Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 73.1% → 84.2%

Time: 17.4s

Alternatives: 26

Speedup: 0.5×

• 58.1% 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: 73.1% 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: 84.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.5%

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

   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 i around 0

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

   5. Simplified 53.1%

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

4. Final simplification 83.9%

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

Alternative 2: 51.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)\\ \mathbf{if}\;c \leq -2 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -1.95 \cdot 10^{-25}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{elif}\;c \leq -3 \cdot 10^{-155}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(y, -j, a \cdot b\right)\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{-47}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, z, i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;c \leq 1.16 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(c, -z, a \cdot i\right), t \cdot \left(c \cdot j\right)\right)\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (fma z (- b) (* t j)))))
   (if (<= c -2e+107)
     t_1
     (if (<= c -1.95e-25)
       (* z (fma c (- b) (* x y)))
       (if (<= c -3e-155)
         (* i (fma y (- j) (* a b)))
         (if (<= c 4.3e-47)
           (* y (fma x z (* i (- j))))
           (if (<= c 1.16e+41)
             (fma b (fma c (- z) (* a i)) (* t (* c j)))
             (if (<= c 7.2e+132) (* x (- (* y z) (* t a))) t_1))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * fma(z, -b, (t * j));
	double tmp;
	if (c <= -2e+107) {
		tmp = t_1;
	} else if (c <= -1.95e-25) {
		tmp = z * fma(c, -b, (x * y));
	} else if (c <= -3e-155) {
		tmp = i * fma(y, -j, (a * b));
	} else if (c <= 4.3e-47) {
		tmp = y * fma(x, z, (i * -j));
	} else if (c <= 1.16e+41) {
		tmp = fma(b, fma(c, -z, (a * i)), (t * (c * j)));
	} else if (c <= 7.2e+132) {
		tmp = x * ((y * z) - (t * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * fma(z, Float64(-b), Float64(t * j)))
	tmp = 0.0
	if (c <= -2e+107)
		tmp = t_1;
	elseif (c <= -1.95e-25)
		tmp = Float64(z * fma(c, Float64(-b), Float64(x * y)));
	elseif (c <= -3e-155)
		tmp = Float64(i * fma(y, Float64(-j), Float64(a * b)));
	elseif (c <= 4.3e-47)
		tmp = Float64(y * fma(x, z, Float64(i * Float64(-j))));
	elseif (c <= 1.16e+41)
		tmp = fma(b, fma(c, Float64(-z), Float64(a * i)), Float64(t * Float64(c * j)));
	elseif (c <= 7.2e+132)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(z * (-b) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2e+107], t$95$1, If[LessEqual[c, -1.95e-25], N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3e-155], N[(i * N[(y * (-j) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.3e-47], N[(y * N[(x * z + N[(i * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.16e+41], N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision] + N[(t * N[(c * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.2e+132], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 6 regimes

2. if c < -1.9999999999999999e107 or 7.20000000000000031e132 < c

   1. Initial program 61.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 58.2

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

   5. Simplified 58.2%

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

   6. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-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. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 74.7

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

   8. Simplified 74.7%

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

   if -1.9999999999999999e107 < c < -1.95e-25

   1. Initial program 89.2%

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 78.4

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

   5. Simplified 78.4%

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

   if -1.95e-25 < c < -2.99999999999999984e-155

   1. Initial program 84.3%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 54.5

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

   5. Simplified 54.5%

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

   if -2.99999999999999984e-155 < c < 4.2999999999999998e-47

   1. Initial program 75.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 74.5

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

   5. Simplified 74.5%

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

   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. lower-*.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. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 57.9

