Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 74.1% → 82.2%

Time: 31.0s

Alternatives: 26

Speedup: 0.5×

• 58.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in a around inf

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

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

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 2.0

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

   5. Simplified 2.0%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

      7. associate-/l* N/A

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

      8. mul-1-neg N/A

         \[\leadsto 0 - a \cdot \left(b \cdot \left(c \cdot \frac{z}{a} + \color{blue}{-1 \cdot i}\right)\right) \]

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

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. --lowering--.f64 53.3

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

   8. Simplified 53.3%

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

4. Final simplification 84.1%

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

Alternative 2: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(t \cdot c - y \cdot i\right) + \mathsf{fma}\left(b, \mathsf{fma}\left(c, 0 - z, a \cdot i\right), y \cdot \left(x \cdot z\right)\right)\\ \mathbf{if}\;j \leq -7.8 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 2.6 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, x \cdot y\right), a \cdot \mathsf{fma}\left(t, 0 - x, b \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (* j (- (* t c) (* y i)))
          (fma b (fma c (- 0.0 z) (* a i)) (* y (* x z))))))
   (if (<= j -7.8e-10)
     t_1
     (if (<= j 2.6e-75)
       (fma z (fma c (- 0.0 b) (* x y)) (* a (fma t (- 0.0 x) (* b i))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (j * ((t * c) - (y * i))) + fma(b, fma(c, (0.0 - z), (a * i)), (y * (x * z)));
	double tmp;
	if (j <= -7.8e-10) {
		tmp = t_1;
	} else if (j <= 2.6e-75) {
		tmp = fma(z, fma(c, (0.0 - b), (x * y)), (a * fma(t, (0.0 - x), (b * i))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(j * Float64(Float64(t * c) - Float64(y * i))) + fma(b, fma(c, Float64(0.0 - z), Float64(a * i)), Float64(y * Float64(x * z))))
	tmp = 0.0
	if (j <= -7.8e-10)
		tmp = t_1;
	elseif (j <= 2.6e-75)
		tmp = fma(z, fma(c, Float64(0.0 - b), Float64(x * y)), Float64(a * fma(t, Float64(0.0 - x), Float64(b * i))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * N[(0.0 - z), $MachinePrecision] + N[(a * i), $MachinePrecision]), $MachinePrecision] + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -7.8e-10], t$95$1, If[LessEqual[j, 2.6e-75], N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(a * N[(t * N[(0.0 - x), $MachinePrecision] + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -7.7999999999999999e-10 or 2.6e-75 < j

   1. Initial program 75.0%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

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

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

      15. mul-1-neg N/A

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

      16. neg-sub0 N/A

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

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

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

      21. associate-*r* N/A

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

      22. *-commutative N/A

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

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

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

      24. *-commutative N/A

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

      25. *-lowering-*.f64 78.6

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

   5. Simplified 78.6%

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

   if -7.7999999999999999e-10 < j < 2.6e-75

   1. Initial program 71.8%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 82.9%

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

   6. Taylor expanded in j around 0

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

   8. Simplified 78.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -7.8 \cdot 10^{-10}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \mathsf{fma}\left(b, \mathsf{fma}\left(c, 0 - z, a \cdot i\right), y \cdot \left(x \cdot z\right)\right)\\ \mathbf{elif}\;j \leq 2.6 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, x \cdot y\right), a \cdot \mathsf{fma}\left(t, 0 - x, b \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \mathsf{fma}\left(b, \mathsf{fma}\left(c, 0 - z, a \cdot i\right), y \cdot \left(x \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -1.15 \cdot 10^{+32}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;j \leq 1.05 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, x \cdot y\right), a \cdot \mathsf{fma}\left(t, 0 - x, b \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 2.7 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, 0 - z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(c, t, y \cdot \left(0 - i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -1.15e+32)
   (- (* j (- (* t c) (* y i))) (* c (* z b)))
   (if (<= j 1.05e-90)
     (fma z (fma c (- 0.0 b) (* x y)) (* a (fma t (- 0.0 x) (* b i))))
     (if (<= j 2.7e+162)
       (fma y (fma j (- 0.0 i) (* x z)) (* b (fma c (- 0.0 z) (* a i))))
       (* j (fma c t (* y (- 0.0 i))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -1.15e+32) {
		tmp = (j * ((t * c) - (y * i))) - (c * (z * b));
	} else if (j <= 1.05e-90) {
		tmp = fma(z, fma(c, (0.0 - b), (x * y)), (a * fma(t, (0.0 - x), (b * i))));
	} else if (j <= 2.7e+162) {
		tmp = fma(y, fma(j, (0.0 - i), (x * z)), (b * fma(c, (0.0 - z), (a * i))));
	} else {
		tmp = j * fma(c, t, (y * (0.0 - i)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -1.15e+32)
		tmp = Float64(Float64(j * Float64(Float64(t * c) - Float64(y * i))) - Float64(c * Float64(z * b)));
	elseif (j <= 1.05e-90)
		tmp = fma(z, fma(c, Float64(0.0 - b), Float64(x * y)), Float64(a * fma(t, Float64(0.0 - x), Float64(b * i))));
	elseif (j <= 2.7e+162)
		tmp = fma(y, fma(j, Float64(0.0 - i), Float64(x * z)), Float64(b * fma(c, Float64(0.0 - z), Float64(a * i))));
	else
		tmp = Float64(j * fma(c, t, Float64(y * Float64(0.0 - i))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -1.15e+32], N[(N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 1.05e-90], N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(a * N[(t * N[(0.0 - x), $MachinePrecision] + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 2.7e+162], N[(y * N[(j * N[(0.0 - i), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * N[(0.0 - z), $MachinePrecision] + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(j * N[(c * t + N[(y * N[(0.0 - i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if j < -1.15e32

   1. Initial program 70.5%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. neg-sub0 N/A

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

      12. --lowering--.f64 77.1

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

   5. Simplified 77.1%

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

   if -1.15e32 < j < 1.05e-90

   1. Initial program 74.5%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 84.2%

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

   6. Taylor expanded in j around 0

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

   8. Simplified 76.3%

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

   if 1.05e-90 < j < 2.7000000000000002e162

   1. Initial program 74.2%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

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

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

      14. *-commutative N/A

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

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

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

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

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

      17. mul-1-neg N/A

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

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

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

      19. mul-1-neg N/A

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

   5. Simplified 76.4%

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

   if 2.7000000000000002e162 < j

   1. Initial program 74.9%

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

   3. Taylor expanded in j around inf

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 69.5

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

   5. Simplified 69.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.15 \cdot 10^{+32}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;j \leq 1.05 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, x \cdot y\right), a \cdot \mathsf{fma}\left(t, 0 - x, b \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 2.7 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, 0 - z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(c, t, y \cdot \left(0 - i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(c \cdot j - x \cdot a\right)\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+275}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, 0 - z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* t (- (* c j) (* x a)))))
   (if (<= t -8.5e+275)
     t_1
     (if (<= t 1e+61)
       (fma y (fma j (- 0.0 i) (* x z)) (* b (fma c (- 0.0 z) (* a i))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = t * ((c * j) - (x * a));
	double tmp;
	if (t <= -8.5e+275) {
		tmp = t_1;
	} else if (t <= 1e+61) {
		tmp = fma(y, fma(j, (0.0 - i), (x * z)), (b * fma(c, (0.0 - z), (a * i))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(t * Float64(Float64(c * j) - Float64(x * a)))
	tmp = 0.0
	if (t <= -8.5e+275)
		tmp = t_1;
	elseif (t <= 1e+61)
		tmp = fma(y, fma(j, Float64(0.0 - i), Float64(x * z)), Float64(b * fma(c, Float64(0.0 - z), Float64(a * i))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(t * N[(N[(c * j), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.5e+275], t$95$1, If[LessEqual[t, 1e+61], N[(y * N[(j * N[(0.0 - i), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * N[(0.0 - z), $MachinePrecision] + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.49999999999999958e275 or 9.99999999999999949e60 < t

   1. Initial program 66.4%

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

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)}^{3} + {\left(j \cdot \left(c \cdot t - i \cdot y\right)\right)}^{3}}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot \left(c \cdot t - i \cdot y\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right) - \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right)\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot \left(c \cdot t - i \cdot y\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right) - \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right)\right)}{{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)}^{3} + {\left(j \cdot \left(c \cdot t - i \cdot y\right)\right)}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot \left(c \cdot t - i \cdot y\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right) - \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right)\right)}{{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)}^{3} + {\left(j \cdot \left(c \cdot t - i \cdot y\right)\right)}^{3}}}} \]

   4. Applied egg-rr 66.4%

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

   5. Taylor expanded in t around inf

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 68.7

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

   7. Simplified 68.7%

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

   if -8.49999999999999958e275 < t < 9.99999999999999949e60

   1. Initial program 75.5%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

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

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

      14. *-commutative N/A

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

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

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

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

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

      17. mul-1-neg N/A

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

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

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

      19. mul-1-neg N/A

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

   5. Simplified 72.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+275}:\\ \;\;\;\;t \cdot \left(c \cdot j - x \cdot a\right)\\ \mathbf{elif}\;t \leq 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, 0 - i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, 0 - z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(c \cdot j - x \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-15}:\\ \;\;\;\;\left(0 - j\right) \cdot \mathsf{fma}\left(b, \frac{z \cdot c}{j}, y \cdot i\right)\\ \mathbf{elif}\;x \leq 0.0305:\\ \;\;\;\;a \cdot \left(b \cdot i\right) + c \cdot \mathsf{fma}\left(b, 0 - z, t \cdot j\right)\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+214}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (- (* y z) (* t a)))))
   (if (<= x -8e+118)
     t_1
     (if (<= x -9.5e-15)
       (* (- 0.0 j) (fma b (/ (* z c) j) (* y i)))
       (if (<= x 0.0305)
         (+ (* a (* b i)) (* c (fma b (- 0.0 z) (* t j))))
         (if (<= x 2.25e+214) t_1 (* z (fma c (- 0.0 b) (* x y)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * ((y * z) - (t * a));
	double tmp;
	if (x <= -8e+118) {
		tmp = t_1;
	} else if (x <= -9.5e-15) {
		tmp = (0.0 - j) * fma(b, ((z * c) / j), (y * i));
	} else if (x <= 0.0305) {
		tmp = (a * (b * i)) + (c * fma(b, (0.0 - z), (t * j)));
	} else if (x <= 2.25e+214) {
		tmp = t_1;
	} else {
		tmp = z * fma(c, (0.0 - b), (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(x * Float64(Float64(y * z) - Float64(t * a)))
	tmp = 0.0
	if (x <= -8e+118)
		tmp = t_1;
	elseif (x <= -9.5e-15)
		tmp = Float64(Float64(0.0 - j) * fma(b, Float64(Float64(z * c) / j), Float64(y * i)));
	elseif (x <= 0.0305)
		tmp = Float64(Float64(a * Float64(b * i)) + Float64(c * fma(b, Float64(0.0 - z), Float64(t * j))));
	elseif (x <= 2.25e+214)
		tmp = t_1;
	else
		tmp = Float64(z * fma(c, Float64(0.0 - b), Float64(x * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e+118], t$95$1, If[LessEqual[x, -9.5e-15], N[(N[(0.0 - j), $MachinePrecision] * N[(b * N[(N[(z * c), $MachinePrecision] / j), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0305], N[(N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision] + N[(c * N[(b * N[(0.0 - z), $MachinePrecision] + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.25e+214], t$95$1, N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.99999999999999973e118 or 0.030499999999999999 < x < 2.24999999999999984e214

