -1.4e+84) {
		tmp = t * (b * fma((0.0 - a), (x / b), i));
	} else if (t <= -5.8e+25) {
		tmp = i * fma(j, (0.0 - y), (t * b));
	} else if (t <= -6e-57) {
		tmp = a * fma(x, (0.0 - t), (c * j));
	} else if (t <= -1.38e-105) {
		tmp = z * fma(y, x, (0.0 - (b * c)));
	} else if (t <= 1.18e-14) {
		tmp = fma(j, fma(a, c, (y * (0.0 - i))), (x * (y * z)));
	} else {
		tmp = t * fma(a, (0.0 - x), (b * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -1.4e+84)
		tmp = Float64(t * Float64(b * fma(Float64(0.0 - a), Float64(x / b), i)));
	elseif (t <= -5.8e+25)
		tmp = Float64(i * fma(j, Float64(0.0 - y), Float64(t * b)));
	elseif (t <= -6e-57)
		tmp = Float64(a * fma(x, Float64(0.0 - t), Float64(c * j)));
	elseif (t <= -1.38e-105)
		tmp = Float64(z * fma(y, x, Float64(0.0 - Float64(b * c))));
	elseif (t <= 1.18e-14)
		tmp = fma(j, fma(a, c, Float64(y * Float64(0.0 - i))), Float64(x * Float64(y * z)));
	else
		tmp = Float64(t * fma(a, Float64(0.0 - x), Float64(b * i)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -1.4e+84], N[(t * N[(b * N[(N[(0.0 - a), $MachinePrecision] * N[(x / b), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5.8e+25], N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -6e-57], N[(a * N[(x * N[(0.0 - t), $MachinePrecision] + N[(c * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.38e-105], N[(z * N[(y * x + N[(0.0 - N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.18e-14], N[(j * N[(a * c + N[(y * N[(0.0 - i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(a * N[(0.0 - x), $MachinePrecision] + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.39999999999999991e84

   1. Initial program 53.6%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 75.5%

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

   6. Taylor expanded in t around inf

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

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

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

      7. *-lowering-*.f64 72.9

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

   8. Simplified 72.9%

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

   9. Taylor expanded in b around inf

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

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

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

       2. +-commutative N/A

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

       3. associate-/l* N/A

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

       4. associate-*r* N/A

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

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

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

       6. mul-1-neg N/A

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

       7. neg-sub0 N/A

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

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

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

       9. /-lowering-/.f64 73.0

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

   11. Simplified 73.0%

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

   if -1.39999999999999991e84 < t < -5.7999999999999998e25

   1. Initial program 64.2%

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

   3. Taylor expanded in i around inf

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

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

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

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

      12. *-lowering-*.f64 82.7

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

   5. Simplified 82.7%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 82.7

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

   7. Applied egg-rr 82.7%

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

   if -5.7999999999999998e25 < t < -6.00000000000000001e-57

   1. Initial program 67.1%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-- N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 67.0

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

   4. Applied egg-rr 67.0%

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

   5. Taylor expanded in a around inf

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

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

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 60.8

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

   7. Simplified 60.8%

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

   if -6.00000000000000001e-57 < t < -1.3800000000000001e-105

   1. Initial program 67.1%

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

   3. Taylor expanded in 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 74.3

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

   5. Simplified 74.3%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. sub0-neg N/A

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

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

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

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

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

      7. *-lowering-*.f64 74.3

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

   7. Applied egg-rr 74.3%

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

   if -1.3800000000000001e-105 < t < 1.17999999999999993e-14

   1. Initial program 88.3%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 72.2%

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

   6. Taylor expanded in y around inf

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 70.3

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

   8. Simplified 70.3%

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

   if 1.17999999999999993e-14 < t

   1. Initial program 63.1%

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

   3. Taylor expanded in t around inf

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

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

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

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

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 58.5

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

   5. Simplified 58.5%

      \[\leadsto \color{blue}{t \cdot \mathsf{fma}\left(a, 0 - x, i \cdot b\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+84}:\\ \;\;\;\;t \cdot \left(b \cdot \mathsf{fma}\left(0 - a, \frac{x}{b}, i\right)\right)\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{+25}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-57}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \mathbf{elif}\;t \leq -1.38 \cdot 10^{-105}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, 0 - b \cdot c\right)\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, y \cdot \left(0 - i\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, 0 - x, b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+145}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(b, 0 - c, x \cdot y\right)\right)\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), z \cdot \left(0 - b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(0 - t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -1.65e+145)
   (* a (fma x (- 0.0 t) (* c j)))
   (if (<= a -4.8e-124)
     (fma t (* b i) (* z (fma b (- 0.0 c) (* x y))))
     (if (<= a 2.2e+156)
       (fma y (fma j (- 0.0 i) (* x z)) (* z (- 0.0 (* b c))))
       (* a (fma j c (* x (- 0.0 t))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -1.65e+145) {
		tmp = a * fma(x, (0.0 - t), (c * j));
	} else if (a <= -4.8e-124) {
		tmp = fma(t, (b * i), (z * fma(b, (0.0 - c), (x * y))));
	} else if (a <= 2.2e+156) {
		tmp = fma(y, fma(j, (0.0 - i), (x * z)), (z * (0.0 - (b * c))));
	} else {
		tmp = a * fma(j, c, (x * (0.0 - t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -1.65e+145)
		tmp = Float64(a * fma(x, Float64(0.0 - t), Float64(c * j)));
	elseif (a <= -4.8e-124)
		tmp = fma(t, Float64(b * i), Float64(z * fma(b, Float64(0.0 - c), Float64(x * y))));
	elseif (a <= 2.2e+156)
		tmp = fma(y, fma(j, Float64(0.0 - i), Float64(x * z)), Float64(z * Float64(0.0 - Float64(b * c))));
	else
		tmp = Float64(a * fma(j, c, Float64(x * Float64(0.0 - t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -1.65e+145], N[(a * N[(x * N[(0.0 - t), $MachinePrecision] + N[(c * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -4.8e-124], N[(t * N[(b * i), $MachinePrecision] + N[(z * N[(b * N[(0.0 - c), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.2e+156], N[(y * N[(j * N[(0.0 - i), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(z * N[(0.0 - N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(j * c + N[(x * N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.65000000000000013e145

   1. Initial program 67.1%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-- N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 67.0

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

   4. Applied egg-rr 67.0%

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

   5. Taylor expanded in a around inf

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

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

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 79.7

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

   7. Simplified 79.7%

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

   if -1.65000000000000013e145 < a < -4.79999999999999985e-124

   1. Initial program 72.2%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

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

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

      14. *-commutative N/A

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

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

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

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

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

      17. mul-1-neg N/A

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

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

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

      19. mul-1-neg N/A

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

   5. Simplified 73.8%

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

   6. Taylor expanded in j around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

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

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

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

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

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

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

   8. Simplified 71.8%

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

   if -4.79999999999999985e-124 < a < 2.20000000000000004e156

   1. Initial program 80.5%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

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

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

      14. *-commutative N/A

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

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

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

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

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

      17. mul-1-neg N/A

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

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

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

      19. mul-1-neg N/A

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

   5. Simplified 77.6%

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

   6. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. neg-sub0 N/A

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

      15. --lowering--.f64 68.6

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

   8. Simplified 68.6%

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

   if 2.20000000000000004e156 < a

   1. Initial program 51.8%

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

   3. Taylor expanded in a around inf

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

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

      11. --lowering--.f64 61.2

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

   5. Simplified 61.2%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+145}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(b, 0 - c, x \cdot y\right)\right)\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), z \cdot \left(0 - b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(0 - t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+142}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, 0 - z, t \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(0 - t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -1.3e+142)
   (* a (fma x (- 0.0 t) (* c j)))
   (if (<= a 5.8e+256)
     (fma y (fma j (- 0.0 i) (* x z)) (* b (fma c (- 0.0 z) (* t i))))
     (* a (fma j c (* x (- 0.0 t)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -1.3e+142) {
		tmp = a * fma(x, (0.0 - t), (c * j));
	} else if (a <= 5.8e+256) {
		tmp = fma(y, fma(j, (0.0 - i), (x * z)), (b * fma(c, (0.0 - z), (t * i))));
	} else {
		tmp = a * fma(j, c, (x * (0.0 - t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -1.3e+142)
		tmp = Float64(a * fma(x, Float64(0.0 - t), Float64(c * j)));
	elseif (a <= 5.8e+256)
		tmp = fma(y, fma(j, Float64(0.0 - i), Float64(x * z)), Float64(b * fma(c, Float64(0.0 - z), Float64(t * i))));
	else
		tmp = Float64(a * fma(j, c, Float64(x * Float64(0.0 - t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -1.3e+142], N[(a * N[(x * N[(0.0 - t), $MachinePrecision] + N[(c * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.8e+256], N[(y * N[(j * N[(0.0 - i), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * N[(0.0 - z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(j * c + N[(x * N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.30000000000000011e142

   1. Initial program 67.1%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-- N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 67.0

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

   4. Applied egg-rr 67.0%

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

   5. Taylor expanded in a around inf

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

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

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 79.7

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

   7. Simplified 79.7%

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

   if -1.30000000000000011e142 < a < 5.8000000000000003e256

   1. Initial program 76.6%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

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

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

      14. *-commutative N/A

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

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

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

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

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

      17. mul-1-neg N/A

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

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

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

      19. mul-1-neg N/A

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

   5. Simplified 75.2%

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

   if 5.8000000000000003e256 < a

   1. Initial program 27.3%

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

   3. Taylor expanded in a around inf

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

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

      11. --lowering--.f64 73.4

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

   5. Simplified 73.4%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+142}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, 0 - z, t \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(0 - t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.36 \cdot 10^{+143}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \mathbf{elif}\;a \leq -1.95 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(b, 0 - c, x \cdot y\right)\right)\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{+156}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(i, 0 - j, x \cdot z\right) - z \cdot \left(b \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(0 - t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -1.36e+143)
   (* a (fma x (- 0.0 t) (* c j)))
   (if (<= a -1.95e-126)
     (fma t (* b i) (* z (fma b (- 0.0 c) (* x y))))
     (if (<= a 2.05e+156)
       (- (* y (fma i (- 0.0 j) (* x z))) (* z (* b c)))
       (* a (fma j c (* x (- 0.0 t))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -1.36e+143) {
		tmp = a * fma(x, (0.0 - t), (c * j));
	} else if (a <= -1.95e-126) {
		tmp = fma(t, (b * i), (z * fma(b, (0.0 - c), (x * y))));
	} else if (a <= 2.05e+156) {
		tmp = (y * fma(i, (0.0 - j), (x * z))) - (z * (b * c));
	} else {
		tmp = a * fma(j, c, (x * (0.0 - t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -1.36e+143)
		tmp = Float64(a * fma(x, Float64(0.0 - t), Float64(c * j)));
	elseif (a <= -1.95e-126)
		tmp = fma(t, Float64(b * i), Float64(z * fma(b, Float64(0.0 - c), Float64(x * y))));
	elseif (a <= 2.05e+156)
		tmp = Float64(Float64(y * fma(i, Float64(0.0 - j), Float64(x * z))) - Float64(z * Float64(b * c)));
	else
		tmp = Float64(a * fma(j, c, Float64(x * Float64(0.0 - t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -1.36e+143], N[(a * N[(x * N[(0.0 - t), $MachinePrecision] + N[(c * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.95e-126], N[(t * N[(b * i), $MachinePrecision] + N[(z * N[(b * N[(0.0 - c), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.05e+156], N[(N[(y * N[(i * N[(0.0 - j), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(j * c + N[(x * N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.3599999999999999e143

   1. Initial program 67.1%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-- N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 67.0