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

   8. Simplified 57.9%

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

   if 4.2999999999999998e-47 < c < 1.16000000000000007e41

   1. Initial program 77.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 73.0

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

   5. Simplified 73.0%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 77.5

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

   8. Simplified 77.5%

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

   if 1.16000000000000007e41 < c < 7.20000000000000031e132

   1. Initial program 80.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.2

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

   5. Simplified 67.2%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2 \cdot 10^{+107}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)\\ \mathbf{elif}\;c \leq -1.95 \cdot 10^{-25}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{elif}\;c \leq -3 \cdot 10^{-155}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(y, -j, a \cdot b\right)\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{-47}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, z, i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;c \leq 1.16 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(c, -z, a \cdot i\right), t \cdot \left(c \cdot j\right)\right)\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma t (fma j c (* x (- a))) (* z (fma c (- b) (* x y))))))
   (if (<= t -1.7e-31)
     t_1
     (if (<= t 1.1e-35)
       (fma y (fma j (- i) (* x z)) (* b (fma c (- 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 = fma(t, fma(j, c, (x * -a)), (z * fma(c, -b, (x * y))));
	double tmp;
	if (t <= -1.7e-31) {
		tmp = t_1;
	} else if (t <= 1.1e-35) {
		tmp = fma(y, fma(j, -i, (x * z)), (b * fma(c, -z, (a * i))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(z * fma(c, Float64(-b), Float64(x * y))))
	tmp = 0.0
	if (t <= -1.7e-31)
		tmp = t_1;
	elseif (t <= 1.1e-35)
		tmp = fma(y, fma(j, Float64(-i), Float64(x * z)), Float64(b * fma(c, Float64(-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[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.7e-31], t$95$1, If[LessEqual[t, 1.1e-35], N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7000000000000001e-31 or 1.09999999999999997e-35 < t

   1. Initial program 67.0%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 77.7%

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

   if -1.7000000000000001e-31 < t < 1.09999999999999997e-35

   1. Initial program 81.3%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

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

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. mul-1-neg N/A

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

      19. sub-neg N/A

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

   5. Simplified 76.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ t_2 := \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{if}\;b \leq -3.9 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(t, t\_2, t\_1\right)\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(t, t\_2, z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma c (- z) (* a i)))) (t_2 (fma j c (* x (- a)))))
   (if (<= b -3.9e+147)
     (fma t t_2 t_1)
     (if (<= b 2.9e+207) (fma t t_2 (* z (fma c (- b) (* x y)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(c, -z, (a * i));
	double t_2 = fma(j, c, (x * -a));
	double tmp;
	if (b <= -3.9e+147) {
		tmp = fma(t, t_2, t_1);
	} else if (b <= 2.9e+207) {
		tmp = fma(t, t_2, (z * fma(c, -b, (x * y))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(c, Float64(-z), Float64(a * i)))
	t_2 = fma(j, c, Float64(x * Float64(-a)))
	tmp = 0.0
	if (b <= -3.9e+147)
		tmp = fma(t, t_2, t_1);
	elseif (b <= 2.9e+207)
		tmp = fma(t, t_2, Float64(z * fma(c, Float64(-b), Float64(x * y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.9e+147], N[(t * t$95$2 + t$95$1), $MachinePrecision], If[LessEqual[b, 2.9e+207], N[(t * t$95$2 + N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.90000000000000016e147