   1. Initial program 78.2%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 75.2

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

   5. Simplified 75.2%

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

   if -7.99999999999999973e118 < x < -9.5000000000000005e-15

   1. Initial program 63.4%

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

   3. Taylor expanded in a around inf

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

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

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 71.0

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

   8. Simplified 71.0%

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

   9. Taylor expanded in t around 0

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

       1. distribute-lft-out N/A

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

       2. mul-1-neg N/A

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

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

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

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

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

       5. *-commutative N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 70.0

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

   11. Simplified 70.0%

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

   12. Taylor expanded in j around inf

       \[\leadsto \mathsf{neg}\left(\color{blue}{j \cdot \left(i \cdot y + \frac{b \cdot \left(c \cdot z\right)}{j}\right)}\right) \]
   13. Step-by-step derivation

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

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

       2. +-commutative N/A

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

       3. associate-/l* N/A

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

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

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

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

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

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

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

       7. *-lowering-*.f64 82.4

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

   14. Simplified 82.4%

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

   if -9.5000000000000005e-15 < x < 0.030499999999999999

   1. Initial program 73.9%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 78.8%

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

   6. Taylor expanded in b around inf

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

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

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

      2. *-lowering-*.f64 61.7

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

   8. Simplified 61.7%

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

   if 2.24999999999999984e214 < x

   1. Initial program 49.7%

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

   3. Taylor expanded in z around inf

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

      1. *-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 83.3

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

   5. Simplified 83.3%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 83.3

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

   7. Applied egg-rr 83.3%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-15}:\\ \;\;\;\;\left(0 - j\right) \cdot \mathsf{fma}\left(b, \frac{z \cdot c}{j}, y \cdot i\right)\\ \mathbf{elif}\;x \leq 0.0305:\\ \;\;\;\;a \cdot \left(b \cdot i\right) + c \cdot \mathsf{fma}\left(b, 0 - z, t \cdot j\right)\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+214}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ t_2 := x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+210}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.022:\\ \;\;\;\;a \cdot \left(b \cdot i\right) + c \cdot \mathsf{fma}\left(b, 0 - z, t \cdot j\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+219}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (fma c (- 0.0 b) (* x y)))) (t_2 (* x (- (* y z) (* t a)))))
   (if (<= x -1.6e+210)
     t_2
     (if (<= x -1.5e+65)
       t_1
       (if (<= x 0.022)
         (+ (* a (* b i)) (* c (fma b (- 0.0 z) (* t j))))
         (if (<= x 8.8e+219) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * fma(c, (0.0 - b), (x * y));
	double t_2 = x * ((y * z) - (t * a));
	double tmp;
	if (x <= -1.6e+210) {
		tmp = t_2;
	} else if (x <= -1.5e+65) {
		tmp = t_1;
	} else if (x <= 0.022) {
		tmp = (a * (b * i)) + (c * fma(b, (0.0 - z), (t * j)));
	} else if (x <= 8.8e+219) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * fma(c, Float64(0.0 - b), Float64(x * y)))
	t_2 = Float64(x * Float64(Float64(y * z) - Float64(t * a)))
	tmp = 0.0
	if (x <= -1.6e+210)
		tmp = t_2;
	elseif (x <= -1.5e+65)
		tmp = t_1;
	elseif (x <= 0.022)
		tmp = Float64(Float64(a * Float64(b * i)) + Float64(c * fma(b, Float64(0.0 - z), Float64(t * j))));
	elseif (x <= 8.8e+219)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6e+210], t$95$2, If[LessEqual[x, -1.5e+65], t$95$1, If[LessEqual[x, 0.022], N[(N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision] + N[(c * N[(b * N[(0.0 - z), $MachinePrecision] + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e+219], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6000000000000001e210 or 0.021999999999999999 < x < 8.8000000000000006e219

   1. Initial program 77.9%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 76.7

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

   5. Simplified 76.7%

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

   if -1.6000000000000001e210 < x < -1.5000000000000001e65 or 8.8000000000000006e219 < x

   1. Initial program 68.2%

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

   3. Taylor expanded in z around inf

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. neg-sub0 N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 85.6

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

   5. Simplified 85.6%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 85.6

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

   7. Applied egg-rr 85.6%

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

   if -1.5000000000000001e65 < x < 0.021999999999999999

   1. Initial program 73.3%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 77.7%

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

   6. Taylor expanded in b around inf

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

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

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

      2. *-lowering-*.f64 60.5

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

   8. Simplified 60.5%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+210}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{+65}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \mathbf{elif}\;x \leq 0.022:\\ \;\;\;\;a \cdot \left(b \cdot i\right) + c \cdot \mathsf{fma}\left(b, 0 - z, t \cdot j\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+219}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+156}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(b, a \cdot i - z \cdot c, y \cdot \mathsf{fma}\left(j, 0 - i, x \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -1.2e+156)
   (* x (- (* y z) (* t a)))
   (if (<= x 1.55e+209)
     (fma b (- (* a i) (* z c)) (* y (fma j (- 0.0 i) (* x z))))
     (* z (fma c (- 0.0 b) (* x y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -1.2e+156) {
		tmp = x * ((y * z) - (t * a));
	} else if (x <= 1.55e+209) {
		tmp = fma(b, ((a * i) - (z * c)), (y * fma(j, (0.0 - i), (x * z))));
	} else {
		tmp = z * fma(c, (0.0 - b), (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -1.2e+156)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	elseif (x <= 1.55e+209)
		tmp = fma(b, Float64(Float64(a * i) - Float64(z * c)), Float64(y * fma(j, Float64(0.0 - i), Float64(x * z))));
	else
		tmp = Float64(z * fma(c, Float64(0.0 - b), Float64(x * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -1.2e+156], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e+209], N[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision] + N[(y * N[(j * N[(0.0 - i), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2000000000000001e156

   1. Initial program 63.9%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 91.8

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

   5. Simplified 91.8%

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

   if -1.2000000000000001e156 < x < 1.55e209

   1. Initial program 76.5%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

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

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

      15. mul-1-neg N/A

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

      16. neg-sub0 N/A

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

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

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

      21. associate-*r* N/A

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

      22. *-commutative N/A

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

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

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

      24. *-commutative N/A

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

      25. *-lowering-*.f64 75.7

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

   5. Simplified 75.7%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

      16. distribute-rgt-in N/A

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

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

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

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

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

   8. Simplified 66.6%

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

   if 1.55e209 < x

   1. Initial program 53.6%

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

   3. Taylor expanded in 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 76.9

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

   5. Simplified 76.9%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 76.9

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

   7. Applied egg-rr 76.9%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+156}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(b, a \cdot i - z \cdot c, y \cdot \mathsf{fma}\left(j, 0 - i, x \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 30.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-65}:\\ \;\;\;\;t \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-117}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-218}:\\ \;\;\;\;c \cdot \left(0 - z \cdot b\right)\\ \mathbf{elif}\;x \leq 0.015:\\ \;\;\;\;i \cdot \left(y \cdot \left(0 - j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (* x y))))
   (if (<= x -3e+58)
     t_1
     (if (<= x -1.45e-65)
       (* t (* c j))
       (if (<= x -1.02e-117)
         (* b (* a i))
         (if (<= x -2.1e-218)
           (* c (- 0.0 (* z b)))
           (if (<= x 0.015) (* i (* y (- 0.0 j))) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (x <= -3e+58) {
		tmp = t_1;
	} else if (x <= -1.45e-65) {
		tmp = t * (c * j);
	} else if (x <= -1.02e-117) {
		tmp = b * (a * i);
	} else if (x <= -2.1e-218) {
		tmp = c * (0.0 - (z * b));
	} else if (x <= 0.015) {
		tmp = i * (y * (0.0 - j));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (x * y)
    if (x <= (-3d+58)) then
        tmp = t_1
    else if (x <= (-1.45d-65)) then
        tmp = t * (c * j)
    else if (x <= (-1.02d-117)) then
        tmp = b * (a * i)
    else if (x <= (-2.1d-218)) then
        tmp = c * (0.0d0 - (z * b))
    else if (x <= 0.015d0) then
        tmp = i * (y * (0.0d0 - j))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (x <= -3e+58) {
		tmp = t_1;
	} else if (x <= -1.45e-65) {
		tmp = t * (c * j);
	} else if (x <= -1.02e-117) {
		tmp = b * (a * i);
	} else if (x <= -2.1e-218) {
		tmp = c * (0.0 - (z * b));
	} else if (x <= 0.015) {
		tmp = i * (y * (0.0 - j));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = z * (x * y)
	tmp = 0
	if x <= -3e+58:
		tmp = t_1
	elif x <= -1.45e-65:
		tmp = t * (c * j)
	elif x <= -1.02e-117:
		tmp = b * (a * i)
	elif x <= -2.1e-218:
		tmp = c * (0.0 - (z * b))
	elif x <= 0.015:
		tmp = i * (y * (0.0 - j))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * Float64(x * y))
	tmp = 0.0
	if (x <= -3e+58)
		tmp = t_1;
	elseif (x <= -1.45e-65)
		tmp = Float64(t * Float64(c * j));
	elseif (x <= -1.02e-117)
		tmp = Float64(b * Float64(a * i));
	elseif (x <= -2.1e-218)
		tmp = Float64(c * Float64(0.0 - Float64(z * b)));
	elseif (x <= 0.015)
		tmp = Float64(i * Float64(y * Float64(0.0 - j)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = z * (x * y);
	tmp = 0.0;
	if (x <= -3e+58)
		tmp = t_1;
	elseif (x <= -1.45e-65)
		tmp = t * (c * j);
	elseif (x <= -1.02e-117)
		tmp = b * (a * i);
	elseif (x <= -2.1e-218)
		tmp = c * (0.0 - (z * b));
	elseif (x <= 0.015)
		tmp = i * (y * (0.0 - j));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3e+58], t$95$1, If[LessEqual[x, -1.45e-65], N[(t * N[(c * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.02e-117], N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.1e-218], N[(c * N[(0.0 - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.015], N[(i * N[(y * N[(0.0 - j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -3.0000000000000002e58 or 0.014999999999999999 < x