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

   4. Applied egg-rr 67.0%

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

   5. Taylor expanded in a around inf

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

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

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 79.7

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

   7. Simplified 79.7%

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

   if -1.3599999999999999e143 < a < -1.9499999999999999e-126

   1. Initial program 72.2%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

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

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

      14. *-commutative N/A

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

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

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

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

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

      17. mul-1-neg N/A

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

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

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

      19. mul-1-neg N/A

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

   5. Simplified 73.8%

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

   6. Taylor expanded in j around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

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

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

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

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

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

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

   8. Simplified 71.8%

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

   if -1.9499999999999999e-126 < a < 2.0500000000000001e156

   1. Initial program 80.5%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

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

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

      14. *-commutative N/A

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

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

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

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

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

      17. mul-1-neg N/A

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

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

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

      19. mul-1-neg N/A

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

   5. Simplified 77.6%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. neg-sub0 N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

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

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

      19. *-lowering-*.f64 67.0

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

   8. Simplified 67.0%

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

   if 2.0500000000000001e156 < a

   1. Initial program 51.8%

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

   3. Taylor expanded in a around inf

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

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

      11. --lowering--.f64 61.2

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

   5. Simplified 61.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.36 \cdot 10^{+143}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \mathbf{elif}\;a \leq -1.95 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(b, 0 - c, x \cdot y\right)\right)\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{+156}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(i, 0 - j, x \cdot z\right) - z \cdot \left(b \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(0 - t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{if}\;j \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;t\_1 - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;j \leq 8 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, b \cdot \mathsf{fma}\left(c, 0 - z, t \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + y \cdot \left(x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* a c) (* y i)))))
   (if (<= j -1.5e+55)
     (- t_1 (* c (* z b)))
     (if (<= j 8e-10)
       (fma x (- (* y z) (* t a)) (* b (fma c (- 0.0 z) (* t i))))
       (+ t_1 (* y (* x z)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((a * c) - (y * i));
	double tmp;
	if (j <= -1.5e+55) {
		tmp = t_1 - (c * (z * b));
	} else if (j <= 8e-10) {
		tmp = fma(x, ((y * z) - (t * a)), (b * fma(c, (0.0 - z), (t * i))));
	} else {
		tmp = t_1 + (y * (x * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(a * c) - Float64(y * i)))
	tmp = 0.0
	if (j <= -1.5e+55)
		tmp = Float64(t_1 - Float64(c * Float64(z * b)));
	elseif (j <= 8e-10)
		tmp = fma(x, Float64(Float64(y * z) - Float64(t * a)), Float64(b * fma(c, Float64(0.0 - z), Float64(t * i))));
	else
		tmp = Float64(t_1 + Float64(y * Float64(x * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -1.5e+55], N[(t$95$1 - N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 8e-10], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * N[(0.0 - z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if j < -1.50000000000000008e55

   1. Initial program 78.3%

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

   3. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

      8. *-lowering-*.f64 71.1

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

   5. Simplified 71.1%

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

   if -1.50000000000000008e55 < j < 8.00000000000000029e-10

   1. Initial program 69.6%

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

   3. Taylor expanded in j around 0

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

      1. sub-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

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

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

      14. mul-1-neg N/A

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

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

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

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

      18. mul-1-neg N/A

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

      19. neg-sub0 N/A

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

      20. --lowering--.f64 N/A

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

      21. *-lowering-*.f64 66.6

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

   5. Simplified 66.6%

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

   if 8.00000000000000029e-10 < j

   1. Initial program 77.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 76.7

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

   5. Simplified 76.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;j \leq 8 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, b \cdot \mathsf{fma}\left(c, 0 - z, t \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) + y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{if}\;j \leq -5.5 \cdot 10^{+37}:\\ \;\;\;\;t\_1 - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;j \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(b, 0 - c, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + y \cdot \left(x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* a c) (* y i)))))
   (if (<= j -5.5e+37)
     (- t_1 (* c (* z b)))
     (if (<= j 7e-10)
       (fma t (* b i) (* z (fma b (- 0.0 c) (* x y))))
       (+ t_1 (* y (* x z)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((a * c) - (y * i));
	double tmp;
	if (j <= -5.5e+37) {
		tmp = t_1 - (c * (z * b));
	} else if (j <= 7e-10) {
		tmp = fma(t, (b * i), (z * fma(b, (0.0 - c), (x * y))));
	} else {
		tmp = t_1 + (y * (x * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(a * c) - Float64(y * i)))
	tmp = 0.0
	if (j <= -5.5e+37)
		tmp = Float64(t_1 - Float64(c * Float64(z * b)));
	elseif (j <= 7e-10)
		tmp = fma(t, Float64(b * i), Float64(z * fma(b, Float64(0.0 - c), Float64(x * y))));
	else
		tmp = Float64(t_1 + Float64(y * Float64(x * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -5.5e+37], N[(t$95$1 - N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 7e-10], N[(t * N[(b * i), $MachinePrecision] + N[(z * N[(b * N[(0.0 - c), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if j < -5.50000000000000016e37

   1. Initial program 78.4%

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

   3. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

      8. *-lowering-*.f64 68.6

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

   5. Simplified 68.6%

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

   if -5.50000000000000016e37 < j < 6.99999999999999961e-10

   1. Initial program 69.2%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

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

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

      14. *-commutative N/A

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

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

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

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

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

      17. mul-1-neg N/A

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

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

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

      19. mul-1-neg N/A

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

   5. Simplified 72.2%

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

   6. Taylor expanded in j around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. distribute-rgt-in N/A

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

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

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

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

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

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

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

   8. Simplified 66.8%

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

   if 6.99999999999999961e-10 < j

   1. Initial program 77.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot z\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. *-lowering-*.f64 76.7

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   5. Simplified 76.7%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -5.5 \cdot 10^{+37}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;j \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(b, 0 - c, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) + y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -2.7 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, y \cdot \left(0 - i\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;j \leq 8 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(b, 0 - c, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) + y \cdot \left(x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -2.7e+37)
   (fma j (fma a c (* y (- 0.0 i))) (* x (* y z)))
   (if (<= j 8e-10)
     (fma t (* b i) (* z (fma b (- 0.0 c) (* x y))))
     (+ (* j (- (* a c) (* y i))) (* y (* x z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -2.7e+37) {
		tmp = fma(j, fma(a, c, (y * (0.0 - i))), (x * (y * z)));
	} else if (j <= 8e-10) {
		tmp = fma(t, (b * i), (z * fma(b, (0.0 - c), (x * y))));
	} else {
		tmp = (j * ((a * c) - (y * i))) + (y * (x * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -2.7e+37)
		tmp = fma(j, fma(a, c, Float64(y * Float64(0.0 - i))), Float64(x * Float64(y * z)));
	elseif (j <= 8e-10)
		tmp = fma(t, Float64(b * i), Float64(z * fma(b, Float64(0.0 - c), Float64(x * y))));
	else
		tmp = Float64(Float64(j * Float64(Float64(a * c) - Float64(y * i))) + Float64(y * Float64(x * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -2.7e+37], N[(j * N[(a * c + N[(y * N[(0.0 - i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 8e-10], N[(t * N[(b * i), $MachinePrecision] + N[(z * N[(b * N[(0.0 - c), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if j < -2.69999999999999986e37

   1. Initial program 78.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), t \cdot \mathsf{fma}\left(a, 0 - x, i \cdot b\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), \color{blue}{x \cdot \left(y \cdot z\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), \color{blue}{x \cdot \left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), x \cdot \color{blue}{\left(z \cdot y\right)}\right) \]

      3. *-lowering-*.f64 60.9

         \[\leadsto \mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), x \cdot \color{blue}{\left(z \cdot y\right)}\right) \]

   8. Simplified 60.9%

      \[\leadsto \mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), \color{blue}{x \cdot \left(z \cdot y\right)}\right) \]

   if -2.69999999999999986e37 < j < 8.00000000000000029e-10

   1. Initial program 69.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z - i \cdot t\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \color{blue}{\left(z \cdot y\right)}\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \color{blue}{\left(x \cdot z\right) \cdot y}\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(\left(i \cdot j\right) \cdot y\right)} + \left(x \cdot z\right) \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(i \cdot j\right)\right) \cdot y} + \left(x \cdot z\right) \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, -1 \cdot \left(i \cdot j\right) + x \cdot z, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(-1 \cdot i\right) \cdot j} + x \cdot z, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{j \cdot \left(-1 \cdot i\right)} + x \cdot z, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(j, -1 \cdot i, x \cdot z\right)}, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(i\right)}, x \cdot z\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), \color{blue}{b \cdot \left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)}\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)}\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)}\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)}\right) \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, 0 - z, i \cdot t\right)\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right) + x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot t + -1 \cdot \left(c \cdot z\right)\right)} + x \cdot \left(y \cdot z\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(i \cdot t\right) + b \cdot \left(-1 \cdot \left(c \cdot z\right)\right)\right)} + x \cdot \left(y \cdot z\right) \]

      3. mul-1-neg N/A

         \[\leadsto \left(b \cdot \left(i \cdot t\right) + b \cdot \color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)}\right) + x \cdot \left(y \cdot z\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \left(b \cdot \left(i \cdot t\right) + \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)}\right) + x \cdot \left(y \cdot z\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(b \cdot \left(i \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) + x \cdot \left(y \cdot z\right) \]

      6. associate-+r+ N/A

         \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(t \cdot i\right)} + \left(-1 \cdot \left(b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(b \cdot t\right) \cdot i} + \left(-1 \cdot \left(b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot b\right)} \cdot i + \left(-1 \cdot \left(b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{t \cdot \left(b \cdot i\right)} + \left(-1 \cdot \left(b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto t \cdot \left(b \cdot i\right) + \left(-1 \cdot \color{blue}{\left(\left(b \cdot c\right) \cdot z\right)} + x \cdot \left(y \cdot z\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto t \cdot \left(b \cdot i\right) + \left(\color{blue}{\left(-1 \cdot \left(b \cdot c\right)\right) \cdot z} + x \cdot \left(y \cdot z\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto t \cdot \left(b \cdot i\right) + \left(\left(-1 \cdot \left(b \cdot c\right)\right) \cdot z + \color{blue}{\left(x \cdot y\right) \cdot z}\right) \]

      14. distribute-rgt-in N/A

          \[\leadsto t \cdot \left(b \cdot i\right) + \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(t, b \cdot i, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{b \cdot i}, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, b \cdot i, \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 66.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(b, 0 - c, x \cdot y\right)\right)} \]

   if 8.00000000000000029e-10 < j

   1. Initial program 77.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot z\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. *-lowering-*.f64 76.7

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   5. Simplified 76.7%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.7 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, y \cdot \left(0 - i\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;j \leq 8 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(b, 0 - c, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) + y \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j, \mathsf{fma}\left(a, c, y \cdot \left(0 - i\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \mathbf{if}\;j \leq -3.6 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 1.5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(b, 0 - c, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma j (fma a c (* y (- 0.0 i))) (* x (* y z)))))
   (if (<= j -3.6e+37)
     t_1
     (if (<= j 1.5e-9) (fma t (* b i) (* z (fma b (- 0.0 c) (* x y)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(j, fma(a, c, (y * (0.0 - i))), (x * (y * z)));
	double tmp;
	if (j <= -3.6e+37) {
		tmp = t_1;
	} else if (j <= 1.5e-9) {
		tmp = fma(t, (b * i), (z * fma(b, (0.0 - c), (x * y))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(j, fma(a, c, Float64(y * Float64(0.0 - i))), Float64(x * Float64(y * z)))
	tmp = 0.0
	if (j <= -3.6e+37)
		tmp = t_1;
	elseif (j <= 1.5e-9)
		tmp = fma(t, Float64(b * i), Float64(z * fma(b, Float64(0.0 - c), Float64(x * y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(a * c + N[(y * N[(0.0 - i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -3.6e+37], t$95$1, If[LessEqual[j, 1.5e-9], N[(t * N[(b * i), $MachinePrecision] + N[(z * N[(b * N[(0.0 - c), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -3.59999999999999998e37 or 1.49999999999999999e-9 < j