   1. Initial program 65.1%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 N/A

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

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

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

   5. Simplified 74.4%

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

   if -3.90000000000000016e147 < b < 2.89999999999999997e207

   1. Initial program 73.6%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 69.7%

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

   if 2.89999999999999997e207 < b

   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 b around inf

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

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

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

      2. remove-double-neg N/A

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

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

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ t_2 := \mathsf{fma}\left(t, t\_1, y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\right)\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{-29}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(t, t\_1, b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma j c (* x (- a))))
        (t_2 (fma t t_1 (* y (fma j (- i) (* x z))))))
   (if (<= y -5.8e-29)
     t_2
     (if (<= y 3.5e-40) (fma t t_1 (* b (fma c (- z) (* a i)))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(j, c, (x * -a));
	double t_2 = fma(t, t_1, (y * fma(j, -i, (x * z))));
	double tmp;
	if (y <= -5.8e-29) {
		tmp = t_2;
	} else if (y <= 3.5e-40) {
		tmp = fma(t, t_1, (b * fma(c, -z, (a * i))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(j, c, Float64(x * Float64(-a)))
	t_2 = fma(t, t_1, Float64(y * fma(j, Float64(-i), Float64(x * z))))
	tmp = 0.0
	if (y <= -5.8e-29)
		tmp = t_2;
	elseif (y <= 3.5e-40)
		tmp = fma(t, t_1, Float64(b * fma(c, Float64(-z), Float64(a * i))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * t$95$1 + N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.8e-29], t$95$2, If[LessEqual[y, 3.5e-40], N[(t * t$95$1 + N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.80000000000000048e-29 or 3.5000000000000002e-40 < y

   1. Initial program 69.1%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 72.1%

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

   if -5.80000000000000048e-29 < y < 3.5000000000000002e-40

   1. Initial program 78.7%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 N/A

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

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

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

   5. Simplified 73.6%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\\ \mathbf{if}\;y \leq -2.75 \cdot 10^{+159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* y (fma j (- i) (* x z)))))
   (if (<= y -2.75e+159)
     t_1
     (if (<= y 6.5e+175)
       (fma t (fma j c (* x (- a))) (* b (fma c (- 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 = y * fma(j, -i, (x * z));
	double tmp;
	if (y <= -2.75e+159) {
		tmp = t_1;
	} else if (y <= 6.5e+175) {
		tmp = fma(t, fma(j, c, (x * -a)), (b * fma(c, -z, (a * i))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(y * fma(j, Float64(-i), Float64(x * z)))
	tmp = 0.0
	if (y <= -2.75e+159)
		tmp = t_1;
	elseif (y <= 6.5e+175)
		tmp = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(b * fma(c, Float64(-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[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.75e+159], t$95$1, If[LessEqual[y, 6.5e+175], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7499999999999999e159 or 6.49999999999999977e175 < y

   1. Initial program 63.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 83.1

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

   5. Simplified 83.1%

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

   if -2.7499999999999999e159 < y < 6.49999999999999977e175

   1. Initial program 75.7%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 N/A

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

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

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

   5. Simplified 65.7%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.75 \cdot 10^{+159}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+22}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, z, i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), -z \cdot \left(b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -4.5e+22)
   (* y (fma x z (* i (- j))))
   (if (<= y 6.5e+175)
     (fma t (fma j c (* x (- a))) (- (* z (* b c))))
     (* y (fma j (- i) (* x z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -4.5e+22) {
		tmp = y * fma(x, z, (i * -j));
	} else if (y <= 6.5e+175) {
		tmp = fma(t, fma(j, c, (x * -a)), -(z * (b * c)));
	} else {
		tmp = y * fma(j, -i, (x * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -4.5e+22)
		tmp = Float64(y * fma(x, z, Float64(i * Float64(-j))));
	elseif (y <= 6.5e+175)
		tmp = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(-Float64(z * Float64(b * c))));
	else
		tmp = Float64(y * fma(j, Float64(-i), Float64(x * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -4.5e+22], N[(y * N[(x * z + N[(i * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+175], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + (-N[(z * N[(b * c), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4999999999999998e22

   1. Initial program 61.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 58.1

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

   5. Simplified 58.1%

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

   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. lower-*.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. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 69.1