   1. Initial program 74.2%

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

   3. Taylor expanded in z around inf

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. neg-sub0 N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in c around 0

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

      1. *-lowering-*.f64 55.2

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

   8. Simplified 55.2%

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

   if -3.0000000000000002e58 < x < -1.4499999999999999e-65

   1. Initial program 64.4%

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

   3. Taylor expanded in a around inf

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

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

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 64.4

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

   5. Simplified 64.4%

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

   6. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 55.3

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

   8. Simplified 55.3%

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

   9. Taylor expanded in t around inf

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 33.7

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

   11. Simplified 33.7%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

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

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

       4. *-lowering-*.f64 43.2

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

   13. Applied egg-rr 43.2%

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

   if -1.4499999999999999e-65 < x < -1.01999999999999993e-117

   1. Initial program 99.9%

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

   3. Taylor expanded in i around inf

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 82.6

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

   5. Simplified 82.6%

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

   6. Taylor expanded in j around 0

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

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

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

      2. *-lowering-*.f64 56.3

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

   8. Simplified 56.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 73.0

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

   10. Applied egg-rr 73.0%

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

   if -1.01999999999999993e-117 < x < -2.09999999999999994e-218

   1. Initial program 69.9%

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

   3. Taylor expanded in a around inf

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

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

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 57.2

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

   5. Simplified 57.2%

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

   6. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 69.8

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

   8. Simplified 69.8%

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

   9. Taylor expanded in t around 0

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

       1. distribute-lft-out N/A

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

       2. mul-1-neg N/A

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

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

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

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

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

       5. *-commutative N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 65.5

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

   11. Simplified 65.5%

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

   12. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 61.7

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

   14. Simplified 61.7%

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

   if -2.09999999999999994e-218 < x < 0.014999999999999999

   1. Initial program 72.6%

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

   3. Taylor expanded in i around inf

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 51.3

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

   5. Simplified 51.3%

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

   6. Taylor expanded in j around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-lowering-*.f64 31.8

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

   8. Simplified 31.8%

      \[\leadsto i \cdot \color{blue}{\left(0 - j \cdot y\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+58}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-65}:\\ \;\;\;\;t \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-117}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-218}:\\ \;\;\;\;c \cdot \left(0 - z \cdot b\right)\\ \mathbf{elif}\;x \leq 0.015:\\ \;\;\;\;i \cdot \left(y \cdot \left(0 - j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-65}:\\ \;\;\;\;t \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-117}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-200}:\\ \;\;\;\;c \cdot \left(0 - z \cdot b\right)\\ \mathbf{elif}\;x \leq 0.0074:\\ \;\;\;\;j \cdot \left(y \cdot \left(0 - i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (* x y))))
   (if (<= x -5.5e+60)
     t_1
     (if (<= x -2.3e-65)
       (* t (* c j))
       (if (<= x -1.1e-117)
         (* b (* a i))
         (if (<= x -1.15e-200)
           (* c (- 0.0 (* z b)))
           (if (<= x 0.0074) (* 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 = z * (x * y);
	double tmp;
	if (x <= -5.5e+60) {
		tmp = t_1;
	} else if (x <= -2.3e-65) {
		tmp = t * (c * j);
	} else if (x <= -1.1e-117) {
		tmp = b * (a * i);
	} else if (x <= -1.15e-200) {
		tmp = c * (0.0 - (z * b));
	} else if (x <= 0.0074) {
		tmp = j * (y * (0.0 - i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (x * y)
    if (x <= (-5.5d+60)) then
        tmp = t_1
    else if (x <= (-2.3d-65)) then
        tmp = t * (c * j)
    else if (x <= (-1.1d-117)) then
        tmp = b * (a * i)
    else if (x <= (-1.15d-200)) then
        tmp = c * (0.0d0 - (z * b))
    else if (x <= 0.0074d0) then
        tmp = j * (y * (0.0d0 - i))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (x <= -5.5e+60) {
		tmp = t_1;
	} else if (x <= -2.3e-65) {
		tmp = t * (c * j);
	} else if (x <= -1.1e-117) {
		tmp = b * (a * i);
	} else if (x <= -1.15e-200) {
		tmp = c * (0.0 - (z * b));
	} else if (x <= 0.0074) {
		tmp = j * (y * (0.0 - i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = z * (x * y)
	tmp = 0
	if x <= -5.5e+60:
		tmp = t_1
	elif x <= -2.3e-65:
		tmp = t * (c * j)
	elif x <= -1.1e-117:
		tmp = b * (a * i)
	elif x <= -1.15e-200:
		tmp = c * (0.0 - (z * b))
	elif x <= 0.0074:
		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(z * Float64(x * y))
	tmp = 0.0
	if (x <= -5.5e+60)
		tmp = t_1;
	elseif (x <= -2.3e-65)
		tmp = Float64(t * Float64(c * j));
	elseif (x <= -1.1e-117)
		tmp = Float64(b * Float64(a * i));
	elseif (x <= -1.15e-200)
		tmp = Float64(c * Float64(0.0 - Float64(z * b)));
	elseif (x <= 0.0074)
		tmp = Float64(j * Float64(y * Float64(0.0 - i)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = z * (x * y);
	tmp = 0.0;
	if (x <= -5.5e+60)
		tmp = t_1;
	elseif (x <= -2.3e-65)
		tmp = t * (c * j);
	elseif (x <= -1.1e-117)
		tmp = b * (a * i);
	elseif (x <= -1.15e-200)
		tmp = c * (0.0 - (z * b));
	elseif (x <= 0.0074)
		tmp = j * (y * (0.0 - i));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e+60], t$95$1, If[LessEqual[x, -2.3e-65], N[(t * N[(c * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.1e-117], N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.15e-200], N[(c * N[(0.0 - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0074], N[(j * N[(y * N[(0.0 - i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -5.5000000000000001e60 or 0.0074000000000000003 < x

   1. Initial program 74.4%

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

   3. Taylor expanded in z around inf

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

      1. *-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 65.4

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

   5. Simplified 65.4%

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

   6. Taylor expanded in c around 0

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

      1. *-lowering-*.f64 54.7

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

   8. Simplified 54.7%

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

   if -5.5000000000000001e60 < x < -2.3e-65

   1. Initial program 64.4%

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

   3. Taylor expanded in a around inf

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

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

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 64.4

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

   5. Simplified 64.4%

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

   6. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 55.3

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

   8. Simplified 55.3%

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

   9. Taylor expanded in t around inf

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 33.7

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

   11. Simplified 33.7%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

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

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

       4. *-lowering-*.f64 43.2

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

   13. Applied egg-rr 43.2%

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

   if -2.3e-65 < x < -1.1000000000000001e-117

   1. Initial program 99.9%

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

   3. Taylor expanded in i around inf

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 82.6

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

   5. Simplified 82.6%

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

   6. Taylor expanded in j around 0

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

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

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

      2. *-lowering-*.f64 56.3

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

   8. Simplified 56.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 73.0

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

   10. Applied egg-rr 73.0%

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

   if -1.1000000000000001e-117 < x < -1.15000000000000004e-200

   1. Initial program 72.7%

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

   3. Taylor expanded in a around inf

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

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

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 67.1

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

   5. Simplified 67.1%

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

   6. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 72.5

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

   8. Simplified 72.5%

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

   9. Taylor expanded in t around 0

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

       1. distribute-lft-out N/A

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

       2. mul-1-neg N/A

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

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

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

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

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

       5. *-commutative N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 67.0

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

   11. Simplified 67.0%

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

   12. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 67.6

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

   14. Simplified 67.6%

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

   if -1.15000000000000004e-200 < x < 0.0074000000000000003

   1. Initial program 71.7%

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

   3. Taylor expanded in a around inf

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

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

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 65.2

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

   5. Simplified 65.2%

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

   6. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 58.3

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

   8. Simplified 58.3%

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

   9. Taylor expanded in t around 0

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

       1. distribute-lft-out N/A

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

       2. mul-1-neg N/A

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

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

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

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

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

       5. *-commutative N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 49.1

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

   11. Simplified 49.1%

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

   12. Taylor expanded in b around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. *-lowering-*.f64 30.7

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

   14. Simplified 30.7%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+60}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-65}:\\ \;\;\;\;t \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-117}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-200}:\\ \;\;\;\;c \cdot \left(0 - z \cdot b\right)\\ \mathbf{elif}\;x \leq 0.0074:\\ \;\;\;\;j \cdot \left(y \cdot \left(0 - i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{if}\;x \leq -9.4 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+125}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+218}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (- (* y z) (* t a)))))
   (if (<= x -9.4e+123)
     t_1
     (if (<= x 4.5e+125)
       (- (* j (- (* t c) (* y i))) (* c (* z b)))
       (if (<= x 3.2e+218) t_1 (* z (fma c (- 0.0 b) (* x y))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * ((y * z) - (t * a));
	double tmp;
	if (x <= -9.4e+123) {
		tmp = t_1;
	} else if (x <= 4.5e+125) {
		tmp = (j * ((t * c) - (y * i))) - (c * (z * b));
	} else if (x <= 3.2e+218) {
		tmp = t_1;
	} else {
		tmp = z * fma(c, (0.0 - b), (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(x * Float64(Float64(y * z) - Float64(t * a)))
	tmp = 0.0
	if (x <= -9.4e+123)
		tmp = t_1;
	elseif (x <= 4.5e+125)
		tmp = Float64(Float64(j * Float64(Float64(t * c) - Float64(y * i))) - Float64(c * Float64(z * b)));
	elseif (x <= 3.2e+218)
		tmp = t_1;
	else
		tmp = Float64(z * fma(c, Float64(0.0 - b), Float64(x * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.4e+123], t$95$1, If[LessEqual[x, 4.5e+125], N[(N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e+218], t$95$1, N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.39999999999999958e123 or 4.5e125 < x < 3.19999999999999987e218

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 84.8

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

   5. Simplified 84.8%

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

   if -9.39999999999999958e123 < x < 4.5e125

   1. Initial program 74.9%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. neg-sub0 N/A

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

      12. --lowering--.f64 61.4

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

   5. Simplified 61.4%

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

   if 3.19999999999999987e218 < x

   1. Initial program 49.7%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-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 83.3

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, y \cdot x\right) \]

      2. neg-lowering-neg.f64 83.3

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{-b}, y \cdot x\right) \]