   1. Initial program 77.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), t \cdot \mathsf{fma}\left(a, 0 - x, i \cdot b\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), \color{blue}{x \cdot \left(y \cdot z\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), \color{blue}{x \cdot \left(y \cdot z\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), x \cdot \color{blue}{\left(z \cdot y\right)}\right) \]

      3. *-lowering-*.f64 65.8

         \[\leadsto \mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), x \cdot \color{blue}{\left(z \cdot y\right)}\right) \]

   8. Simplified 65.8%

      \[\leadsto \mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), \color{blue}{x \cdot \left(z \cdot y\right)}\right) \]

   if -3.59999999999999998e37 < j < 1.49999999999999999e-9

   1. Initial program 69.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z - i \cdot t\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \color{blue}{\left(z \cdot y\right)}\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \color{blue}{\left(x \cdot z\right) \cdot y}\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(\left(i \cdot j\right) \cdot y\right)} + \left(x \cdot z\right) \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(i \cdot j\right)\right) \cdot y} + \left(x \cdot z\right) \cdot y\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, -1 \cdot \left(i \cdot j\right) + x \cdot z, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(-1 \cdot i\right) \cdot j} + x \cdot z, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{j \cdot \left(-1 \cdot i\right)} + x \cdot z, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(j, -1 \cdot i, x \cdot z\right)}, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(i\right)}, x \cdot z\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), \color{blue}{b \cdot \left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)}\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)}\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)}\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)}\right) \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, 0 - z, i \cdot t\right)\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right) + x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot t + -1 \cdot \left(c \cdot z\right)\right)} + x \cdot \left(y \cdot z\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(i \cdot t\right) + b \cdot \left(-1 \cdot \left(c \cdot z\right)\right)\right)} + x \cdot \left(y \cdot z\right) \]

      3. mul-1-neg N/A

         \[\leadsto \left(b \cdot \left(i \cdot t\right) + b \cdot \color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)}\right) + x \cdot \left(y \cdot z\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \left(b \cdot \left(i \cdot t\right) + \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)}\right) + x \cdot \left(y \cdot z\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(b \cdot \left(i \cdot t\right) + \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) + x \cdot \left(y \cdot z\right) \]

      6. associate-+r+ N/A

         \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(t \cdot i\right)} + \left(-1 \cdot \left(b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(b \cdot t\right) \cdot i} + \left(-1 \cdot \left(b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot b\right)} \cdot i + \left(-1 \cdot \left(b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{t \cdot \left(b \cdot i\right)} + \left(-1 \cdot \left(b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto t \cdot \left(b \cdot i\right) + \left(-1 \cdot \color{blue}{\left(\left(b \cdot c\right) \cdot z\right)} + x \cdot \left(y \cdot z\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto t \cdot \left(b \cdot i\right) + \left(\color{blue}{\left(-1 \cdot \left(b \cdot c\right)\right) \cdot z} + x \cdot \left(y \cdot z\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto t \cdot \left(b \cdot i\right) + \left(\left(-1 \cdot \left(b \cdot c\right)\right) \cdot z + \color{blue}{\left(x \cdot y\right) \cdot z}\right) \]

      14. distribute-rgt-in N/A

          \[\leadsto t \cdot \left(b \cdot i\right) + \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(t, b \cdot i, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{b \cdot i}, z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, b \cdot i, \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)}\right) \]

   8. Simplified 66.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(b, 0 - c, x \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -3.6 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, y \cdot \left(0 - i\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;j \leq 1.5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t, b \cdot i, z \cdot \mathsf{fma}\left(b, 0 - c, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, y \cdot \left(0 - i\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+26}:\\ \;\;\;\;0 - i \cdot \left(y \cdot j\right)\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-50}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-290}:\\ \;\;\;\;b \cdot \left(c \cdot \left(0 - z\right)\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;t \cdot \left(a \cdot \left(0 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot \left(i \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -2.3e+26)
   (- 0.0 (* i (* y j)))
   (if (<= y -1.2e-50)
     (* b (* t i))
     (if (<= y 2.8e-290)
       (* b (* c (- 0.0 z)))
       (if (<= y 2.7e+14) (* t (* a (- 0.0 x))) (- 0.0 (* y (* i 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 (y <= -2.3e+26) {
		tmp = 0.0 - (i * (y * j));
	} else if (y <= -1.2e-50) {
		tmp = b * (t * i);
	} else if (y <= 2.8e-290) {
		tmp = b * (c * (0.0 - z));
	} else if (y <= 2.7e+14) {
		tmp = t * (a * (0.0 - x));
	} else {
		tmp = 0.0 - (y * (i * j));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (y <= (-2.3d+26)) then
        tmp = 0.0d0 - (i * (y * j))
    else if (y <= (-1.2d-50)) then
        tmp = b * (t * i)
    else if (y <= 2.8d-290) then
        tmp = b * (c * (0.0d0 - z))
    else if (y <= 2.7d+14) then
        tmp = t * (a * (0.0d0 - x))
    else
        tmp = 0.0d0 - (y * (i * j))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -2.3e+26) {
		tmp = 0.0 - (i * (y * j));
	} else if (y <= -1.2e-50) {
		tmp = b * (t * i);
	} else if (y <= 2.8e-290) {
		tmp = b * (c * (0.0 - z));
	} else if (y <= 2.7e+14) {
		tmp = t * (a * (0.0 - x));
	} else {
		tmp = 0.0 - (y * (i * j));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if y <= -2.3e+26:
		tmp = 0.0 - (i * (y * j))
	elif y <= -1.2e-50:
		tmp = b * (t * i)
	elif y <= 2.8e-290:
		tmp = b * (c * (0.0 - z))
	elif y <= 2.7e+14:
		tmp = t * (a * (0.0 - x))
	else:
		tmp = 0.0 - (y * (i * j))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -2.3e+26)
		tmp = Float64(0.0 - Float64(i * Float64(y * j)));
	elseif (y <= -1.2e-50)
		tmp = Float64(b * Float64(t * i));
	elseif (y <= 2.8e-290)
		tmp = Float64(b * Float64(c * Float64(0.0 - z)));
	elseif (y <= 2.7e+14)
		tmp = Float64(t * Float64(a * Float64(0.0 - x)));
	else
		tmp = Float64(0.0 - Float64(y * Float64(i * j)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (y <= -2.3e+26)
		tmp = 0.0 - (i * (y * j));
	elseif (y <= -1.2e-50)
		tmp = b * (t * i);
	elseif (y <= 2.8e-290)
		tmp = b * (c * (0.0 - z));
	elseif (y <= 2.7e+14)
		tmp = t * (a * (0.0 - x));
	else
		tmp = 0.0 - (y * (i * j));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -2.3e+26], N[(0.0 - N[(i * N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.2e-50], N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e-290], N[(b * N[(c * N[(0.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+14], N[(t * N[(a * N[(0.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(y * N[(i * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -2.3000000000000001e26

   1. Initial program 65.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 48.7

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 48.7%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto 0 - i \cdot \color{blue}{\left(y \cdot j\right)} \]

      6. *-lowering-*.f64 38.2

         \[\leadsto 0 - i \cdot \color{blue}{\left(y \cdot j\right)} \]

   8. Simplified 38.2%

      \[\leadsto \color{blue}{0 - i \cdot \left(y \cdot j\right)} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(i \cdot \left(y \cdot j\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(y \cdot j\right) \cdot i}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot j\right)\right) \cdot i} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot j\right)\right) \cdot i} \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot j\right)\right)} \cdot i \]

      6. *-lowering-*.f64 38.2

         \[\leadsto \left(-\color{blue}{y \cdot j}\right) \cdot i \]

   10. Applied egg-rr 38.2%

       \[\leadsto \color{blue}{\left(-y \cdot j\right) \cdot i} \]

   if -2.3000000000000001e26 < y < -1.20000000000000001e-50

   1. Initial program 64.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 47.0

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 47.0%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

      2. *-lowering-*.f64 37.8

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   8. Simplified 37.8%

      \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(i \cdot t\right) \cdot b} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(i \cdot t\right) \cdot b} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot i\right)} \cdot b \]

      4. *-lowering-*.f64 42.1

         \[\leadsto \color{blue}{\left(t \cdot i\right)} \cdot b \]

   10. Applied egg-rr 42.1%

       \[\leadsto \color{blue}{\left(t \cdot i\right) \cdot b} \]

   if -1.20000000000000001e-50 < y < 2.79999999999999997e-290

   1. Initial program 88.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j - b \cdot z\right)} \]

      3. *-commutative N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      5. *-lowering-*.f64 56.7

         \[\leadsto c \cdot \left(j \cdot a - \color{blue}{b \cdot z}\right) \]

   5. Simplified 56.7%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \color{blue}{\left(b \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(b \cdot \left(-1 \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      6. neg-sub0 N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(0 - z\right)}\right) \]

      7. --lowering--.f64 36.4

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(0 - z\right)}\right) \]

   8. Simplified 36.4%

      \[\leadsto c \cdot \color{blue}{\left(b \cdot \left(0 - z\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot b\right) \cdot \left(0 - z\right)} \]

      2. sub0-neg N/A

         \[\leadsto \left(c \cdot b\right) \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(c \cdot b\right) \cdot z\right)} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{z \cdot \left(c \cdot b\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(z \cdot c\right) \cdot b}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(z \cdot c\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot c\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot c\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      9. neg-lowering-neg.f64 36.4

         \[\leadsto \left(z \cdot c\right) \cdot \color{blue}{\left(-b\right)} \]

   10. Applied egg-rr 36.4%

       \[\leadsto \color{blue}{\left(z \cdot c\right) \cdot \left(-b\right)} \]

   if 2.79999999999999997e-290 < y < 2.7e14

   1. Initial program 82.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 85.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), t \cdot \mathsf{fma}\left(a, 0 - x, i \cdot b\right)\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(b \cdot i + -1 \cdot \left(a \cdot x\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto t \cdot \left(b \cdot i + \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(b \cdot i - a \cdot x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto t \cdot \color{blue}{\left(b \cdot i - a \cdot x\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto t \cdot \left(\color{blue}{b \cdot i} - a \cdot x\right) \]

      7. *-lowering-*.f64 49.9

         \[\leadsto t \cdot \left(b \cdot i - \color{blue}{a \cdot x}\right) \]

   8. Simplified 49.9%

      \[\leadsto \color{blue}{t \cdot \left(b \cdot i - a \cdot x\right)} \]

   9. Taylor expanded in b around 0

      \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(a \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto t \cdot \left(\mathsf{neg}\left(\color{blue}{x \cdot a}\right)\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto t \cdot \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(a\right)\right)\right)} \]

       4. mul-1-neg N/A

          \[\leadsto t \cdot \left(x \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto t \cdot \color{blue}{\left(x \cdot \left(-1 \cdot a\right)\right)} \]

       6. mul-1-neg N/A

          \[\leadsto t \cdot \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

       7. neg-sub0 N/A

          \[\leadsto t \cdot \left(x \cdot \color{blue}{\left(0 - a\right)}\right) \]

       8. --lowering--.f64 38.4

          \[\leadsto t \cdot \left(x \cdot \color{blue}{\left(0 - a\right)}\right) \]