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

   8. Simplified 69.1%

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

   if -4.4999999999999998e22 < y < 6.49999999999999977e175

   1. Initial program 77.1%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 64.8%

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

   6. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 60.1

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

   8. Simplified 60.1%

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

   if 6.49999999999999977e175 < y

   1. Initial program 77.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 77.8

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

   5. Simplified 77.8%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+22}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, z, i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), -z \cdot \left(b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-246}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(y, -j, a \cdot b\right)\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-291}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;z \leq 10^{+42}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(i, -y, t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (fma x y (* b (- c))))))
   (if (<= z -1.9e+21)
     t_1
     (if (<= z -1.02e-246)
       (* i (fma y (- j) (* a b)))
       (if (<= z -4.3e-291)
         (* t (fma j c (* x (- a))))
         (if (<= z 1e+42) (* j (fma i (- y) (* t c))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * fma(x, y, (b * -c));
	double tmp;
	if (z <= -1.9e+21) {
		tmp = t_1;
	} else if (z <= -1.02e-246) {
		tmp = i * fma(y, -j, (a * b));
	} else if (z <= -4.3e-291) {
		tmp = t * fma(j, c, (x * -a));
	} else if (z <= 1e+42) {
		tmp = j * fma(i, -y, (t * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * fma(x, y, Float64(b * Float64(-c))))
	tmp = 0.0
	if (z <= -1.9e+21)
		tmp = t_1;
	elseif (z <= -1.02e-246)
		tmp = Float64(i * fma(y, Float64(-j), Float64(a * b)));
	elseif (z <= -4.3e-291)
		tmp = Float64(t * fma(j, c, Float64(x * Float64(-a))));
	elseif (z <= 1e+42)
		tmp = Float64(j * fma(i, Float64(-y), Float64(t * 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[(z * N[(x * y + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.9e+21], t$95$1, If[LessEqual[z, -1.02e-246], N[(i * N[(y * (-j) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -4.3e-291], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e+42], N[(j * N[(i * (-y) + N[(t * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.9e21 or 1.00000000000000004e42 < z

   1. Initial program 61.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 66.7

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

   5. Simplified 66.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 66.7

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 67.6

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

   7. Applied egg-rr 67.6%

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

   if -1.9e21 < z < -1.02e-246

   1. Initial program 80.7%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 51.6

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

   5. Simplified 51.6%

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

   if -1.02e-246 < z < -4.30000000000000035e-291

   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 t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 85.4

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

   5. Simplified 85.4%

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

   if -4.30000000000000035e-291 < z < 1.00000000000000004e42

   1. Initial program 85.5%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.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. +-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 62.2

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

   5. Simplified 62.2%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+21}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-246}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(y, -j, a \cdot b\right)\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-291}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;z \leq 10^{+42}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(i, -y, t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{-101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-246}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(y, -j, a \cdot b\right)\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-291}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;z \leq 10^{+42}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(i, -y, t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (fma c (- b) (* x y)))))
   (if (<= z -1.05e-101)
     t_1
     (if (<= z -1.02e-246)
       (* i (fma y (- j) (* a b)))
       (if (<= z -4.3e-291)
         (* t (fma j c (* x (- a))))
         (if (<= z 1e+42) (* j (fma i (- y) (* t c))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * fma(c, -b, (x * y));
	double tmp;
	if (z <= -1.05e-101) {
		tmp = t_1;
	} else if (z <= -1.02e-246) {
		tmp = i * fma(y, -j, (a * b));
	} else if (z <= -4.3e-291) {
		tmp = t * fma(j, c, (x * -a));
	} else if (z <= 1e+42) {
		tmp = j * fma(i, -y, (t * c));
	} 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(-b), Float64(x * y)))
	tmp = 0.0
	if (z <= -1.05e-101)
		tmp = t_1;
	elseif (z <= -1.02e-246)
		tmp = Float64(i * fma(y, Float64(-j), Float64(a * b)));
	elseif (z <= -4.3e-291)
		tmp = Float64(t * fma(j, c, Float64(x * Float64(-a))));
	elseif (z <= 1e+42)
		tmp = Float64(j * fma(i, Float64(-y), Float64(t * 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[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.05e-101], t$95$1, If[LessEqual[z, -1.02e-246], N[(i * N[(y * (-j) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -4.3e-291], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e+42], N[(j * N[(i * (-y) + N[(t * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.05000000000000008e-101 or 1.00000000000000004e42 < z

   1. Initial program 67.2%

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 63.0

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

   5. Simplified 63.0%

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

   if -1.05000000000000008e-101 < z < -1.02e-246

   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 i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 56.0