   7. Applied egg-rr 83.3%

      \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{-b}, y \cdot x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.4 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+125}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+218}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-65}:\\ \;\;\;\;t \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-117}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-298}:\\ \;\;\;\;c \cdot \left(0 - z \cdot b\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-85}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (* x y))))
   (if (<= x -4e+59)
     t_1
     (if (<= x -1.25e-65)
       (* t (* c j))
       (if (<= x -1.05e-117)
         (* b (* a i))
         (if (<= x 9.2e-298)
           (* c (- 0.0 (* z b)))
           (if (<= x 2.3e-85) (* c (* t j)) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (x <= -4e+59) {
		tmp = t_1;
	} else if (x <= -1.25e-65) {
		tmp = t * (c * j);
	} else if (x <= -1.05e-117) {
		tmp = b * (a * i);
	} else if (x <= 9.2e-298) {
		tmp = c * (0.0 - (z * b));
	} else if (x <= 2.3e-85) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (x * y)
    if (x <= (-4d+59)) then
        tmp = t_1
    else if (x <= (-1.25d-65)) then
        tmp = t * (c * j)
    else if (x <= (-1.05d-117)) then
        tmp = b * (a * i)
    else if (x <= 9.2d-298) then
        tmp = c * (0.0d0 - (z * b))
    else if (x <= 2.3d-85) then
        tmp = c * (t * j)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (x <= -4e+59) {
		tmp = t_1;
	} else if (x <= -1.25e-65) {
		tmp = t * (c * j);
	} else if (x <= -1.05e-117) {
		tmp = b * (a * i);
	} else if (x <= 9.2e-298) {
		tmp = c * (0.0 - (z * b));
	} else if (x <= 2.3e-85) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = z * (x * y)
	tmp = 0
	if x <= -4e+59:
		tmp = t_1
	elif x <= -1.25e-65:
		tmp = t * (c * j)
	elif x <= -1.05e-117:
		tmp = b * (a * i)
	elif x <= 9.2e-298:
		tmp = c * (0.0 - (z * b))
	elif x <= 2.3e-85:
		tmp = c * (t * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * Float64(x * y))
	tmp = 0.0
	if (x <= -4e+59)
		tmp = t_1;
	elseif (x <= -1.25e-65)
		tmp = Float64(t * Float64(c * j));
	elseif (x <= -1.05e-117)
		tmp = Float64(b * Float64(a * i));
	elseif (x <= 9.2e-298)
		tmp = Float64(c * Float64(0.0 - Float64(z * b)));
	elseif (x <= 2.3e-85)
		tmp = Float64(c * Float64(t * j));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = z * (x * y);
	tmp = 0.0;
	if (x <= -4e+59)
		tmp = t_1;
	elseif (x <= -1.25e-65)
		tmp = t * (c * j);
	elseif (x <= -1.05e-117)
		tmp = b * (a * i);
	elseif (x <= 9.2e-298)
		tmp = c * (0.0 - (z * b));
	elseif (x <= 2.3e-85)
		tmp = c * (t * j);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e+59], t$95$1, If[LessEqual[x, -1.25e-65], N[(t * N[(c * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.05e-117], N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.2e-298], N[(c * N[(0.0 - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-85], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -3.99999999999999989e59 or 2.3e-85 < x

   1. Initial program 74.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 60.4

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 60.4%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 48.0

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   8. Simplified 48.0%

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   if -3.99999999999999989e59 < x < -1.24999999999999996e-65

   1. Initial program 64.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. sub-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\left(\frac{c \cdot z}{a} + \left(\mathsf{neg}\left(i\right)\right)\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(\color{blue}{c \cdot \frac{z}{a}} + \left(\mathsf{neg}\left(i\right)\right)\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(c \cdot \frac{z}{a} + \color{blue}{-1 \cdot i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(c, \frac{z}{a}, -1 \cdot i\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \color{blue}{\frac{z}{a}}, -1 \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{\mathsf{neg}\left(i\right)}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 64.4

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 64.4%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, 0 - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right)} \cdot \left(-1 \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 55.3

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   8. Simplified 55.3%

      \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

       2. *-commutative N/A

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

       3. *-lowering-*.f64 33.7

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   11. Simplified 33.7%

       \[\leadsto \color{blue}{c \cdot \left(t \cdot j\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(c \cdot j\right) \cdot t} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(c \cdot j\right) \cdot t} \]

       4. *-lowering-*.f64 43.2

          \[\leadsto \color{blue}{\left(c \cdot j\right)} \cdot t \]

   13. Applied egg-rr 43.2%

       \[\leadsto \color{blue}{\left(c \cdot j\right) \cdot t} \]

   if -1.24999999999999996e-65 < x < -1.05e-117

   1. Initial program 99.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 82.6

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 82.6%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]

      2. *-lowering-*.f64 56.3

         \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]

   8. Simplified 56.3%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(a \cdot i\right) \cdot b} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot a\right)} \cdot b \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(i \cdot a\right) \cdot b} \]

      5. *-lowering-*.f64 73.0

         \[\leadsto \color{blue}{\left(i \cdot a\right)} \cdot b \]

   10. Applied egg-rr 73.0%

       \[\leadsto \color{blue}{\left(i \cdot a\right) \cdot b} \]

   if -1.05e-117 < x < 9.2000000000000003e-298

   1. Initial program 65.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. sub-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\left(\frac{c \cdot z}{a} + \left(\mathsf{neg}\left(i\right)\right)\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(\color{blue}{c \cdot \frac{z}{a}} + \left(\mathsf{neg}\left(i\right)\right)\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(c \cdot \frac{z}{a} + \color{blue}{-1 \cdot i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(c, \frac{z}{a}, -1 \cdot i\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \color{blue}{\frac{z}{a}}, -1 \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{\mathsf{neg}\left(i\right)}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 57.1

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 57.1%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, 0 - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right)} \cdot \left(-1 \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 63.3

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   8. Simplified 63.3%

      \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right) + i \cdot \left(j \cdot y\right)\right)} \]

       2. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\left(b \cdot \left(c \cdot z\right) + i \cdot \left(j \cdot y\right)\right)\right)} \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\left(b \cdot \left(c \cdot z\right) + i \cdot \left(j \cdot y\right)\right)\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(b, c \cdot z, i \cdot \left(j \cdot y\right)\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(b, \color{blue}{z \cdot c}, i \cdot \left(j \cdot y\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(b, \color{blue}{z \cdot c}, i \cdot \left(j \cdot y\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(b, z \cdot c, \color{blue}{i \cdot \left(j \cdot y\right)}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(b, z \cdot c, i \cdot \color{blue}{\left(y \cdot j\right)}\right)\right) \]

       9. *-lowering-*.f64 53.4

          \[\leadsto -\mathsf{fma}\left(b, z \cdot c, i \cdot \color{blue}{\left(y \cdot j\right)}\right) \]

   11. Simplified 53.4%

       \[\leadsto \color{blue}{-\mathsf{fma}\left(b, z \cdot c, i \cdot \left(y \cdot j\right)\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto \mathsf{neg}\left(\color{blue}{b \cdot \left(c \cdot z\right)}\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(c \cdot z\right) \cdot b}\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{c \cdot \left(z \cdot b\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(c \cdot \color{blue}{\left(b \cdot z\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{c \cdot \left(b \cdot z\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(c \cdot \color{blue}{\left(z \cdot b\right)}\right) \]

       6. *-lowering-*.f64 43.9

          \[\leadsto -c \cdot \color{blue}{\left(z \cdot b\right)} \]

   14. Simplified 43.9%

       \[\leadsto -\color{blue}{c \cdot \left(z \cdot b\right)} \]

   if 9.2000000000000003e-298 < x < 2.3e-85

   1. Initial program 76.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. sub-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\left(\frac{c \cdot z}{a} + \left(\mathsf{neg}\left(i\right)\right)\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(\color{blue}{c \cdot \frac{z}{a}} + \left(\mathsf{neg}\left(i\right)\right)\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(c \cdot \frac{z}{a} + \color{blue}{-1 \cdot i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(c, \frac{z}{a}, -1 \cdot i\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \color{blue}{\frac{z}{a}}, -1 \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{\mathsf{neg}\left(i\right)}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 71.0

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 71.0%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, 0 - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right)} \cdot \left(-1 \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 65.6

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   8. Simplified 65.6%

      \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

       2. *-commutative N/A

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

       3. *-lowering-*.f64 35.1

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   11. Simplified 35.1%

       \[\leadsto \color{blue}{c \cdot \left(t \cdot j\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+59}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-65}:\\ \;\;\;\;t \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-117}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-298}:\\ \;\;\;\;c \cdot \left(0 - z \cdot b\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-85}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{if}\;i \leq -1.8 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -2.05 \cdot 10^{-12}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(c, t, y \cdot \left(0 - i\right)\right)\\ \mathbf{elif}\;i \leq 1.2 \cdot 10^{+69}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- 0.0 y) (* a b)))))
   (if (<= i -1.8e+139)
     t_1
     (if (<= i -2.05e-12)
       (* j (fma c t (* y (- 0.0 i))))
       (if (<= i 1.2e+69) (* z (fma c (- 0.0 b) (* x y))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * fma(j, (0.0 - y), (a * b));
	double tmp;
	if (i <= -1.8e+139) {
		tmp = t_1;
	} else if (i <= -2.05e-12) {
		tmp = j * fma(c, t, (y * (0.0 - i)));
	} else if (i <= 1.2e+69) {
		tmp = z * fma(c, (0.0 - b), (x * y));
	} 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(a * b)))
	tmp = 0.0
	if (i <= -1.8e+139)
		tmp = t_1;
	elseif (i <= -2.05e-12)
		tmp = Float64(j * fma(c, t, Float64(y * Float64(0.0 - i))));
	elseif (i <= 1.2e+69)
		tmp = Float64(z * fma(c, Float64(0.0 - b), Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.8e+139], t$95$1, If[LessEqual[i, -2.05e-12], N[(j * N[(c * t + N[(y * N[(0.0 - i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.2e+69], N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.79999999999999993e139 or 1.2000000000000001e69 < i

   1. Initial program 73.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 72.5

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 72.5%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot a\right) \]

      2. neg-lowering-neg.f64 72.5

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot a\right) \]

   7. Applied egg-rr 72.5%

      \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot a\right) \]

   if -1.79999999999999993e139 < i < -2.04999999999999995e-12

   1. Initial program 69.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 52.8

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 52.8%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]

   if -2.04999999999999995e-12 < i < 1.2000000000000001e69

   1. Initial program 75.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 58.0

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 58.0%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, y \cdot x\right) \]

      2. neg-lowering-neg.f64 58.0

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{-b}, y \cdot x\right) \]