   11. Simplified 38.4%

       \[\leadsto t \cdot \color{blue}{\left(x \cdot \left(0 - a\right)\right)} \]

   if 2.7e14 < y

   1. Initial program 63.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 57.1

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 57.1%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto 0 - i \cdot \color{blue}{\left(y \cdot j\right)} \]

      6. *-lowering-*.f64 44.0

         \[\leadsto 0 - i \cdot \color{blue}{\left(y \cdot j\right)} \]

   8. Simplified 44.0%

      \[\leadsto \color{blue}{0 - i \cdot \left(y \cdot j\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 0 - i \cdot \color{blue}{\left(j \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto 0 - \color{blue}{\left(i \cdot j\right) \cdot y} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\left(i \cdot j\right) \cdot y} \]

      4. *-lowering-*.f64 47.1

         \[\leadsto 0 - \color{blue}{\left(i \cdot j\right)} \cdot y \]

   10. Applied egg-rr 47.1%

       \[\leadsto 0 - \color{blue}{\left(i \cdot j\right) \cdot y} \]
3. Recombined 5 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+26}:\\ \;\;\;\;0 - i \cdot \left(y \cdot j\right)\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-50}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-290}:\\ \;\;\;\;b \cdot \left(c \cdot \left(0 - z\right)\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;t \cdot \left(a \cdot \left(0 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot \left(i \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -8 \cdot 10^{+30}:\\ \;\;\;\;i \cdot \left(t \cdot \left(b - \frac{y \cdot j}{t}\right)\right)\\ \mathbf{elif}\;i \leq 1.2 \cdot 10^{-118}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, 0 - b \cdot c\right)\\ \mathbf{elif}\;i \leq 3 \cdot 10^{+91}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, 0 - i, x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -8e+30)
   (* i (* t (- b (/ (* y j) t))))
   (if (<= i 1.2e-118)
     (* z (fma y x (- 0.0 (* b c))))
     (if (<= i 3e+91)
       (* y (fma j (- 0.0 i) (* x z)))
       (* i (fma j (- 0.0 y) (* t b)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -8e+30) {
		tmp = i * (t * (b - ((y * j) / t)));
	} else if (i <= 1.2e-118) {
		tmp = z * fma(y, x, (0.0 - (b * c)));
	} else if (i <= 3e+91) {
		tmp = y * fma(j, (0.0 - i), (x * z));
	} else {
		tmp = i * fma(j, (0.0 - y), (t * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -8e+30)
		tmp = Float64(i * Float64(t * Float64(b - Float64(Float64(y * j) / t))));
	elseif (i <= 1.2e-118)
		tmp = Float64(z * fma(y, x, Float64(0.0 - Float64(b * c))));
	elseif (i <= 3e+91)
		tmp = Float64(y * fma(j, Float64(0.0 - i), Float64(x * z)));
	else
		tmp = Float64(i * fma(j, Float64(0.0 - y), Float64(t * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -8e+30], N[(i * N[(t * N[(b - N[(N[(y * j), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.2e-118], N[(z * N[(y * x + N[(0.0 - N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3e+91], N[(y * N[(j * N[(0.0 - i), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -8.0000000000000002e30

   1. Initial program 67.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 66.0

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto i \cdot \color{blue}{\left(t \cdot \left(b + -1 \cdot \frac{j \cdot y}{t}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto i \cdot \color{blue}{\left(t \cdot \left(b + -1 \cdot \frac{j \cdot y}{t}\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto i \cdot \left(t \cdot \left(b + \color{blue}{\left(\mathsf{neg}\left(\frac{j \cdot y}{t}\right)\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto i \cdot \left(t \cdot \color{blue}{\left(b - \frac{j \cdot y}{t}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto i \cdot \left(t \cdot \color{blue}{\left(b - \frac{j \cdot y}{t}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto i \cdot \left(t \cdot \left(b - \color{blue}{\frac{j \cdot y}{t}}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto i \cdot \left(t \cdot \left(b - \frac{\color{blue}{y \cdot j}}{t}\right)\right) \]

      7. *-lowering-*.f64 67.5

         \[\leadsto i \cdot \left(t \cdot \left(b - \frac{\color{blue}{y \cdot j}}{t}\right)\right) \]

   8. Simplified 67.5%

      \[\leadsto i \cdot \color{blue}{\left(t \cdot \left(b - \frac{y \cdot j}{t}\right)\right)} \]

   if -8.0000000000000002e30 < i < 1.2000000000000001e-118

   1. Initial program 81.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-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 52.6

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 52.6%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x + c \cdot \left(0 - b\right)\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(y, x, c \cdot \left(0 - b\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\left(0 - b\right) \cdot c}\right) \]

      4. sub0-neg N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot c\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\mathsf{neg}\left(b \cdot c\right)}\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\mathsf{neg}\left(b \cdot c\right)}\right) \]

      7. *-lowering-*.f64 54.4

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, -\color{blue}{b \cdot c}\right) \]

   7. Applied egg-rr 54.4%

      \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(y, x, -b \cdot c\right)} \]

   if 1.2000000000000001e-118 < i < 3.00000000000000006e91

   1. Initial program 68.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(-1 \cdot i\right) \cdot j} + x \cdot z\right) \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{j \cdot \left(-1 \cdot i\right)} + x \cdot z\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot i, x \cdot z\right)} \]

      5. mul-1-neg N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(i\right)}, x \cdot z\right) \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right) \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right) \]

      9. *-lowering-*.f64 52.4

         \[\leadsto y \cdot \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right) \]

   5. Simplified 52.4%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(j, 0 - i, z \cdot x\right)} \]

   if 3.00000000000000006e91 < i

   1. Initial program 64.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 78.5

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 78.5%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      2. neg-lowering-neg.f64 78.5

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot t\right) \]

   7. Applied egg-rr 78.5%

      \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot t\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -8 \cdot 10^{+30}:\\ \;\;\;\;i \cdot \left(t \cdot \left(b - \frac{y \cdot j}{t}\right)\right)\\ \mathbf{elif}\;i \leq 1.2 \cdot 10^{-118}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, 0 - b \cdot c\right)\\ \mathbf{elif}\;i \leq 3 \cdot 10^{+91}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, 0 - i, x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \mathbf{if}\;i \leq -3.2 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 9 \cdot 10^{-117}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, 0 - b \cdot c\right)\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{+91}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, 0 - i, x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- 0.0 y) (* t b)))))
   (if (<= i -3.2e+57)
     t_1
     (if (<= i 9e-117)
       (* z (fma y x (- 0.0 (* b c))))
       (if (<= i 3.1e+91) (* y (fma j (- 0.0 i) (* 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 = i * fma(j, (0.0 - y), (t * b));
	double tmp;
	if (i <= -3.2e+57) {
		tmp = t_1;
	} else if (i <= 9e-117) {
		tmp = z * fma(y, x, (0.0 - (b * c)));
	} else if (i <= 3.1e+91) {
		tmp = y * fma(j, (0.0 - i), (x * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * fma(j, Float64(0.0 - y), Float64(t * b)))
	tmp = 0.0
	if (i <= -3.2e+57)
		tmp = t_1;
	elseif (i <= 9e-117)
		tmp = Float64(z * fma(y, x, Float64(0.0 - Float64(b * c))));
	elseif (i <= 3.1e+91)
		tmp = Float64(y * fma(j, Float64(0.0 - i), Float64(x * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -3.2e+57], t$95$1, If[LessEqual[i, 9e-117], N[(z * N[(y * x + N[(0.0 - N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.1e+91], N[(y * N[(j * N[(0.0 - i), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -3.20000000000000029e57 or 3.09999999999999998e91 < i

   1. Initial program 65.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 74.4

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 74.4%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      2. neg-lowering-neg.f64 74.4

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot t\right) \]

   7. Applied egg-rr 74.4%

      \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot t\right) \]

   if -3.20000000000000029e57 < i < 8.99999999999999939e-117

   1. Initial program 80.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 52.1

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 52.1%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x + c \cdot \left(0 - b\right)\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(y, x, c \cdot \left(0 - b\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\left(0 - b\right) \cdot c}\right) \]

      4. sub0-neg N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot c\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\mathsf{neg}\left(b \cdot c\right)}\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\mathsf{neg}\left(b \cdot c\right)}\right) \]

      7. *-lowering-*.f64 53.8

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, -\color{blue}{b \cdot c}\right) \]

   7. Applied egg-rr 53.8%

      \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(y, x, -b \cdot c\right)} \]

   if 8.99999999999999939e-117 < i < 3.09999999999999998e91

   1. Initial program 68.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(-1 \cdot i\right) \cdot j} + x \cdot z\right) \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{j \cdot \left(-1 \cdot i\right)} + x \cdot z\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot i, x \cdot z\right)} \]

      5. mul-1-neg N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(i\right)}, x \cdot z\right) \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right) \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right) \]

      9. *-lowering-*.f64 52.4

         \[\leadsto y \cdot \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right) \]

   5. Simplified 52.4%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(j, 0 - i, z \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.2 \cdot 10^{+57}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \mathbf{elif}\;i \leq 9 \cdot 10^{-117}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, 0 - b \cdot c\right)\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{+91}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, 0 - i, x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \mathbf{if}\;i \leq -1.9 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-19}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, 0 - b \cdot c\right)\\ \mathbf{elif}\;i \leq 4.1 \cdot 10^{+57}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(0 - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- 0.0 y) (* t b)))))
   (if (<= i -1.9e+56)
     t_1
     (if (<= i 2.5e-19)
       (* z (fma y x (- 0.0 (* b c))))
       (if (<= i 4.1e+57) (* a (fma j c (* x (- 0.0 t)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * fma(j, (0.0 - y), (t * b));
	double tmp;
	if (i <= -1.9e+56) {
		tmp = t_1;
	} else if (i <= 2.5e-19) {
		tmp = z * fma(y, x, (0.0 - (b * c)));
	} else if (i <= 4.1e+57) {
		tmp = a * fma(j, c, (x * (0.0 - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * fma(j, Float64(0.0 - y), Float64(t * b)))
	tmp = 0.0
	if (i <= -1.9e+56)
		tmp = t_1;
	elseif (i <= 2.5e-19)
		tmp = Float64(z * fma(y, x, Float64(0.0 - Float64(b * c))));
	elseif (i <= 4.1e+57)
		tmp = Float64(a * fma(j, c, Float64(x * Float64(0.0 - t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.9e+56], t$95$1, If[LessEqual[i, 2.5e-19], N[(z * N[(y * x + N[(0.0 - N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 4.1e+57], N[(a * N[(j * c + N[(x * N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.89999999999999998e56 or 4.1e57 < i

   1. Initial program 66.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 71.6

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      2. neg-lowering-neg.f64 71.6

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot t\right) \]

   7. Applied egg-rr 71.6%

      \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot t\right) \]

   if -1.89999999999999998e56 < i < 2.5000000000000002e-19

   1. Initial program 80.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 51.6

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 51.6%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x + c \cdot \left(0 - b\right)\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(y, x, c \cdot \left(0 - b\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\left(0 - b\right) \cdot c}\right) \]

      4. sub0-neg N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot c\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\mathsf{neg}\left(b \cdot c\right)}\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\mathsf{neg}\left(b \cdot c\right)}\right) \]

      7. *-lowering-*.f64 53.1

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, -\color{blue}{b \cdot c}\right) \]