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

   5. Simplified 56.0%

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

   if -1.02e-246 < z < -4.30000000000000035e-291

   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 t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 85.4

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

   5. Simplified 85.4%

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

   if -4.30000000000000035e-291 < z < 1.00000000000000004e42

   1. Initial program 85.5%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.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. +-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 62.2

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

   5. Simplified 62.2%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-101}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-246}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(y, -j, a \cdot b\right)\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-291}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;z \leq 10^{+42}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(i, -y, t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{if}\;a \leq -6.6 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -1.4 \cdot 10^{-269}:\\ \;\;\;\;-c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-218}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{-17}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (fma t (- x) (* b i)))))
   (if (<= a -6.6e+25)
     t_1
     (if (<= a -1.4e-269)
       (- (* c (* z b)))
       (if (<= a 3.7e-218)
         (* y (* x z))
         (if (<= a 1.02e-17) (* 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 * fma(t, -x, (b * i));
	double tmp;
	if (a <= -6.6e+25) {
		tmp = t_1;
	} else if (a <= -1.4e-269) {
		tmp = -(c * (z * b));
	} else if (a <= 3.7e-218) {
		tmp = y * (x * z);
	} else if (a <= 1.02e-17) {
		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 * fma(t, Float64(-x), Float64(b * i)))
	tmp = 0.0
	if (a <= -6.6e+25)
		tmp = t_1;
	elseif (a <= -1.4e-269)
		tmp = Float64(-Float64(c * Float64(z * b)));
	elseif (a <= 3.7e-218)
		tmp = Float64(y * Float64(x * z));
	elseif (a <= 1.02e-17)
		tmp = Float64(c * Float64(t * j));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.6e+25], t$95$1, If[LessEqual[a, -1.4e-269], (-N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision]), If[LessEqual[a, 3.7e-218], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.02e-17], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6.6000000000000002e25 or 1.01999999999999997e-17 < a

   1. Initial program 71.2%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 58.2

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

   5. Simplified 58.2%

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

   if -6.6000000000000002e25 < a < -1.39999999999999997e-269

   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 z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 55.5

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

   5. Simplified 55.5%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 48.8

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

   8. Simplified 48.8%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 41.1

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

   11. Simplified 41.1%

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

   if -1.39999999999999997e-269 < a < 3.7000000000000002e-218

   1. Initial program 80.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 50.2

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

   5. Simplified 50.2%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.6

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

   8. Simplified 49.6%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 43.7

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

   11. Simplified 43.7%

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

   if 3.7000000000000002e-218 < a < 1.01999999999999997e-17

   1. Initial program 76.7%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.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. +-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 61.4

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

   5. Simplified 61.4%

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

   6. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 39.3

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

   8. Simplified 39.3%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 39.4

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

   10. Applied egg-rr 39.4%

       \[\leadsto \color{blue}{\left(t \cdot j\right) \cdot c} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{+25}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{elif}\;a \leq -1.4 \cdot 10^{-269}:\\ \;\;\;\;-c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-218}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{-17}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \mathsf{fma}\left(c, j, x \cdot \left(-a\right)\right), x \cdot \left(y \cdot z\right)\right)\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(c, -z, a \cdot i\right), t \cdot \left(c \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma t (fma c j (* x (- a))) (* x (* y z)))))
   (if (<= x -1.55e-83)
     t_1
     (if (<= x 1.08e-13) (fma b (fma c (- z) (* a i)) (* t (* c j))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(t, fma(c, j, (x * -a)), (x * (y * z)));
	double tmp;
	if (x <= -1.55e-83) {
		tmp = t_1;
	} else if (x <= 1.08e-13) {
		tmp = fma(b, fma(c, -z, (a * i)), (t * (c * j)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(t, fma(c, j, Float64(x * Float64(-a))), Float64(x * Float64(y * z)))
	tmp = 0.0
	if (x <= -1.55e-83)
		tmp = t_1;
	elseif (x <= 1.08e-13)
		tmp = fma(b, fma(c, Float64(-z), Float64(a * i)), Float64(t * Float64(c * 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[(t * N[(c * j + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.55e-83], t$95$1, If[LessEqual[x, 1.08e-13], N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision] + N[(t * N[(c * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.54999999999999996e-83 or 1.0799999999999999e-13 < x