   7. Applied egg-rr 58.0%

      \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{-b}, y \cdot x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.8 \cdot 10^{+139}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -2.05 \cdot 10^{-12}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(c, t, y \cdot \left(0 - i\right)\right)\\ \mathbf{elif}\;i \leq 1.2 \cdot 10^{+69}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{if}\;i \leq -5.5 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -3.9 \cdot 10^{+83}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, 0 - z, t \cdot j\right)\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{+68}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- 0.0 y) (* a b)))))
   (if (<= i -5.5e+134)
     t_1
     (if (<= i -3.9e+83)
       (* c (fma b (- 0.0 z) (* t j)))
       (if (<= i 2.5e+68) (* z (fma c (- 0.0 b) (* x y))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * fma(j, (0.0 - y), (a * b));
	double tmp;
	if (i <= -5.5e+134) {
		tmp = t_1;
	} else if (i <= -3.9e+83) {
		tmp = c * fma(b, (0.0 - z), (t * j));
	} else if (i <= 2.5e+68) {
		tmp = z * fma(c, (0.0 - b), (x * y));
	} 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(a * b)))
	tmp = 0.0
	if (i <= -5.5e+134)
		tmp = t_1;
	elseif (i <= -3.9e+83)
		tmp = Float64(c * fma(b, Float64(0.0 - z), Float64(t * j)));
	elseif (i <= 2.5e+68)
		tmp = Float64(z * fma(c, Float64(0.0 - b), Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -5.5e+134], t$95$1, If[LessEqual[i, -3.9e+83], N[(c * N[(b * N[(0.0 - z), $MachinePrecision] + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.5e+68], N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -5.4999999999999999e134 or 2.5000000000000002e68 < i

   1. Initial program 71.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 72.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 72.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot a\right) \]

      2. neg-lowering-neg.f64 72.3

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot a\right) \]

   7. Applied egg-rr 72.3%

      \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot a\right) \]

   if -5.4999999999999999e134 < i < -3.9000000000000002e83

   1. Initial program 60.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} + j \cdot t\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + j \cdot t\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)} + j \cdot t\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, j \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{0 - z}, j \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{0 - z}, j \cdot t\right) \]

      12. *-lowering-*.f64 90.1

          \[\leadsto c \cdot \mathsf{fma}\left(b, 0 - z, \color{blue}{j \cdot t}\right) \]

   5. Simplified 90.1%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(b, 0 - z, j \cdot t\right)} \]

   if -3.9000000000000002e83 < i < 2.5000000000000002e68

   1. Initial program 76.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 54.1

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 54.1%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, y \cdot x\right) \]

      2. neg-lowering-neg.f64 54.1

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{-b}, y \cdot x\right) \]

   7. Applied egg-rr 54.1%

      \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{-b}, y \cdot x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5.5 \cdot 10^{+134}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -3.9 \cdot 10^{+83}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, 0 - z, t \cdot j\right)\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{+68}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(i, a, 0 - z \cdot c\right)\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-131}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+50}:\\ \;\;\;\;t \cdot \left(c \cdot j - x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma i a (- 0.0 (* z c))))))
   (if (<= b -1.7e-41)
     t_1
     (if (<= b 2.3e-131)
       (* x (- (* y z) (* t a)))
       (if (<= b 9e+50) (* t (- (* c j) (* x a))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(i, a, (0.0 - (z * c)));
	double tmp;
	if (b <= -1.7e-41) {
		tmp = t_1;
	} else if (b <= 2.3e-131) {
		tmp = x * ((y * z) - (t * a));
	} else if (b <= 9e+50) {
		tmp = t * ((c * j) - (x * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(i, a, Float64(0.0 - Float64(z * c))))
	tmp = 0.0
	if (b <= -1.7e-41)
		tmp = t_1;
	elseif (b <= 2.3e-131)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	elseif (b <= 9e+50)
		tmp = Float64(t * Float64(Float64(c * j) - Float64(x * a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(i * a + N[(0.0 - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.7e-41], t$95$1, If[LessEqual[b, 2.3e-131], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9e+50], N[(t * N[(N[(c * j), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.6999999999999999e-41 or 9.00000000000000027e50 < b

   1. Initial program 73.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \]

      2. remove-double-neg N/A

         \[\leadsto b \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)} + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(a \cdot i\right)}\right)\right) + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \left(a \cdot i\right) + c \cdot z\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + -1 \cdot \left(a \cdot i\right)\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(c \cdot z + \color{blue}{\left(\mathsf{neg}\left(a \cdot i\right)\right)}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \]

      11. sub-neg N/A

          \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(c \cdot z + \color{blue}{-1 \cdot \left(a \cdot i\right)}\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot i\right)\right)\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot i\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot z\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot i\right)\right)\right)\right) \]

      16. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot i\right)\right)}\right)\right)\right) \]

      17. remove-double-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \color{blue}{a \cdot i}\right) \]

      18. accelerator-lowering-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot z, a \cdot i\right)} \]

      19. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      20. neg-sub0 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{0 - z}, a \cdot i\right) \]

      21. --lowering--.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{0 - z}, a \cdot i\right) \]

      22. *-commutative N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, 0 - z, \color{blue}{i \cdot a}\right) \]

      23. *-lowering-*.f64 64.0

          \[\leadsto b \cdot \mathsf{fma}\left(c, 0 - z, \color{blue}{i \cdot a}\right) \]

   5. Simplified 64.0%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(c, 0 - z, i \cdot a\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot a + c \cdot \left(0 - z\right)\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, a, c \cdot \left(0 - z\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\left(0 - z\right) \cdot c}\right) \]

      4. sub0-neg N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot c\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\mathsf{neg}\left(z \cdot c\right)}\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\mathsf{neg}\left(z \cdot c\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \mathsf{neg}\left(\color{blue}{c \cdot z}\right)\right) \]

      8. *-lowering-*.f64 64.0

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, -\color{blue}{c \cdot z}\right) \]

   7. Applied egg-rr 64.0%

      \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, a, -c \cdot z\right)} \]

   if -1.6999999999999999e-41 < b < 2.30000000000000022e-131

   1. Initial program 74.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 58.2

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 58.2%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   if 2.30000000000000022e-131 < b < 9.00000000000000027e50

   1. Initial program 72.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)}^{3} + {\left(j \cdot \left(c \cdot t - i \cdot y\right)\right)}^{3}}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot \left(c \cdot t - i \cdot y\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right) - \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right)\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot \left(c \cdot t - i \cdot y\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right) - \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right)\right)}{{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)}^{3} + {\left(j \cdot \left(c \cdot t - i \cdot y\right)\right)}^{3}}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot \left(c \cdot t - i \cdot y\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right) - \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right)\right)}{{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)}^{3} + {\left(j \cdot \left(c \cdot t - i \cdot y\right)\right)}^{3}}}} \]

   4. Applied egg-rr 75.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(j, t \cdot c - y \cdot i, x \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(z \cdot c - a \cdot i\right)}}} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(a \cdot x\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto t \cdot \left(c \cdot j + \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(c \cdot j - a \cdot x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto t \cdot \color{blue}{\left(c \cdot j - a \cdot x\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto t \cdot \left(\color{blue}{c \cdot j} - a \cdot x\right) \]

      7. *-commutative N/A

         \[\leadsto t \cdot \left(c \cdot j - \color{blue}{x \cdot a}\right) \]

      8. *-lowering-*.f64 48.8

         \[\leadsto t \cdot \left(c \cdot j - \color{blue}{x \cdot a}\right) \]

   7. Simplified 48.8%

      \[\leadsto \color{blue}{t \cdot \left(c \cdot j - x \cdot a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-41}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, a, 0 - z \cdot c\right)\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-131}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+50}:\\ \;\;\;\;t \cdot \left(c \cdot j - x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, a, 0 - z \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(c \cdot j\right)\\ t_2 := z \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+61}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-217}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* t (* c j))) (t_2 (* z (* x y))))
   (if (<= x -1.05e+61)
     t_2
     (if (<= x -3.2e-65)
       t_1
       (if (<= x 3.2e-217) (* i (* a b)) (if (<= x 2.4e-85) 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 = t * (c * j);
	double t_2 = z * (x * y);
	double tmp;
	if (x <= -1.05e+61) {
		tmp = t_2;
	} else if (x <= -3.2e-65) {
		tmp = t_1;
	} else if (x <= 3.2e-217) {
		tmp = i * (a * b);
	} else if (x <= 2.4e-85) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t * (c * j)
    t_2 = z * (x * y)
    if (x <= (-1.05d+61)) then
        tmp = t_2
    else if (x <= (-3.2d-65)) then
        tmp = t_1
    else if (x <= 3.2d-217) then
        tmp = i * (a * b)
    else if (x <= 2.4d-85) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = t * (c * j);
	double t_2 = z * (x * y);
	double tmp;
	if (x <= -1.05e+61) {
		tmp = t_2;
	} else if (x <= -3.2e-65) {
		tmp = t_1;
	} else if (x <= 3.2e-217) {
		tmp = i * (a * b);
	} else if (x <= 2.4e-85) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = t * (c * j)
	t_2 = z * (x * y)
	tmp = 0
	if x <= -1.05e+61:
		tmp = t_2
	elif x <= -3.2e-65:
		tmp = t_1
	elif x <= 3.2e-217:
		tmp = i * (a * b)
	elif x <= 2.4e-85:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(t * Float64(c * j))
	t_2 = Float64(z * Float64(x * y))
	tmp = 0.0
	if (x <= -1.05e+61)
		tmp = t_2;
	elseif (x <= -3.2e-65)
		tmp = t_1;
	elseif (x <= 3.2e-217)
		tmp = Float64(i * Float64(a * b));
	elseif (x <= 2.4e-85)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = t * (c * j);
	t_2 = z * (x * y);
	tmp = 0.0;
	if (x <= -1.05e+61)
		tmp = t_2;
	elseif (x <= -3.2e-65)
		tmp = t_1;
	elseif (x <= 3.2e-217)
		tmp = i * (a * b);
	elseif (x <= 2.4e-85)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(t * N[(c * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05e+61], t$95$2, If[LessEqual[x, -3.2e-65], t$95$1, If[LessEqual[x, 3.2e-217], N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-85], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0500000000000001e61 or 2.4000000000000001e-85 < x

   1. Initial program 74.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 60.4

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 60.4%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 48.0

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   8. Simplified 48.0%

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   if -1.0500000000000001e61 < x < -3.1999999999999999e-65 or 3.2000000000000001e-217 < x < 2.4000000000000001e-85

   1. Initial program 68.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. sub-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\left(\frac{c \cdot z}{a} + \left(\mathsf{neg}\left(i\right)\right)\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(\color{blue}{c \cdot \frac{z}{a}} + \left(\mathsf{neg}\left(i\right)\right)\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(c \cdot \frac{z}{a} + \color{blue}{-1 \cdot i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(c, \frac{z}{a}, -1 \cdot i\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \color{blue}{\frac{z}{a}}, -1 \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{\mathsf{neg}\left(i\right)}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 66.1