   7. Applied egg-rr 53.1%

      \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(y, x, -b \cdot c\right)} \]

   if 2.5000000000000002e-19 < i < 4.1e57

   1. Initial program 57.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(0 - x\right)}\right) \]

      11. --lowering--.f64 65.3

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(0 - x\right)}\right) \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(0 - x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.9 \cdot 10^{+56}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-19}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, 0 - b \cdot c\right)\\ \mathbf{elif}\;i \leq 4.1 \cdot 10^{+57}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(0 - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \mathbf{if}\;i \leq -3.2 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 5.9 \cdot 10^{-151}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;i \leq 10^{+85}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- 0.0 y) (* t b)))))
   (if (<= i -3.2e-39)
     t_1
     (if (<= i 5.9e-151)
       (* c (- (* a j) (* z b)))
       (if (<= i 1e+85) (* x (- (* y z) (* t a))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * fma(j, (0.0 - y), (t * b));
	double tmp;
	if (i <= -3.2e-39) {
		tmp = t_1;
	} else if (i <= 5.9e-151) {
		tmp = c * ((a * j) - (z * b));
	} else if (i <= 1e+85) {
		tmp = x * ((y * z) - (t * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * fma(j, Float64(0.0 - y), Float64(t * b)))
	tmp = 0.0
	if (i <= -3.2e-39)
		tmp = t_1;
	elseif (i <= 5.9e-151)
		tmp = Float64(c * Float64(Float64(a * j) - Float64(z * b)));
	elseif (i <= 1e+85)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -3.2e-39], t$95$1, If[LessEqual[i, 5.9e-151], N[(c * N[(N[(a * j), $MachinePrecision] - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1e+85], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -3.1999999999999998e-39 or 1e85 < i

   1. Initial program 66.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 66.5

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 66.5%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      2. neg-lowering-neg.f64 66.5

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot t\right) \]

   7. Applied egg-rr 66.5%

      \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot t\right) \]

   if -3.1999999999999998e-39 < i < 5.8999999999999999e-151

   1. Initial program 83.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j - b \cdot z\right)} \]

      3. *-commutative N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      5. *-lowering-*.f64 56.7

         \[\leadsto c \cdot \left(j \cdot a - \color{blue}{b \cdot z}\right) \]

   5. Simplified 56.7%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot a - b \cdot z\right)} \]

   if 5.8999999999999999e-151 < i < 1e85

   1. Initial program 70.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 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. *-lowering-*.f64 47.4

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{a \cdot t}\right) \]

   5. Simplified 47.4%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.2 \cdot 10^{-39}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \mathbf{elif}\;i \leq 5.9 \cdot 10^{-151}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;i \leq 10^{+85}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \mathbf{if}\;a \leq -1.45 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-123}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;a \leq 54:\\ \;\;\;\;j \cdot \left(y \cdot \left(0 - i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (fma x (- 0.0 t) (* c j)))))
   (if (<= a -1.45e+60)
     t_1
     (if (<= a -6e-123)
       (* i (* t b))
       (if (<= a 54.0) (* j (* 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 = a * fma(x, (0.0 - t), (c * j));
	double tmp;
	if (a <= -1.45e+60) {
		tmp = t_1;
	} else if (a <= -6e-123) {
		tmp = i * (t * b);
	} else if (a <= 54.0) {
		tmp = j * (y * (0.0 - i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * fma(x, Float64(0.0 - t), Float64(c * j)))
	tmp = 0.0
	if (a <= -1.45e+60)
		tmp = t_1;
	elseif (a <= -6e-123)
		tmp = Float64(i * Float64(t * b));
	elseif (a <= 54.0)
		tmp = Float64(j * 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[(a * N[(x * N[(0.0 - t), $MachinePrecision] + N[(c * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.45e+60], t$95$1, If[LessEqual[a, -6e-123], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 54.0], N[(j * N[(y * N[(0.0 - i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.45e60 or 54 < a

   1. Initial program 64.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(x \cdot \color{blue}{\frac{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}{y \cdot z + t \cdot a}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. clear-num N/A

         \[\leadsto \left(x \cdot \color{blue}{\frac{1}{\frac{y \cdot z + t \cdot a}{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. un-div-inv N/A

         \[\leadsto \left(\color{blue}{\frac{x}{\frac{y \cdot z + t \cdot a}{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{x}{\frac{y \cdot z + t \cdot a}{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{x}{\color{blue}{\frac{1}{\frac{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}{y \cdot z + t \cdot a}}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. flip-- N/A

         \[\leadsto \left(\frac{x}{\frac{1}{\color{blue}{y \cdot z - t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{x}{\color{blue}{\frac{1}{y \cdot z - t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\frac{x}{\frac{1}{\color{blue}{y \cdot z - t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{x}{\frac{1}{\color{blue}{y \cdot z} - t \cdot a}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. *-lowering-*.f64 64.4

          \[\leadsto \left(\frac{x}{\frac{1}{y \cdot z - \color{blue}{t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Applied egg-rr 64.4%

      \[\leadsto \left(\color{blue}{\frac{x}{\frac{1}{y \cdot z - t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + c \cdot j\right) \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\left(\mathsf{neg}\left(\color{blue}{x \cdot t}\right)\right) + c \cdot j\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(t\right)\right)} + c \cdot j\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(x \cdot \color{blue}{\left(-1 \cdot t\right)} + c \cdot j\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(x, -1 \cdot t, c \cdot j\right)} \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{neg}\left(t\right)}, c \cdot j\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{neg}\left(t\right)}, c \cdot j\right) \]

      9. *-commutative N/A

         \[\leadsto a \cdot \mathsf{fma}\left(x, \mathsf{neg}\left(t\right), \color{blue}{j \cdot c}\right) \]

      10. *-lowering-*.f64 57.7

          \[\leadsto a \cdot \mathsf{fma}\left(x, -t, \color{blue}{j \cdot c}\right) \]

   7. Simplified 57.7%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(x, -t, j \cdot c\right)} \]

   if -1.45e60 < a < -5.99999999999999968e-123

   1. Initial program 75.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 59.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 59.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

      2. *-lowering-*.f64 45.5

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   8. Simplified 45.5%

      \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   if -5.99999999999999968e-123 < a < 54

   1. Initial program 81.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 47.5

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 47.5%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto 0 - i \cdot \color{blue}{\left(y \cdot j\right)} \]

      6. *-lowering-*.f64 32.1

         \[\leadsto 0 - i \cdot \color{blue}{\left(y \cdot j\right)} \]

   8. Simplified 32.1%

      \[\leadsto \color{blue}{0 - i \cdot \left(y \cdot j\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto 0 - \color{blue}{\left(i \cdot y\right) \cdot j} \]

      2. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot i\right)} \cdot j \]

      3. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot i\right) \cdot j} \]

      4. *-lowering-*.f64 33.7

         \[\leadsto 0 - \color{blue}{\left(y \cdot i\right)} \cdot j \]

   10. Applied egg-rr 33.7%

       \[\leadsto 0 - \color{blue}{\left(y \cdot i\right) \cdot j} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+60}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-123}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;a \leq 54:\\ \;\;\;\;j \cdot \left(y \cdot \left(0 - i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 28.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{+68}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-9}:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-105}:\\ \;\;\;\;c \cdot \left(0 - z \cdot b\right)\\ \mathbf{elif}\;t \leq 7.3 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -6.8e+68)
   (* i (* t b))
   (if (<= t -7.5e-9)
     (* a (* c j))
     (if (<= t -2.3e-105)
       (* c (- 0.0 (* z b)))
       (if (<= t 7.3e-14) (* x (* y z)) (* t (* 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 (t <= -6.8e+68) {
		tmp = i * (t * b);
	} else if (t <= -7.5e-9) {
		tmp = a * (c * j);
	} else if (t <= -2.3e-105) {
		tmp = c * (0.0 - (z * b));
	} else if (t <= 7.3e-14) {
		tmp = x * (y * z);
	} else {
		tmp = t * (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 (t <= (-6.8d+68)) then
        tmp = i * (t * b)
    else if (t <= (-7.5d-9)) then
        tmp = a * (c * j)
    else if (t <= (-2.3d-105)) then
        tmp = c * (0.0d0 - (z * b))
    else if (t <= 7.3d-14) then
        tmp = x * (y * z)
    else
        tmp = t * (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 (t <= -6.8e+68) {
		tmp = i * (t * b);
	} else if (t <= -7.5e-9) {
		tmp = a * (c * j);
	} else if (t <= -2.3e-105) {
		tmp = c * (0.0 - (z * b));
	} else if (t <= 7.3e-14) {
		tmp = x * (y * z);
	} else {
		tmp = t * (b * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if t <= -6.8e+68:
		tmp = i * (t * b)
	elif t <= -7.5e-9:
		tmp = a * (c * j)
	elif t <= -2.3e-105:
		tmp = c * (0.0 - (z * b))
	elif t <= 7.3e-14:
		tmp = x * (y * z)
	else:
		tmp = t * (b * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -6.8e+68)
		tmp = Float64(i * Float64(t * b));
	elseif (t <= -7.5e-9)
		tmp = Float64(a * Float64(c * j));
	elseif (t <= -2.3e-105)
		tmp = Float64(c * Float64(0.0 - Float64(z * b)));
	elseif (t <= 7.3e-14)
		tmp = Float64(x * Float64(y * z));
	else
		tmp = Float64(t * 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 (t <= -6.8e+68)
		tmp = i * (t * b);
	elseif (t <= -7.5e-9)
		tmp = a * (c * j);
	elseif (t <= -2.3e-105)
		tmp = c * (0.0 - (z * b));
	elseif (t <= 7.3e-14)
		tmp = x * (y * z);
	else
		tmp = t * (b * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -6.8e+68], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.5e-9], N[(a * N[(c * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.3e-105], N[(c * N[(0.0 - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.3e-14], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(t * N[(b * i), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -6.8000000000000003e68

   1. Initial program 53.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 54.9

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 54.9%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

      2. *-lowering-*.f64 46.4

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   8. Simplified 46.4%

      \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   if -6.8000000000000003e68 < t < -7.49999999999999933e-9

   1. Initial program 67.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j - b \cdot z\right)} \]

      3. *-commutative N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      5. *-lowering-*.f64 48.6

         \[\leadsto c \cdot \left(j \cdot a - \color{blue}{b \cdot z}\right) \]

   5. Simplified 48.6%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 41.9

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]

   8. Simplified 41.9%

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot a\right) \cdot j} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot a\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(j \cdot c\right) \cdot a} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(j \cdot c\right) \cdot a} \]

      5. *-lowering-*.f64 48.0

         \[\leadsto \color{blue}{\left(j \cdot c\right)} \cdot a \]

   10. Applied egg-rr 48.0%

       \[\leadsto \color{blue}{\left(j \cdot c\right) \cdot a} \]

   if -7.49999999999999933e-9 < t < -2.3000000000000001e-105

   1. Initial program 72.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j - b \cdot z\right)} \]

      3. *-commutative N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      5. *-lowering-*.f64 53.2

         \[\leadsto c \cdot \left(j \cdot a - \color{blue}{b \cdot z}\right) \]

   5. Simplified 53.2%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \color{blue}{\left(b \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(b \cdot \left(-1 \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      6. neg-sub0 N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(0 - z\right)}\right) \]

      7. --lowering--.f64 41.2

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(0 - z\right)}\right) \]

   8. Simplified 41.2%

      \[\leadsto c \cdot \color{blue}{\left(b \cdot \left(0 - z\right)\right)} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      2. neg-lowering-neg.f64 41.2

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-z\right)}\right) \]

   10. Applied egg-rr 41.2%

       \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-z\right)}\right) \]

   if -2.3000000000000001e-105 < t < 7.30000000000000045e-14

   1. Initial program 87.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 42.0

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 42.0%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]

      2. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

      3. *-lowering-*.f64 33.0

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

   8. Simplified 33.0%

      \[\leadsto \color{blue}{x \cdot \left(z \cdot y\right)} \]

   if 7.30000000000000045e-14 < t

   1. Initial program 63.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 52.0

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 52.0%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(t \cdot i\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(b \cdot t\right) \cdot i} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot b\right)} \cdot i \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{t \cdot \left(b \cdot i\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(b \cdot i\right)} \]