   1. Initial program 71.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 65.5

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

   5. Simplified 65.5%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. associate-+l+ N/A

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

      14. associate-*r* N/A

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

      15. distribute-rgt-in N/A

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

      16. lower-fma.f64 N/A

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

   8. Simplified 65.1%

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

   if -1.54999999999999996e-83 < x < 1.0799999999999999e-13

   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 c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 68.5

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

   5. Simplified 68.5%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 60.3

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

   8. Simplified 60.3%

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

Alternative 12: 30.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.3 \cdot 10^{+63}:\\ \;\;\;\;-y \cdot \left(i \cdot j\right)\\ \mathbf{elif}\;i \leq -9.8 \cdot 10^{-183}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;i \leq 1.15 \cdot 10^{-167}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;i \leq 5 \cdot 10^{+84}:\\ \;\;\;\;-a \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -1.3e+63)
   (- (* y (* i j)))
   (if (<= i -9.8e-183)
     (* j (* t c))
     (if (<= i 1.15e-167)
       (* y (* x z))
       (if (<= i 5e+84) (- (* a (* x t))) (* a (* b i)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -1.3e+63) {
		tmp = -(y * (i * j));
	} else if (i <= -9.8e-183) {
		tmp = j * (t * c);
	} else if (i <= 1.15e-167) {
		tmp = y * (x * z);
	} else if (i <= 5e+84) {
		tmp = -(a * (x * t));
	} else {
		tmp = a * (b * i);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (i <= (-1.3d+63)) then
        tmp = -(y * (i * j))
    else if (i <= (-9.8d-183)) then
        tmp = j * (t * c)
    else if (i <= 1.15d-167) then
        tmp = y * (x * z)
    else if (i <= 5d+84) then
        tmp = -(a * (x * t))
    else
        tmp = a * (b * i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -1.3e+63) {
		tmp = -(y * (i * j));
	} else if (i <= -9.8e-183) {
		tmp = j * (t * c);
	} else if (i <= 1.15e-167) {
		tmp = y * (x * z);
	} else if (i <= 5e+84) {
		tmp = -(a * (x * t));
	} else {
		tmp = a * (b * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if i <= -1.3e+63:
		tmp = -(y * (i * j))
	elif i <= -9.8e-183:
		tmp = j * (t * c)
	elif i <= 1.15e-167:
		tmp = y * (x * z)
	elif i <= 5e+84:
		tmp = -(a * (x * t))
	else:
		tmp = a * (b * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -1.3e+63)
		tmp = Float64(-Float64(y * Float64(i * j)));
	elseif (i <= -9.8e-183)
		tmp = Float64(j * Float64(t * c));
	elseif (i <= 1.15e-167)
		tmp = Float64(y * Float64(x * z));
	elseif (i <= 5e+84)
		tmp = Float64(-Float64(a * Float64(x * t)));
	else
		tmp = Float64(a * Float64(b * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (i <= -1.3e+63)
		tmp = -(y * (i * j));
	elseif (i <= -9.8e-183)
		tmp = j * (t * c);
	elseif (i <= 1.15e-167)
		tmp = y * (x * z);
	elseif (i <= 5e+84)
		tmp = -(a * (x * t));
	else
		tmp = a * (b * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -1.3e+63], (-N[(y * N[(i * j), $MachinePrecision]), $MachinePrecision]), If[LessEqual[i, -9.8e-183], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.15e-167], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5e+84], (-N[(a * N[(x * t), $MachinePrecision]), $MachinePrecision]), N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if i < -1.3000000000000001e63

   1. Initial program 64.8%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.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. +-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 51.0