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 66.1%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, 0 - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right)} \cdot \left(-1 \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 62.7

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   8. Simplified 62.7%

      \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

       2. *-commutative N/A

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

       3. *-lowering-*.f64 38.5

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   11. Simplified 38.5%

       \[\leadsto \color{blue}{c \cdot \left(t \cdot j\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(c \cdot j\right) \cdot t} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(c \cdot j\right) \cdot t} \]

       4. *-lowering-*.f64 42.2

          \[\leadsto \color{blue}{\left(c \cdot j\right)} \cdot t \]

   13. Applied egg-rr 42.2%

       \[\leadsto \color{blue}{\left(c \cdot j\right) \cdot t} \]

   if -3.1999999999999999e-65 < x < 3.2000000000000001e-217

   1. Initial program 75.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 52.2

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 52.2%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(a \cdot b\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

      2. *-lowering-*.f64 31.8

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   8. Simplified 31.8%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+61}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-65}:\\ \;\;\;\;t \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-217}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-85}:\\ \;\;\;\;t \cdot \left(c \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+121}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-214}:\\ \;\;\;\;z \cdot \left(b \cdot \left(0 - c\right)\right)\\ \mathbf{elif}\;x \leq 0.0235:\\ \;\;\;\;i \cdot \left(y \cdot \left(0 - j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -3.1e+121)
   (* x (* y z))
   (if (<= x -6.4e-214)
     (* z (* b (- 0.0 c)))
     (if (<= x 0.0235) (* i (* y (- 0.0 j))) (* z (* x y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -3.1e+121) {
		tmp = x * (y * z);
	} else if (x <= -6.4e-214) {
		tmp = z * (b * (0.0 - c));
	} else if (x <= 0.0235) {
		tmp = i * (y * (0.0 - j));
	} else {
		tmp = z * (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (x <= (-3.1d+121)) then
        tmp = x * (y * z)
    else if (x <= (-6.4d-214)) then
        tmp = z * (b * (0.0d0 - c))
    else if (x <= 0.0235d0) then
        tmp = i * (y * (0.0d0 - j))
    else
        tmp = z * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -3.1e+121) {
		tmp = x * (y * z);
	} else if (x <= -6.4e-214) {
		tmp = z * (b * (0.0 - c));
	} else if (x <= 0.0235) {
		tmp = i * (y * (0.0 - j));
	} else {
		tmp = z * (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if x <= -3.1e+121:
		tmp = x * (y * z)
	elif x <= -6.4e-214:
		tmp = z * (b * (0.0 - c))
	elif x <= 0.0235:
		tmp = i * (y * (0.0 - j))
	else:
		tmp = z * (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -3.1e+121)
		tmp = Float64(x * Float64(y * z));
	elseif (x <= -6.4e-214)
		tmp = Float64(z * Float64(b * Float64(0.0 - c)));
	elseif (x <= 0.0235)
		tmp = Float64(i * Float64(y * Float64(0.0 - j)));
	else
		tmp = Float64(z * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (x <= -3.1e+121)
		tmp = x * (y * z);
	elseif (x <= -6.4e-214)
		tmp = z * (b * (0.0 - c));
	elseif (x <= 0.0235)
		tmp = i * (y * (0.0 - j));
	else
		tmp = z * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -3.1e+121], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.4e-214], N[(z * N[(b * N[(0.0 - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0235], N[(i * N[(y * N[(0.0 - j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.10000000000000008e121

   1. Initial program 68.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 86.7

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 86.7%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

      2. *-lowering-*.f64 70.6

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

   8. Simplified 70.6%

      \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

   if -3.10000000000000008e121 < x < -6.40000000000000027e-214

   1. Initial program 73.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 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 48.3

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 48.3%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)} \]

      2. neg-sub0 N/A

         \[\leadsto z \cdot \color{blue}{\left(0 - b \cdot c\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto z \cdot \color{blue}{\left(0 - b \cdot c\right)} \]

      4. *-commutative N/A

         \[\leadsto z \cdot \left(0 - \color{blue}{c \cdot b}\right) \]

      5. *-lowering-*.f64 42.9

         \[\leadsto z \cdot \left(0 - \color{blue}{c \cdot b}\right) \]

   8. Simplified 42.9%

      \[\leadsto z \cdot \color{blue}{\left(0 - c \cdot b\right)} \]

   if -6.40000000000000027e-214 < x < 0.0235

   1. Initial program 72.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 51.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 51.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} \]

      2. neg-sub0 N/A

         \[\leadsto i \cdot \color{blue}{\left(0 - j \cdot y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto i \cdot \color{blue}{\left(0 - j \cdot y\right)} \]

      4. *-lowering-*.f64 31.8

         \[\leadsto i \cdot \left(0 - \color{blue}{j \cdot y}\right) \]

   8. Simplified 31.8%

      \[\leadsto i \cdot \color{blue}{\left(0 - j \cdot y\right)} \]

   if 0.0235 < x

   1. Initial program 79.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 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 56.6

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 56.6%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 43.0

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   8. Simplified 43.0%

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+121}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-214}:\\ \;\;\;\;z \cdot \left(b \cdot \left(0 - c\right)\right)\\ \mathbf{elif}\;x \leq 0.0235:\\ \;\;\;\;i \cdot \left(y \cdot \left(0 - j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{if}\;i \leq -1.9 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 4.2 \cdot 10^{+67}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- 0.0 y) (* a b)))))
   (if (<= i -1.9e-12)
     t_1
     (if (<= i 4.2e+67) (* z (fma c (- 0.0 b) (* x y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * fma(j, (0.0 - y), (a * b));
	double tmp;
	if (i <= -1.9e-12) {
		tmp = t_1;
	} else if (i <= 4.2e+67) {
		tmp = z * fma(c, (0.0 - b), (x * y));
	} 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(a * b)))
	tmp = 0.0
	if (i <= -1.9e-12)
		tmp = t_1;
	elseif (i <= 4.2e+67)
		tmp = Float64(z * fma(c, Float64(0.0 - b), Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.9e-12], t$95$1, If[LessEqual[i, 4.2e+67], N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.89999999999999998e-12 or 4.2000000000000003e67 < i

   1. Initial program 72.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 63.6

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 63.6%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot a\right) \]

      2. neg-lowering-neg.f64 63.6

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot a\right) \]

   7. Applied egg-rr 63.6%

      \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot a\right) \]

   if -1.89999999999999998e-12 < i < 4.2000000000000003e67

   1. Initial program 75.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 58.0

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 58.0%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, y \cdot x\right) \]

      2. neg-lowering-neg.f64 58.0

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{-b}, y \cdot x\right) \]

   7. Applied egg-rr 58.0%

      \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{-b}, y \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.9 \cdot 10^{-12}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{elif}\;i \leq 4.2 \cdot 10^{+67}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, 0 - b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(i, a, 0 - z \cdot c\right)\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{-52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 0.044:\\ \;\;\;\;y \cdot \left(x \cdot z - i \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma i a (- 0.0 (* z c))))))
   (if (<= b -4.5e-52) t_1 (if (<= b 0.044) (* y (- (* x z) (* i j))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(i, a, (0.0 - (z * c)));
	double tmp;
	if (b <= -4.5e-52) {
		tmp = t_1;
	} else if (b <= 0.044) {
		tmp = y * ((x * z) - (i * j));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(i, a, Float64(0.0 - Float64(z * c))))
	tmp = 0.0
	if (b <= -4.5e-52)
		tmp = t_1;
	elseif (b <= 0.044)
		tmp = Float64(y * Float64(Float64(x * z) - Float64(i * j)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(i * a + N[(0.0 - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.5e-52], t$95$1, If[LessEqual[b, 0.044], N[(y * N[(N[(x * z), $MachinePrecision] - N[(i * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5e-52 or 0.043999999999999997 < b

   1. Initial program 74.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \]

      2. remove-double-neg N/A

         \[\leadsto b \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)} + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(a \cdot i\right)}\right)\right) + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \left(a \cdot i\right) + c \cdot z\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + -1 \cdot \left(a \cdot i\right)\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(c \cdot z + \color{blue}{\left(\mathsf{neg}\left(a \cdot i\right)\right)}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \]

      11. sub-neg N/A

          \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(c \cdot z + \color{blue}{-1 \cdot \left(a \cdot i\right)}\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot i\right)\right)\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot i\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot z\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot i\right)\right)\right)\right) \]

      16. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot i\right)\right)}\right)\right)\right) \]

      17. remove-double-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \color{blue}{a \cdot i}\right) \]

      18. accelerator-lowering-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot z, a \cdot i\right)} \]

      19. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      20. neg-sub0 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{0 - z}, a \cdot i\right) \]

      21. --lowering--.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{0 - z}, a \cdot i\right) \]

      22. *-commutative N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, 0 - z, \color{blue}{i \cdot a}\right) \]

      23. *-lowering-*.f64 61.8

          \[\leadsto b \cdot \mathsf{fma}\left(c, 0 - z, \color{blue}{i \cdot a}\right) \]

   5. Simplified 61.8%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(c, 0 - z, i \cdot a\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot a + c \cdot \left(0 - z\right)\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, a, c \cdot \left(0 - z\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\left(0 - z\right) \cdot c}\right) \]

      4. sub0-neg N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot c\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\mathsf{neg}\left(z \cdot c\right)}\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\mathsf{neg}\left(z \cdot c\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \mathsf{neg}\left(\color{blue}{c \cdot z}\right)\right) \]

      8. *-lowering-*.f64 61.8

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, -\color{blue}{c \cdot z}\right) \]

   7. Applied egg-rr 61.8%

      \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, a, -c \cdot z\right)} \]

   if -4.5e-52 < b < 0.043999999999999997

   1. Initial program 72.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(j \cdot t\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), x \cdot \left(y \cdot z - t \cdot a\right)\right) + c \cdot \mathsf{fma}\left(b, 0 - z, j \cdot t\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   7. 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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z + -1 \cdot \left(i \cdot j\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(x \cdot z + \color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z - i \cdot j\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z - i \cdot j\right)} \]

      6. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{z \cdot x} - i \cdot j\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \left(\color{blue}{z \cdot x} - i \cdot j\right) \]

      8. *-commutative N/A

         \[\leadsto y \cdot \left(z \cdot x - \color{blue}{j \cdot i}\right) \]

      9. *-lowering-*.f64 54.3

         \[\leadsto y \cdot \left(z \cdot x - \color{blue}{j \cdot i}\right) \]