      6. *-lowering-*.f64 38.5

         \[\leadsto t \cdot \color{blue}{\left(b \cdot i\right)} \]

   8. Simplified 38.5%

      \[\leadsto \color{blue}{t \cdot \left(b \cdot i\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{+68}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-9}:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-105}:\\ \;\;\;\;c \cdot \left(0 - z \cdot b\right)\\ \mathbf{elif}\;t \leq 7.3 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+84}:\\ \;\;\;\;t \cdot \left(a \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-121}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+27}:\\ \;\;\;\;j \cdot \left(y \cdot \left(0 - i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -5.2e+84)
   (* t (* a (- 0.0 x)))
   (if (<= a -1e-121)
     (* i (* t b))
     (if (<= a 1.35e+27) (* j (* y (- 0.0 i))) (* c (* a j))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -5.2e+84) {
		tmp = t * (a * (0.0 - x));
	} else if (a <= -1e-121) {
		tmp = i * (t * b);
	} else if (a <= 1.35e+27) {
		tmp = j * (y * (0.0 - i));
	} else {
		tmp = c * (a * j);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (a <= (-5.2d+84)) then
        tmp = t * (a * (0.0d0 - x))
    else if (a <= (-1d-121)) then
        tmp = i * (t * b)
    else if (a <= 1.35d+27) then
        tmp = j * (y * (0.0d0 - i))
    else
        tmp = c * (a * j)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -5.2e+84) {
		tmp = t * (a * (0.0 - x));
	} else if (a <= -1e-121) {
		tmp = i * (t * b);
	} else if (a <= 1.35e+27) {
		tmp = j * (y * (0.0 - i));
	} else {
		tmp = c * (a * j);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if a <= -5.2e+84:
		tmp = t * (a * (0.0 - x))
	elif a <= -1e-121:
		tmp = i * (t * b)
	elif a <= 1.35e+27:
		tmp = j * (y * (0.0 - i))
	else:
		tmp = c * (a * j)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -5.2e+84)
		tmp = Float64(t * Float64(a * Float64(0.0 - x)));
	elseif (a <= -1e-121)
		tmp = Float64(i * Float64(t * b));
	elseif (a <= 1.35e+27)
		tmp = Float64(j * Float64(y * Float64(0.0 - i)));
	else
		tmp = Float64(c * Float64(a * j));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (a <= -5.2e+84)
		tmp = t * (a * (0.0 - x));
	elseif (a <= -1e-121)
		tmp = i * (t * b);
	elseif (a <= 1.35e+27)
		tmp = j * (y * (0.0 - i));
	else
		tmp = c * (a * j);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -5.2e+84], N[(t * N[(a * N[(0.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1e-121], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.35e+27], N[(j * N[(y * N[(0.0 - i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(a * j), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -5.2000000000000002e84

   1. Initial program 65.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), t \cdot \mathsf{fma}\left(a, 0 - x, i \cdot b\right)\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(b \cdot i + -1 \cdot \left(a \cdot x\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto t \cdot \left(b \cdot i + \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(b \cdot i - a \cdot x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto t \cdot \color{blue}{\left(b \cdot i - a \cdot x\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto t \cdot \left(\color{blue}{b \cdot i} - a \cdot x\right) \]

      7. *-lowering-*.f64 50.0

         \[\leadsto t \cdot \left(b \cdot i - \color{blue}{a \cdot x}\right) \]

   8. Simplified 50.0%

      \[\leadsto \color{blue}{t \cdot \left(b \cdot i - a \cdot x\right)} \]

   9. Taylor expanded in b around 0

      \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(a \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto t \cdot \left(\mathsf{neg}\left(\color{blue}{x \cdot a}\right)\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto t \cdot \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(a\right)\right)\right)} \]

       4. mul-1-neg N/A

          \[\leadsto t \cdot \left(x \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto t \cdot \color{blue}{\left(x \cdot \left(-1 \cdot a\right)\right)} \]

       6. mul-1-neg N/A

          \[\leadsto t \cdot \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

       7. neg-sub0 N/A

          \[\leadsto t \cdot \left(x \cdot \color{blue}{\left(0 - a\right)}\right) \]

       8. --lowering--.f64 50.1

          \[\leadsto t \cdot \left(x \cdot \color{blue}{\left(0 - a\right)}\right) \]

   11. Simplified 50.1%

       \[\leadsto t \cdot \color{blue}{\left(x \cdot \left(0 - a\right)\right)} \]

   if -5.2000000000000002e84 < a < -9.9999999999999998e-122

   1. Initial program 73.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 60.4

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 60.4%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

      2. *-lowering-*.f64 44.6

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   8. Simplified 44.6%

      \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   if -9.9999999999999998e-122 < a < 1.3499999999999999e27

   1. Initial program 81.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 46.8

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 46.8%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto 0 - i \cdot \color{blue}{\left(y \cdot j\right)} \]

      6. *-lowering-*.f64 31.3

         \[\leadsto 0 - i \cdot \color{blue}{\left(y \cdot j\right)} \]

   8. Simplified 31.3%

      \[\leadsto \color{blue}{0 - i \cdot \left(y \cdot j\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto 0 - \color{blue}{\left(i \cdot y\right) \cdot j} \]

      2. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot i\right)} \cdot j \]

      3. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot i\right) \cdot j} \]

      4. *-lowering-*.f64 32.9

         \[\leadsto 0 - \color{blue}{\left(y \cdot i\right)} \cdot j \]

   10. Applied egg-rr 32.9%

       \[\leadsto 0 - \color{blue}{\left(y \cdot i\right) \cdot j} \]

   if 1.3499999999999999e27 < a

   1. Initial program 63.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j - b \cdot z\right)} \]

      3. *-commutative N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      5. *-lowering-*.f64 48.5

         \[\leadsto c \cdot \left(j \cdot a - \color{blue}{b \cdot z}\right) \]

   5. Simplified 48.5%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 37.2

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]

   8. Simplified 37.2%

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+84}:\\ \;\;\;\;t \cdot \left(a \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-121}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+27}:\\ \;\;\;\;j \cdot \left(y \cdot \left(0 - i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 29.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+81}:\\ \;\;\;\;t \cdot \left(a \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-121}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+107}:\\ \;\;\;\;0 - i \cdot \left(y \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -8.6e+81)
   (* t (* a (- 0.0 x)))
   (if (<= a -3.7e-121)
     (* i (* t b))
     (if (<= a 1.1e+107) (- 0.0 (* i (* y j))) (* c (* a j))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -8.6e+81) {
		tmp = t * (a * (0.0 - x));
	} else if (a <= -3.7e-121) {
		tmp = i * (t * b);
	} else if (a <= 1.1e+107) {
		tmp = 0.0 - (i * (y * j));
	} else {
		tmp = c * (a * j);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (a <= (-8.6d+81)) then
        tmp = t * (a * (0.0d0 - x))
    else if (a <= (-3.7d-121)) then
        tmp = i * (t * b)
    else if (a <= 1.1d+107) then
        tmp = 0.0d0 - (i * (y * j))
    else
        tmp = c * (a * j)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -8.6e+81) {
		tmp = t * (a * (0.0 - x));
	} else if (a <= -3.7e-121) {
		tmp = i * (t * b);
	} else if (a <= 1.1e+107) {
		tmp = 0.0 - (i * (y * j));
	} else {
		tmp = c * (a * j);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if a <= -8.6e+81:
		tmp = t * (a * (0.0 - x))
	elif a <= -3.7e-121:
		tmp = i * (t * b)
	elif a <= 1.1e+107:
		tmp = 0.0 - (i * (y * j))
	else:
		tmp = c * (a * j)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -8.6e+81)
		tmp = Float64(t * Float64(a * Float64(0.0 - x)));
	elseif (a <= -3.7e-121)
		tmp = Float64(i * Float64(t * b));
	elseif (a <= 1.1e+107)
		tmp = Float64(0.0 - Float64(i * Float64(y * j)));
	else
		tmp = Float64(c * Float64(a * j));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (a <= -8.6e+81)
		tmp = t * (a * (0.0 - x));
	elseif (a <= -3.7e-121)
		tmp = i * (t * b);
	elseif (a <= 1.1e+107)
		tmp = 0.0 - (i * (y * j));
	else
		tmp = c * (a * j);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -8.6e+81], N[(t * N[(a * N[(0.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -3.7e-121], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.1e+107], N[(0.0 - N[(i * N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(a * j), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -8.6000000000000003e81

   1. Initial program 65.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), t \cdot \mathsf{fma}\left(a, 0 - x, i \cdot b\right)\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(b \cdot i + -1 \cdot \left(a \cdot x\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto t \cdot \left(b \cdot i + \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(b \cdot i - a \cdot x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto t \cdot \color{blue}{\left(b \cdot i - a \cdot x\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto t \cdot \left(\color{blue}{b \cdot i} - a \cdot x\right) \]

      7. *-lowering-*.f64 50.0

         \[\leadsto t \cdot \left(b \cdot i - \color{blue}{a \cdot x}\right) \]

   8. Simplified 50.0%

      \[\leadsto \color{blue}{t \cdot \left(b \cdot i - a \cdot x\right)} \]

   9. Taylor expanded in b around 0

      \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(a \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto t \cdot \left(\mathsf{neg}\left(\color{blue}{x \cdot a}\right)\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto t \cdot \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(a\right)\right)\right)} \]

       4. mul-1-neg N/A

          \[\leadsto t \cdot \left(x \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto t \cdot \color{blue}{\left(x \cdot \left(-1 \cdot a\right)\right)} \]

       6. mul-1-neg N/A

          \[\leadsto t \cdot \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

       7. neg-sub0 N/A

          \[\leadsto t \cdot \left(x \cdot \color{blue}{\left(0 - a\right)}\right) \]

       8. --lowering--.f64 50.1

          \[\leadsto t \cdot \left(x \cdot \color{blue}{\left(0 - a\right)}\right) \]

   11. Simplified 50.1%

       \[\leadsto t \cdot \color{blue}{\left(x \cdot \left(0 - a\right)\right)} \]

   if -8.6000000000000003e81 < a < -3.7000000000000002e-121

   1. Initial program 73.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 61.6

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 61.6%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

      2. *-lowering-*.f64 45.5

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   8. Simplified 45.5%

      \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   if -3.7000000000000002e-121 < a < 1.1e107

   1. Initial program 81.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 46.2

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 46.2%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto 0 - i \cdot \color{blue}{\left(y \cdot j\right)} \]

      6. *-lowering-*.f64 31.0

         \[\leadsto 0 - i \cdot \color{blue}{\left(y \cdot j\right)} \]

   8. Simplified 31.0%

      \[\leadsto \color{blue}{0 - i \cdot \left(y \cdot j\right)} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(i \cdot \left(y \cdot j\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(y \cdot j\right) \cdot i}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot j\right)\right) \cdot i} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot j\right)\right) \cdot i} \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot j\right)\right)} \cdot i \]

      6. *-lowering-*.f64 31.0

         \[\leadsto \left(-\color{blue}{y \cdot j}\right) \cdot i \]

   10. Applied egg-rr 31.0%

       \[\leadsto \color{blue}{\left(-y \cdot j\right) \cdot i} \]

   if 1.1e107 < a

   1. Initial program 58.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j - b \cdot z\right)} \]

      3. *-commutative N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      5. *-lowering-*.f64 52.0

         \[\leadsto c \cdot \left(j \cdot a - \color{blue}{b \cdot z}\right) \]

   5. Simplified 52.0%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 39.3

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]