   8. Simplified 54.3%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot x - j \cdot i\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-52}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, a, 0 - z \cdot c\right)\\ \mathbf{elif}\;b \leq 0.044:\\ \;\;\;\;y \cdot \left(x \cdot z - i \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, a, 0 - z \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(c \cdot j - x \cdot a\right)\\ \mathbf{if}\;t \leq -5 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+36}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* t (- (* c j) (* x a)))))
   (if (<= t -5e+83)
     t_1
     (if (<= t 4.8e+36) (* i (fma j (- 0.0 y) (* a b))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = t * ((c * j) - (x * a));
	double tmp;
	if (t <= -5e+83) {
		tmp = t_1;
	} else if (t <= 4.8e+36) {
		tmp = i * fma(j, (0.0 - y), (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(t * Float64(Float64(c * j) - Float64(x * a)))
	tmp = 0.0
	if (t <= -5e+83)
		tmp = t_1;
	elseif (t <= 4.8e+36)
		tmp = Float64(i * fma(j, Float64(0.0 - y), Float64(a * b)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(t * N[(N[(c * j), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5e+83], t$95$1, If[LessEqual[t, 4.8e+36], N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.00000000000000029e83 or 4.79999999999999985e36 < t

   1. Initial program 66.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)}^{3} + {\left(j \cdot \left(c \cdot t - i \cdot y\right)\right)}^{3}}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot \left(c \cdot t - i \cdot y\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right) - \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right)\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot \left(c \cdot t - i \cdot y\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right) - \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right)\right)}{{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)}^{3} + {\left(j \cdot \left(c \cdot t - i \cdot y\right)\right)}^{3}}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot \left(c \cdot t - i \cdot y\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right) - \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \left(j \cdot \left(c \cdot t - i \cdot y\right)\right)\right)}{{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)}^{3} + {\left(j \cdot \left(c \cdot t - i \cdot y\right)\right)}^{3}}}} \]

   4. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(j, t \cdot c - y \cdot i, x \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(z \cdot c - a \cdot i\right)}}} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(a \cdot x\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto t \cdot \left(c \cdot j + \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(c \cdot j - a \cdot x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto t \cdot \color{blue}{\left(c \cdot j - a \cdot x\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto t \cdot \left(\color{blue}{c \cdot j} - a \cdot x\right) \]

      7. *-commutative N/A

         \[\leadsto t \cdot \left(c \cdot j - \color{blue}{x \cdot a}\right) \]

      8. *-lowering-*.f64 60.9

         \[\leadsto t \cdot \left(c \cdot j - \color{blue}{x \cdot a}\right) \]

   7. Simplified 60.9%

      \[\leadsto \color{blue}{t \cdot \left(c \cdot j - x \cdot a\right)} \]

   if -5.00000000000000029e83 < t < 4.79999999999999985e36

   1. Initial program 77.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 49.2

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 49.2%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot a\right) \]

      2. neg-lowering-neg.f64 49.2

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot a\right) \]

   7. Applied egg-rr 49.2%

      \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+83}:\\ \;\;\;\;t \cdot \left(c \cdot j - x \cdot a\right)\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+36}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(c \cdot j - x \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{if}\;j \leq -1.65 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 1.05 \cdot 10^{+91}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, a, 0 - z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- 0.0 y) (* a b)))))
   (if (<= j -1.65e+61)
     t_1
     (if (<= j 1.05e+91) (* b (fma i a (- 0.0 (* 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 = i * fma(j, (0.0 - y), (a * b));
	double tmp;
	if (j <= -1.65e+61) {
		tmp = t_1;
	} else if (j <= 1.05e+91) {
		tmp = b * fma(i, a, (0.0 - (z * 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(a * b)))
	tmp = 0.0
	if (j <= -1.65e+61)
		tmp = t_1;
	elseif (j <= 1.05e+91)
		tmp = Float64(b * fma(i, a, Float64(0.0 - 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[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -1.65e+61], t$95$1, If[LessEqual[j, 1.05e+91], N[(b * N[(i * a + N[(0.0 - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -1.6499999999999999e61 or 1.05000000000000004e91 < j

   1. Initial program 73.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 57.4

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 57.4%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot a\right) \]

      2. neg-lowering-neg.f64 57.4

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot a\right) \]

   7. Applied egg-rr 57.4%

      \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{-y}, b \cdot a\right) \]

   if -1.6499999999999999e61 < j < 1.05000000000000004e91

   1. Initial program 73.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \]

      2. remove-double-neg N/A

         \[\leadsto b \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)} + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(a \cdot i\right)}\right)\right) + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \left(a \cdot i\right) + c \cdot z\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + -1 \cdot \left(a \cdot i\right)\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(c \cdot z + \color{blue}{\left(\mathsf{neg}\left(a \cdot i\right)\right)}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \]

      11. sub-neg N/A

          \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(c \cdot z + \color{blue}{-1 \cdot \left(a \cdot i\right)}\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot i\right)\right)\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot i\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot z\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot i\right)\right)\right)\right) \]

      16. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot i\right)\right)}\right)\right)\right) \]

      17. remove-double-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \color{blue}{a \cdot i}\right) \]

      18. accelerator-lowering-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot z, a \cdot i\right)} \]

      19. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      20. neg-sub0 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{0 - z}, a \cdot i\right) \]

      21. --lowering--.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{0 - z}, a \cdot i\right) \]

      22. *-commutative N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, 0 - z, \color{blue}{i \cdot a}\right) \]

      23. *-lowering-*.f64 50.1

          \[\leadsto b \cdot \mathsf{fma}\left(c, 0 - z, \color{blue}{i \cdot a}\right) \]

   5. Simplified 50.1%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(c, 0 - z, i \cdot a\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot a + c \cdot \left(0 - z\right)\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, a, c \cdot \left(0 - z\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\left(0 - z\right) \cdot c}\right) \]

      4. sub0-neg N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot c\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\mathsf{neg}\left(z \cdot c\right)}\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\mathsf{neg}\left(z \cdot c\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \mathsf{neg}\left(\color{blue}{c \cdot z}\right)\right) \]

      8. *-lowering-*.f64 50.1

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, -\color{blue}{c \cdot z}\right) \]

   7. Applied egg-rr 50.1%

      \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, a, -c \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.65 \cdot 10^{+61}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \mathbf{elif}\;j \leq 1.05 \cdot 10^{+91}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, a, 0 - z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, 0 - y, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 41.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+146}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+110}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, a, 0 - z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -1.95e+146)
   (* x (* y z))
   (if (<= x 2.3e+110) (* b (fma i a (- 0.0 (* z c)))) (* z (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -1.95e+146) {
		tmp = x * (y * z);
	} else if (x <= 2.3e+110) {
		tmp = b * fma(i, a, (0.0 - (z * c)));
	} else {
		tmp = z * (x * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -1.95e+146)
		tmp = Float64(x * Float64(y * z));
	elseif (x <= 2.3e+110)
		tmp = Float64(b * fma(i, a, Float64(0.0 - Float64(z * c))));
	else
		tmp = Float64(z * Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -1.95e+146], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e+110], N[(b * N[(i * a + N[(0.0 - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95e146

   1. Initial program 67.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 90.3

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 90.3%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

      2. *-lowering-*.f64 72.5

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

   8. Simplified 72.5%

      \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

   if -1.95e146 < x < 2.3e110

   1. Initial program 74.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \]

      2. remove-double-neg N/A

         \[\leadsto b \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)} + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(a \cdot i\right)}\right)\right) + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \left(a \cdot i\right) + c \cdot z\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + -1 \cdot \left(a \cdot i\right)\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(c \cdot z + \color{blue}{\left(\mathsf{neg}\left(a \cdot i\right)\right)}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \]

      11. sub-neg N/A

          \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto b \cdot \left(\mathsf{neg}\left(\left(c \cdot z + \color{blue}{-1 \cdot \left(a \cdot i\right)}\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot i\right)\right)\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot i\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot z\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot i\right)\right)\right)\right) \]

      16. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot i\right)\right)}\right)\right)\right) \]

      17. remove-double-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \color{blue}{a \cdot i}\right) \]

      18. accelerator-lowering-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot z, a \cdot i\right)} \]

      19. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      20. neg-sub0 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{0 - z}, a \cdot i\right) \]

      21. --lowering--.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{0 - z}, a \cdot i\right) \]

      22. *-commutative N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, 0 - z, \color{blue}{i \cdot a}\right) \]

      23. *-lowering-*.f64 47.4

          \[\leadsto b \cdot \mathsf{fma}\left(c, 0 - z, \color{blue}{i \cdot a}\right) \]

   5. Simplified 47.4%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(c, 0 - z, i \cdot a\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot a + c \cdot \left(0 - z\right)\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, a, c \cdot \left(0 - z\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\left(0 - z\right) \cdot c}\right) \]

      4. sub0-neg N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot c\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\mathsf{neg}\left(z \cdot c\right)}\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \color{blue}{\mathsf{neg}\left(z \cdot c\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, \mathsf{neg}\left(\color{blue}{c \cdot z}\right)\right) \]

      8. *-lowering-*.f64 47.9

         \[\leadsto b \cdot \mathsf{fma}\left(i, a, -\color{blue}{c \cdot z}\right) \]

   7. Applied egg-rr 47.9%

      \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, a, -c \cdot z\right)} \]

   if 2.3e110 < x

   1. Initial program 75.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 59.5

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 59.5%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 45.7

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   8. Simplified 45.7%

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+146}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+110}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, a, 0 - z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 30.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-298}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{-86}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (* x y))))
   (if (<= x -3.6e+58)
     t_1
     (if (<= x 2.65e-298)
       (* i (* a b))
       (if (<= x 6.3e-86) (* c (* t j)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (x <= -3.6e+58) {
		tmp = t_1;
	} else if (x <= 2.65e-298) {
		tmp = i * (a * b);
	} else if (x <= 6.3e-86) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (x * y)
    if (x <= (-3.6d+58)) then
        tmp = t_1
    else if (x <= 2.65d-298) then
        tmp = i * (a * b)
    else if (x <= 6.3d-86) then
        tmp = c * (t * j)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (x <= -3.6e+58) {
		tmp = t_1;
	} else if (x <= 2.65e-298) {
		tmp = i * (a * b);
	} else if (x <= 6.3e-86) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = z * (x * y)
	tmp = 0
	if x <= -3.6e+58:
		tmp = t_1
	elif x <= 2.65e-298:
		tmp = i * (a * b)
	elif x <= 6.3e-86:
		tmp = c * (t * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * Float64(x * y))
	tmp = 0.0
	if (x <= -3.6e+58)
		tmp = t_1;
	elseif (x <= 2.65e-298)
		tmp = Float64(i * Float64(a * b));
	elseif (x <= 6.3e-86)
		tmp = Float64(c * Float64(t * j));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = z * (x * y);
	tmp = 0.0;
	if (x <= -3.6e+58)
		tmp = t_1;
	elseif (x <= 2.65e-298)
		tmp = i * (a * b);
	elseif (x <= 6.3e-86)
		tmp = c * (t * j);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.6e+58], t$95$1, If[LessEqual[x, 2.65e-298], N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.3e-86], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.59999999999999996e58 or 6.2999999999999999e-86 < x

   1. Initial program 74.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 60.4

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 60.4%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 48.0

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   8. Simplified 48.0%

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   if -3.59999999999999996e58 < x < 2.65000000000000001e-298

   1. Initial program 69.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 49.5

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 49.5%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(a \cdot b\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

      2. *-lowering-*.f64 30.5

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   8. Simplified 30.5%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   if 2.65000000000000001e-298 < x < 6.2999999999999999e-86

   1. Initial program 76.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. sub-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\left(\frac{c \cdot z}{a} + \left(\mathsf{neg}\left(i\right)\right)\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(\color{blue}{c \cdot \frac{z}{a}} + \left(\mathsf{neg}\left(i\right)\right)\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(c \cdot \frac{z}{a} + \color{blue}{-1 \cdot i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(c, \frac{z}{a}, -1 \cdot i\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \color{blue}{\frac{z}{a}}, -1 \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{\mathsf{neg}\left(i\right)}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 71.0

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 71.0%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, 0 - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right)} \cdot \left(-1 \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 65.6

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   8. Simplified 65.6%

      \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

       2. *-commutative N/A

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

       3. *-lowering-*.f64 35.1

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   11. Simplified 35.1%

       \[\leadsto \color{blue}{c \cdot \left(t \cdot j\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+58}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-298}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{-86}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 29.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-298}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-85}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (* y z))))
   (if (<= x -4e+59)
     t_1
     (if (<= x 7.4e-298)
       (* i (* a b))
       (if (<= x 1.45e-85) (* c (* t j)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (x <= -4e+59) {
		tmp = t_1;
	} else if (x <= 7.4e-298) {
		tmp = i * (a * b);
	} else if (x <= 1.45e-85) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y * z)
    if (x <= (-4d+59)) then
        tmp = t_1
    else if (x <= 7.4d-298) then
        tmp = i * (a * b)
    else if (x <= 1.45d-85) then
        tmp = c * (t * j)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (x <= -4e+59) {
		tmp = t_1;
	} else if (x <= 7.4e-298) {
		tmp = i * (a * b);
	} else if (x <= 1.45e-85) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = x * (y * z)
	tmp = 0
	if x <= -4e+59:
		tmp = t_1
	elif x <= 7.4e-298:
		tmp = i * (a * b)
	elif x <= 1.45e-85:
		tmp = c * (t * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(x * Float64(y * z))
	tmp = 0.0
	if (x <= -4e+59)
		tmp = t_1;
	elseif (x <= 7.4e-298)
		tmp = Float64(i * Float64(a * b));
	elseif (x <= 1.45e-85)
		tmp = Float64(c * Float64(t * j));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = x * (y * z);
	tmp = 0.0;
	if (x <= -4e+59)
		tmp = t_1;
	elseif (x <= 7.4e-298)
		tmp = i * (a * b);
	elseif (x <= 1.45e-85)
		tmp = c * (t * j);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e+59], t$95$1, If[LessEqual[x, 7.4e-298], N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-85], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999989e59 or 1.4500000000000001e-85 < x

   1. Initial program 74.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 63.7

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 63.7%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

      2. *-lowering-*.f64 47.1

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

   8. Simplified 47.1%

      \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

   if -3.99999999999999989e59 < x < 7.3999999999999996e-298

   1. Initial program 69.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 49.5

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 49.5%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(a \cdot b\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

      2. *-lowering-*.f64 30.5

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   8. Simplified 30.5%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   if 7.3999999999999996e-298 < x < 1.4500000000000001e-85

   1. Initial program 76.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. sub-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\left(\frac{c \cdot z}{a} + \left(\mathsf{neg}\left(i\right)\right)\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(\color{blue}{c \cdot \frac{z}{a}} + \left(\mathsf{neg}\left(i\right)\right)\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(c \cdot \frac{z}{a} + \color{blue}{-1 \cdot i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(c, \frac{z}{a}, -1 \cdot i\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \color{blue}{\frac{z}{a}}, -1 \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{\mathsf{neg}\left(i\right)}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 71.0

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 71.0%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, 0 - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right)} \cdot \left(-1 \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 65.6

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   8. Simplified 65.6%

      \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

       2. *-commutative N/A

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

       3. *-lowering-*.f64 35.1

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   11. Simplified 35.1%

       \[\leadsto \color{blue}{c \cdot \left(t \cdot j\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+59}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-298}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-85}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 29.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(a \cdot b\right)\\ \mathbf{if}\;a \leq -4.4 \cdot 10^{-82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-53}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (* a b))))
   (if (<= a -4.4e-82) t_1 (if (<= a 1.7e-53) (* c (* t j)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * (a * b);
	double tmp;
	if (a <= -4.4e-82) {
		tmp = t_1;
	} else if (a <= 1.7e-53) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = i * (a * b)
    if (a <= (-4.4d-82)) then
        tmp = t_1
    else if (a <= 1.7d-53) then
        tmp = c * (t * j)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * (a * b);
	double tmp;
	if (a <= -4.4e-82) {
		tmp = t_1;
	} else if (a <= 1.7e-53) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = i * (a * b)
	tmp = 0
	if a <= -4.4e-82:
		tmp = t_1
	elif a <= 1.7e-53:
		tmp = c * (t * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * Float64(a * b))
	tmp = 0.0
	if (a <= -4.4e-82)
		tmp = t_1;
	elseif (a <= 1.7e-53)
		tmp = Float64(c * Float64(t * j));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = i * (a * b);
	tmp = 0.0;
	if (a <= -4.4e-82)
		tmp = t_1;
	elseif (a <= 1.7e-53)
		tmp = c * (t * j);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.4e-82], t$95$1, If[LessEqual[a, 1.7e-53], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.39999999999999971e-82 or 1.7e-53 < a

   1. Initial program 69.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 51.0

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 51.0%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(a \cdot b\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

      2. *-lowering-*.f64 36.3

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   8. Simplified 36.3%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   if -4.39999999999999971e-82 < a < 1.7e-53

   1. Initial program 78.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. sub-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\left(\frac{c \cdot z}{a} + \left(\mathsf{neg}\left(i\right)\right)\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(\color{blue}{c \cdot \frac{z}{a}} + \left(\mathsf{neg}\left(i\right)\right)\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(c \cdot \frac{z}{a} + \color{blue}{-1 \cdot i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(c, \frac{z}{a}, -1 \cdot i\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \color{blue}{\frac{z}{a}}, -1 \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{\mathsf{neg}\left(i\right)}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 64.6

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 64.6%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, 0 - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right)} \cdot \left(-1 \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 62.4

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   8. Simplified 62.4%

      \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

       2. *-commutative N/A

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

       3. *-lowering-*.f64 26.3

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   11. Simplified 26.3%

       \[\leadsto \color{blue}{c \cdot \left(t \cdot j\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{-82}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-53}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 29.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot i\right)\\ \mathbf{if}\;a \leq -8.4 \cdot 10^{-86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-53}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (* b i))))
   (if (<= a -8.4e-86) t_1 (if (<= a 8.2e-53) (* c (* t j)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (b * i);
	double tmp;
	if (a <= -8.4e-86) {
		tmp = t_1;
	} else if (a <= 8.2e-53) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * (b * i)
    if (a <= (-8.4d-86)) then
        tmp = t_1
    else if (a <= 8.2d-53) then
        tmp = c * (t * j)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (b * i);
	double tmp;
	if (a <= -8.4e-86) {
		tmp = t_1;
	} else if (a <= 8.2e-53) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = a * (b * i)
	tmp = 0
	if a <= -8.4e-86:
		tmp = t_1
	elif a <= 8.2e-53:
		tmp = c * (t * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * Float64(b * i))
	tmp = 0.0
	if (a <= -8.4e-86)
		tmp = t_1;
	elseif (a <= 8.2e-53)
		tmp = Float64(c * Float64(t * j));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = a * (b * i);
	tmp = 0.0;
	if (a <= -8.4e-86)
		tmp = t_1;
	elseif (a <= 8.2e-53)
		tmp = c * (t * j);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -8.4e-86], t$95$1, If[LessEqual[a, 8.2e-53], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.4e-86 or 8.2000000000000001e-53 < a

   1. Initial program 69.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 51.0

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 51.0%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]

      2. *-lowering-*.f64 34.1

         \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]

   8. Simplified 34.1%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]

   if -8.4e-86 < a < 8.2000000000000001e-53

   1. Initial program 78.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot z}{a} - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. sub-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\left(\frac{c \cdot z}{a} + \left(\mathsf{neg}\left(i\right)\right)\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(\color{blue}{c \cdot \frac{z}{a}} + \left(\mathsf{neg}\left(i\right)\right)\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \left(c \cdot \frac{z}{a} + \color{blue}{-1 \cdot i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(c, \frac{z}{a}, -1 \cdot i\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \color{blue}{\frac{z}{a}}, -1 \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{\mathsf{neg}\left(i\right)}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 64.6

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, \color{blue}{0 - i}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 64.6%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(a \cdot \mathsf{fma}\left(c, \frac{z}{a}, 0 - i\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z\right) \cdot b}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(-1 \cdot b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot z\right)} \cdot \left(-1 \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. --lowering--.f64 62.4

         \[\leadsto \left(c \cdot z\right) \cdot \color{blue}{\left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   8. Simplified 62.4%

      \[\leadsto \color{blue}{\left(c \cdot z\right) \cdot \left(0 - b\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

       2. *-commutative N/A

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

       3. *-lowering-*.f64 26.3

          \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

   11. Simplified 26.3%

       \[\leadsto \color{blue}{c \cdot \left(t \cdot j\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 22.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(b \cdot i\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* a (* b i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (b * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = a * (b * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (b * i);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return a * (b * i)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(a * Float64(b * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = a * (b * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in i around inf

   \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

   3. mul-1-neg N/A

      \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

   7. remove-double-neg N/A

      \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

   9. mul-1-neg N/A

      \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

   10. neg-sub0 N/A

       \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

   12. *-commutative N/A

       \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   13. *-lowering-*.f64 42.4

       \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

5. Simplified 42.4%

   \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

6. Taylor expanded in j around 0

   \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]

   2. *-lowering-*.f64 22.6

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]

8. Simplified 22.6%

   \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
9. Add Preprocessing

Developer Target 1: 69.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1015122364899489/125000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -942510763643697/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -238547917063487/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 10535888557455487/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))