   8. Simplified 39.3%

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+81}:\\ \;\;\;\;t \cdot \left(a \cdot \left(0 - x\right)\right)\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-121}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+107}:\\ \;\;\;\;0 - i \cdot \left(y \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 29.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+75}:\\ \;\;\;\;a \cdot \left(x \cdot \left(0 - t\right)\right)\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-121}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;a \leq 9 \cdot 10^{+106}:\\ \;\;\;\;0 - i \cdot \left(y \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -3.4e+75)
   (* a (* x (- 0.0 t)))
   (if (<= a -4.8e-121)
     (* i (* t b))
     (if (<= a 9e+106) (- 0.0 (* i (* y j))) (* c (* a j))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -3.4e+75) {
		tmp = a * (x * (0.0 - t));
	} else if (a <= -4.8e-121) {
		tmp = i * (t * b);
	} else if (a <= 9e+106) {
		tmp = 0.0 - (i * (y * j));
	} else {
		tmp = c * (a * j);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (a <= (-3.4d+75)) then
        tmp = a * (x * (0.0d0 - t))
    else if (a <= (-4.8d-121)) then
        tmp = i * (t * b)
    else if (a <= 9d+106) then
        tmp = 0.0d0 - (i * (y * j))
    else
        tmp = c * (a * j)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -3.4e+75) {
		tmp = a * (x * (0.0 - t));
	} else if (a <= -4.8e-121) {
		tmp = i * (t * b);
	} else if (a <= 9e+106) {
		tmp = 0.0 - (i * (y * j));
	} else {
		tmp = c * (a * j);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if a <= -3.4e+75:
		tmp = a * (x * (0.0 - t))
	elif a <= -4.8e-121:
		tmp = i * (t * b)
	elif a <= 9e+106:
		tmp = 0.0 - (i * (y * j))
	else:
		tmp = c * (a * j)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -3.4e+75)
		tmp = Float64(a * Float64(x * Float64(0.0 - t)));
	elseif (a <= -4.8e-121)
		tmp = Float64(i * Float64(t * b));
	elseif (a <= 9e+106)
		tmp = Float64(0.0 - Float64(i * Float64(y * j)));
	else
		tmp = Float64(c * Float64(a * j));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (a <= -3.4e+75)
		tmp = a * (x * (0.0 - t));
	elseif (a <= -4.8e-121)
		tmp = i * (t * b);
	elseif (a <= 9e+106)
		tmp = 0.0 - (i * (y * j));
	else
		tmp = c * (a * j);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -3.4e+75], N[(a * N[(x * N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -4.8e-121], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9e+106], N[(0.0 - N[(i * N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(a * j), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -3.40000000000000011e75

   1. Initial program 65.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(x \cdot \color{blue}{\frac{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}{y \cdot z + t \cdot a}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. clear-num N/A

         \[\leadsto \left(x \cdot \color{blue}{\frac{1}{\frac{y \cdot z + t \cdot a}{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. un-div-inv N/A

         \[\leadsto \left(\color{blue}{\frac{x}{\frac{y \cdot z + t \cdot a}{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{x}{\frac{y \cdot z + t \cdot a}{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{x}{\color{blue}{\frac{1}{\frac{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}{y \cdot z + t \cdot a}}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. flip-- N/A

         \[\leadsto \left(\frac{x}{\frac{1}{\color{blue}{y \cdot z - t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{x}{\color{blue}{\frac{1}{y \cdot z - t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\frac{x}{\frac{1}{\color{blue}{y \cdot z - t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{x}{\frac{1}{\color{blue}{y \cdot z} - t \cdot a}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. *-lowering-*.f64 65.0

          \[\leadsto \left(\frac{x}{\frac{1}{y \cdot z - \color{blue}{t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Applied egg-rr 65.0%

      \[\leadsto \left(\color{blue}{\frac{x}{\frac{1}{y \cdot z - t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   5. Taylor expanded in c around inf

      \[\leadsto \left(\frac{x}{\frac{1}{y \cdot z - t \cdot a}} - \color{blue}{b \cdot \left(c \cdot z\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{x}{\frac{1}{y \cdot z - t \cdot a}} - \color{blue}{\left(c \cdot z\right) \cdot b}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{x}{\frac{1}{y \cdot z - t \cdot a}} - \color{blue}{\left(z \cdot c\right)} \cdot b\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{x}{\frac{1}{y \cdot z - t \cdot a}} - \color{blue}{z \cdot \left(c \cdot b\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{x}{\frac{1}{y \cdot z - t \cdot a}} - z \cdot \color{blue}{\left(b \cdot c\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{x}{\frac{1}{y \cdot z - t \cdot a}} - \color{blue}{z \cdot \left(b \cdot c\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{x}{\frac{1}{y \cdot z - t \cdot a}} - z \cdot \color{blue}{\left(c \cdot b\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. *-lowering-*.f64 73.9

         \[\leadsto \left(\frac{x}{\frac{1}{y \cdot z - t \cdot a}} - z \cdot \color{blue}{\left(c \cdot b\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   7. Simplified 73.9%

      \[\leadsto \left(\frac{x}{\frac{1}{y \cdot z - t \cdot a}} - \color{blue}{z \cdot \left(c \cdot b\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right)} \]
   9. 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. *-commutative N/A

         \[\leadsto a \cdot \left(\mathsf{neg}\left(\color{blue}{x \cdot t}\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \color{blue}{\left(x \cdot \left(\mathsf{neg}\left(t\right)\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto a \cdot \left(x \cdot \color{blue}{\left(-1 \cdot t\right)}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(x \cdot \left(-1 \cdot t\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto a \cdot \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) \]

      11. neg-lowering-neg.f64 47.9

          \[\leadsto a \cdot \left(x \cdot \color{blue}{\left(-t\right)}\right) \]

   10. Simplified 47.9%

       \[\leadsto \color{blue}{a \cdot \left(x \cdot \left(-t\right)\right)} \]

   if -3.40000000000000011e75 < a < -4.80000000000000007e-121

   1. Initial program 73.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 61.6

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 61.6%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

      2. *-lowering-*.f64 45.5

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   8. Simplified 45.5%

      \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   if -4.80000000000000007e-121 < a < 8.9999999999999994e106

   1. Initial program 81.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 46.2

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 46.2%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto 0 - i \cdot \color{blue}{\left(y \cdot j\right)} \]

      6. *-lowering-*.f64 31.0

         \[\leadsto 0 - i \cdot \color{blue}{\left(y \cdot j\right)} \]

   8. Simplified 31.0%

      \[\leadsto \color{blue}{0 - i \cdot \left(y \cdot j\right)} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(i \cdot \left(y \cdot j\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(y \cdot j\right) \cdot i}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot j\right)\right) \cdot i} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot j\right)\right) \cdot i} \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot j\right)\right)} \cdot i \]

      6. *-lowering-*.f64 31.0

         \[\leadsto \left(-\color{blue}{y \cdot j}\right) \cdot i \]

   10. Applied egg-rr 31.0%

       \[\leadsto \color{blue}{\left(-y \cdot j\right) \cdot i} \]

   if 8.9999999999999994e106 < a

   1. Initial program 58.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j - b \cdot z\right)} \]

      3. *-commutative N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      5. *-lowering-*.f64 52.0

         \[\leadsto c \cdot \left(j \cdot a - \color{blue}{b \cdot z}\right) \]

   5. Simplified 52.0%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 39.3

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]

   8. Simplified 39.3%

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+75}:\\ \;\;\;\;a \cdot \left(x \cdot \left(0 - t\right)\right)\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-121}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;a \leq 9 \cdot 10^{+106}:\\ \;\;\;\;0 - i \cdot \left(y \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 52.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \mathbf{if}\;i \leq -7.2 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 1.65 \cdot 10^{-18}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, 0 - b \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- 0.0 y) (* t b)))))
   (if (<= i -7.2e+55)
     t_1
     (if (<= i 1.65e-18) (* z (fma y x (- 0.0 (* b 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 = i * fma(j, (0.0 - y), (t * b));
	double tmp;
	if (i <= -7.2e+55) {
		tmp = t_1;
	} else if (i <= 1.65e-18) {
		tmp = z * fma(y, x, (0.0 - (b * c)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * fma(j, Float64(0.0 - y), Float64(t * b)))
	tmp = 0.0
	if (i <= -7.2e+55)
		tmp = t_1;
	elseif (i <= 1.65e-18)
		tmp = Float64(z * fma(y, x, Float64(0.0 - Float64(b * c))));
	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[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -7.2e+55], t$95$1, If[LessEqual[i, 1.65e-18], N[(z * N[(y * x + N[(0.0 - N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -7.19999999999999975e55 or 1.6500000000000001e-18 < i

   1. Initial program 65.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 67.4

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      2. neg-lowering-neg.f64 67.4

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot t\right) \]

   7. Applied egg-rr 67.4%

      \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot t\right) \]

   if -7.19999999999999975e55 < i < 1.6500000000000001e-18

   1. Initial program 80.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 51.6

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 51.6%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x + c \cdot \left(0 - b\right)\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(y, x, c \cdot \left(0 - b\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\left(0 - b\right) \cdot c}\right) \]

      4. sub0-neg N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot c\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\mathsf{neg}\left(b \cdot c\right)}\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{\mathsf{neg}\left(b \cdot c\right)}\right) \]

      7. *-lowering-*.f64 53.1

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, -\color{blue}{b \cdot c}\right) \]

   7. Applied egg-rr 53.1%

      \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(y, x, -b \cdot c\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.2 \cdot 10^{+55}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \mathbf{elif}\;i \leq 1.65 \cdot 10^{-18}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, 0 - b \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 50.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \mathbf{if}\;i \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 9.8 \cdot 10^{-117}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- 0.0 y) (* t b)))))
   (if (<= i -3.5e-39)
     t_1
     (if (<= i 9.8e-117) (* c (- (* a j) (* z b))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * fma(j, (0.0 - y), (t * b));
	double tmp;
	if (i <= -3.5e-39) {
		tmp = t_1;
	} else if (i <= 9.8e-117) {
		tmp = c * ((a * j) - (z * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * fma(j, Float64(0.0 - y), Float64(t * b)))
	tmp = 0.0
	if (i <= -3.5e-39)
		tmp = t_1;
	elseif (i <= 9.8e-117)
		tmp = Float64(c * Float64(Float64(a * j) - Float64(z * b)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -3.5e-39], t$95$1, If[LessEqual[i, 9.8e-117], N[(c * N[(N[(a * j), $MachinePrecision] - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -3.5e-39 or 9.7999999999999995e-117 < i

   1. Initial program 67.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 60.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 60.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      2. neg-lowering-neg.f64 60.3

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot t\right) \]

   7. Applied egg-rr 60.3%

      \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot t\right) \]

   if -3.5e-39 < i < 9.7999999999999995e-117

   1. Initial program 82.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j - b \cdot z\right)} \]

      3. *-commutative N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      5. *-lowering-*.f64 55.4

         \[\leadsto c \cdot \left(j \cdot a - \color{blue}{b \cdot z}\right) \]

   5. Simplified 55.4%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot a - b \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \mathbf{elif}\;i \leq 9.8 \cdot 10^{-117}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, t \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 52.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \mathbf{if}\;a \leq -9 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-45}:\\ \;\;\;\;b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (fma x (- 0.0 t) (* c j)))))
   (if (<= a -9e+65) t_1 (if (<= a 7.5e-45) (* b (- (* t i) (* z 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 * fma(x, (0.0 - t), (c * j));
	double tmp;
	if (a <= -9e+65) {
		tmp = t_1;
	} else if (a <= 7.5e-45) {
		tmp = b * ((t * i) - (z * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * fma(x, Float64(0.0 - t), Float64(c * j)))
	tmp = 0.0
	if (a <= -9e+65)
		tmp = t_1;
	elseif (a <= 7.5e-45)
		tmp = Float64(b * Float64(Float64(t * i) - Float64(z * c)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(x * N[(0.0 - t), $MachinePrecision] + N[(c * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -9e+65], t$95$1, If[LessEqual[a, 7.5e-45], N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -9e65 or 7.5000000000000006e-45 < a

   1. Initial program 67.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(x \cdot \color{blue}{\frac{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}{y \cdot z + t \cdot a}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. clear-num N/A

         \[\leadsto \left(x \cdot \color{blue}{\frac{1}{\frac{y \cdot z + t \cdot a}{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. un-div-inv N/A

         \[\leadsto \left(\color{blue}{\frac{x}{\frac{y \cdot z + t \cdot a}{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{x}{\frac{y \cdot z + t \cdot a}{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. clear-num N/A

         \[\leadsto \left(\frac{x}{\color{blue}{\frac{1}{\frac{\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(t \cdot a\right) \cdot \left(t \cdot a\right)}{y \cdot z + t \cdot a}}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. flip-- N/A

         \[\leadsto \left(\frac{x}{\frac{1}{\color{blue}{y \cdot z - t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \left(\frac{x}{\color{blue}{\frac{1}{y \cdot z - t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\frac{x}{\frac{1}{\color{blue}{y \cdot z - t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{x}{\frac{1}{\color{blue}{y \cdot z} - t \cdot a}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. *-lowering-*.f64 67.0

          \[\leadsto \left(\frac{x}{\frac{1}{y \cdot z - \color{blue}{t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Applied egg-rr 67.0%

      \[\leadsto \left(\color{blue}{\frac{x}{\frac{1}{y \cdot z - t \cdot a}}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + c \cdot j\right) \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\left(\mathsf{neg}\left(\color{blue}{x \cdot t}\right)\right) + c \cdot j\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(t\right)\right)} + c \cdot j\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(x \cdot \color{blue}{\left(-1 \cdot t\right)} + c \cdot j\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(x, -1 \cdot t, c \cdot j\right)} \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{neg}\left(t\right)}, c \cdot j\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{neg}\left(t\right)}, c \cdot j\right) \]

      9. *-commutative N/A

         \[\leadsto a \cdot \mathsf{fma}\left(x, \mathsf{neg}\left(t\right), \color{blue}{j \cdot c}\right) \]

      10. *-lowering-*.f64 56.8

          \[\leadsto a \cdot \mathsf{fma}\left(x, -t, \color{blue}{j \cdot c}\right) \]

   7. Simplified 56.8%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(x, -t, j \cdot c\right)} \]

   if -9e65 < a < 7.5000000000000006e-45

   1. Initial program 78.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, \mathsf{fma}\left(a, c, i \cdot \left(0 - y\right)\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), t \cdot \mathsf{fma}\left(a, 0 - x, i \cdot b\right)\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot t + -1 \cdot \left(c \cdot z\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto b \cdot \left(i \cdot t + \color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot t - c \cdot z\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot t - c \cdot z\right)} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \left(\color{blue}{t \cdot i} - c \cdot z\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto b \cdot \left(\color{blue}{t \cdot i} - c \cdot z\right) \]

      8. *-commutative N/A

         \[\leadsto b \cdot \left(t \cdot i - \color{blue}{z \cdot c}\right) \]

      9. *-lowering-*.f64 42.8

         \[\leadsto b \cdot \left(t \cdot i - \color{blue}{z \cdot c}\right) \]

   8. Simplified 42.8%

      \[\leadsto \color{blue}{b \cdot \left(t \cdot i - z \cdot c\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+65}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-45}:\\ \;\;\;\;b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, 0 - t, c \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 29.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{+68}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-110}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -6.4e+68)
   (* i (* t b))
   (if (<= t -9.5e-110)
     (* c (* a j))
     (if (<= t 4.2e-14) (* x (* y z)) (* t (* 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 (t <= -6.4e+68) {
		tmp = i * (t * b);
	} else if (t <= -9.5e-110) {
		tmp = c * (a * j);
	} else if (t <= 4.2e-14) {
		tmp = x * (y * z);
	} else {
		tmp = t * (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 (t <= (-6.4d+68)) then
        tmp = i * (t * b)
    else if (t <= (-9.5d-110)) then
        tmp = c * (a * j)
    else if (t <= 4.2d-14) then
        tmp = x * (y * z)
    else
        tmp = t * (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 (t <= -6.4e+68) {
		tmp = i * (t * b);
	} else if (t <= -9.5e-110) {
		tmp = c * (a * j);
	} else if (t <= 4.2e-14) {
		tmp = x * (y * z);
	} else {
		tmp = t * (b * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if t <= -6.4e+68:
		tmp = i * (t * b)
	elif t <= -9.5e-110:
		tmp = c * (a * j)
	elif t <= 4.2e-14:
		tmp = x * (y * z)
	else:
		tmp = t * (b * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -6.4e+68)
		tmp = Float64(i * Float64(t * b));
	elseif (t <= -9.5e-110)
		tmp = Float64(c * Float64(a * j));
	elseif (t <= 4.2e-14)
		tmp = Float64(x * Float64(y * z));
	else
		tmp = Float64(t * 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 (t <= -6.4e+68)
		tmp = i * (t * b);
	elseif (t <= -9.5e-110)
		tmp = c * (a * j);
	elseif (t <= 4.2e-14)
		tmp = x * (y * z);
	else
		tmp = t * (b * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -6.4e+68], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -9.5e-110], N[(c * N[(a * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-14], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(t * N[(b * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -6.39999999999999989e68

   1. Initial program 53.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 54.9

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 54.9%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

      2. *-lowering-*.f64 46.4

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   8. Simplified 46.4%

      \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   if -6.39999999999999989e68 < t < -9.50000000000000004e-110

   1. Initial program 67.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j - b \cdot z\right)} \]

      3. *-commutative N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      5. *-lowering-*.f64 51.5

         \[\leadsto c \cdot \left(j \cdot a - \color{blue}{b \cdot z}\right) \]

   5. Simplified 51.5%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 31.0

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]

   8. Simplified 31.0%

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]

   if -9.50000000000000004e-110 < t < 4.1999999999999998e-14

   1. Initial program 89.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 41.8

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 41.8%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]

      2. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

      3. *-lowering-*.f64 33.5

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

   8. Simplified 33.5%

      \[\leadsto \color{blue}{x \cdot \left(z \cdot y\right)} \]

   if 4.1999999999999998e-14 < t

   1. Initial program 63.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 52.0

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 52.0%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(t \cdot i\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(b \cdot t\right) \cdot i} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot b\right)} \cdot i \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{t \cdot \left(b \cdot i\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(b \cdot i\right)} \]

      6. *-lowering-*.f64 38.5

         \[\leadsto t \cdot \color{blue}{\left(b \cdot i\right)} \]

   8. Simplified 38.5%

      \[\leadsto \color{blue}{t \cdot \left(b \cdot i\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{+68}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-110}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 29.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(a \cdot j\right)\\ \mathbf{if}\;j \leq -1.55 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 2.9 \cdot 10^{+88}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* a j))))
   (if (<= j -1.55e+103) t_1 (if (<= j 2.9e+88) (* i (* t b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (a * j);
	double tmp;
	if (j <= -1.55e+103) {
		tmp = t_1;
	} else if (j <= 2.9e+88) {
		tmp = i * (t * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (a * j)
    if (j <= (-1.55d+103)) then
        tmp = t_1
    else if (j <= 2.9d+88) then
        tmp = i * (t * b)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (a * j);
	double tmp;
	if (j <= -1.55e+103) {
		tmp = t_1;
	} else if (j <= 2.9e+88) {
		tmp = i * (t * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (a * j)
	tmp = 0
	if j <= -1.55e+103:
		tmp = t_1
	elif j <= 2.9e+88:
		tmp = i * (t * b)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(a * j))
	tmp = 0.0
	if (j <= -1.55e+103)
		tmp = t_1;
	elseif (j <= 2.9e+88)
		tmp = Float64(i * Float64(t * b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (a * j);
	tmp = 0.0;
	if (j <= -1.55e+103)
		tmp = t_1;
	elseif (j <= 2.9e+88)
		tmp = i * (t * b);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(a * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -1.55e+103], t$95$1, If[LessEqual[j, 2.9e+88], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -1.5500000000000001e103 or 2.9e88 < j

   1. Initial program 76.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j - b \cdot z\right)} \]

      3. *-commutative N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

      5. *-lowering-*.f64 49.0

         \[\leadsto c \cdot \left(j \cdot a - \color{blue}{b \cdot z}\right) \]

   5. Simplified 49.0%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 40.5

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]

   8. Simplified 40.5%

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]

   if -1.5500000000000001e103 < j < 2.9e88

   1. Initial program 71.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, b \cdot t\right) \]

      12. *-lowering-*.f64 40.8

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 40.8%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

      2. *-lowering-*.f64 28.9

         \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]

   8. Simplified 28.9%

      \[\leadsto i \cdot \color{blue}{\left(t \cdot b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 22.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(a \cdot j\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* c (* a j)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return c * (a * j);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = c * (a * j)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return c * (a * j);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return c * (a * j)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(c * Float64(a * j))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = c * (a * j);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(c * N[(a * j), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.0%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in c around inf

   \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]

   2. --lowering--.f64 N/A

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j - b \cdot z\right)} \]

   3. *-commutative N/A

      \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto c \cdot \left(\color{blue}{j \cdot a} - b \cdot z\right) \]

   5. *-lowering-*.f64 35.0

      \[\leadsto c \cdot \left(j \cdot a - \color{blue}{b \cdot z}\right) \]

5. Simplified 35.0%

   \[\leadsto \color{blue}{c \cdot \left(j \cdot a - b \cdot z\right)} \]

6. Taylor expanded in j around inf

   \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 19.2

      \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]

8. Simplified 19.2%

   \[\leadsto c \cdot \color{blue}{\left(a \cdot j\right)} \]
9. Add Preprocessing

Developer Target 1: 60.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_2 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + t\_1\\ \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* c a) (* y i))))
        (t_2
         (+
          (-
           (* x (- (* y z) (* t a)))
           (/
            (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0)))
            (+ (* c z) (* t i))))
          t_1)))
   (if (< x -1.469694296777705e-64)
     t_2
     (if (< x 3.2113527362226803e-147)
       (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) t_1))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (pow((c * z), 2.0) - pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = j * ((c * a) - (y * i))
    t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ** 2.0d0) - ((t * i) ** 2.0d0))) / ((c * z) + (t * i)))) + t_1
    if (x < (-1.469694296777705d-64)) then
        tmp = t_2
    else if (x < 3.2113527362226803d-147) then
        tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (Math.pow((c * z), 2.0) - Math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((c * a) - (y * i))
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (math.pow((c * z), 2.0) - math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1
	tmp = 0
	if x < -1.469694296777705e-64:
		tmp = t_2
	elif x < 3.2113527362226803e-147:
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_2 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(b * Float64((Float64(c * z) ^ 2.0) - (Float64(t * i) ^ 2.0))) / Float64(Float64(c * z) + Float64(t * i)))) + t_1)
	tmp = 0.0
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = Float64(Float64(Float64(Float64(b * i) - Float64(x * a)) * t) - Float64(Float64(z * Float64(c * b)) - t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * ((c * a) - (y * i));
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ^ 2.0) - ((t * i) ^ 2.0))) / ((c * z) + (t * i)))) + t_1;
	tmp = 0.0;
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[Power[N[(c * z), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(t * i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Less[x, -1.469694296777705e-64], t$95$2, If[Less[x, 3.2113527362226803e-147], N[(N[(N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] - N[(N[(z * N[(c * b), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024196 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -293938859355541/2000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 32113527362226803/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))