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

   5. Simplified 51.0%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 40.9

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

   8. Simplified 40.9%

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

   if -1.3000000000000001e63 < i < -9.799999999999999e-183

   1. Initial program 87.8%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.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. +-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 50.6

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

   5. Simplified 50.6%

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

   6. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 43.0

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

   8. Simplified 43.0%

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

   if -9.799999999999999e-183 < i < 1.1500000000000001e-167

   1. Initial program 76.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 57.4

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

   5. Simplified 57.4%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 63.4

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

   8. Simplified 63.4%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 49.8

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

   11. Simplified 49.8%

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

   if 1.1500000000000001e-167 < i < 5.0000000000000001e84

   1. Initial program 81.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 41.5

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

   5. Simplified 41.5%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 39.4

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

   8. Simplified 39.4%

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

   if 5.0000000000000001e84 < i

   1. Initial program 57.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 42.9

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

   5. Simplified 42.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 40.6

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

   8. Simplified 40.6%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.3 \cdot 10^{+63}:\\ \;\;\;\;-y \cdot \left(i \cdot j\right)\\ \mathbf{elif}\;i \leq -9.8 \cdot 10^{-183}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;i \leq 1.15 \cdot 10^{-167}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;i \leq 5 \cdot 10^{+84}:\\ \;\;\;\;-a \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -c \cdot \left(z \cdot b\right)\\ \mathbf{if}\;z \leq -8 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+42}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+93}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (- (* c (* z b)))))
   (if (<= z -8e+188)
     t_1
     (if (<= z -9.5e-8)
       (* z (* x y))
       (if (<= z 2.05e+42)
         (* c (* t j))
         (if (<= z 3e+93) (* t (* 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 = -(c * (z * b));
	double tmp;
	if (z <= -8e+188) {
		tmp = t_1;
	} else if (z <= -9.5e-8) {
		tmp = z * (x * y);
	} else if (z <= 2.05e+42) {
		tmp = c * (t * j);
	} else if (z <= 3e+93) {
		tmp = t * (x * -a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -(c * (z * b))
    if (z <= (-8d+188)) then
        tmp = t_1
    else if (z <= (-9.5d-8)) then
        tmp = z * (x * y)
    else if (z <= 2.05d+42) then
        tmp = c * (t * j)
    else if (z <= 3d+93) then
        tmp = t * (x * -a)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = -(c * (z * b));
	double tmp;
	if (z <= -8e+188) {
		tmp = t_1;
	} else if (z <= -9.5e-8) {
		tmp = z * (x * y);
	} else if (z <= 2.05e+42) {
		tmp = c * (t * j);
	} else if (z <= 3e+93) {
		tmp = t * (x * -a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = -(c * (z * b))
	tmp = 0
	if z <= -8e+188:
		tmp = t_1
	elif z <= -9.5e-8:
		tmp = z * (x * y)
	elif z <= 2.05e+42:
		tmp = c * (t * j)
	elif z <= 3e+93:
		tmp = t * (x * -a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(-Float64(c * Float64(z * b)))
	tmp = 0.0
	if (z <= -8e+188)
		tmp = t_1;
	elseif (z <= -9.5e-8)
		tmp = Float64(z * Float64(x * y));
	elseif (z <= 2.05e+42)
		tmp = Float64(c * Float64(t * j));
	elseif (z <= 3e+93)
		tmp = Float64(t * Float64(x * Float64(-a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = -(c * (z * b));
	tmp = 0.0;
	if (z <= -8e+188)
		tmp = t_1;
	elseif (z <= -9.5e-8)
		tmp = z * (x * y);
	elseif (z <= 2.05e+42)
		tmp = c * (t * j);
	elseif (z <= 3e+93)
		tmp = t * (x * -a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = (-N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[z, -8e+188], t$95$1, If[LessEqual[z, -9.5e-8], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.05e+42], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3e+93], N[(t * N[(x * (-a)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8.0000000000000002e188 or 2.99999999999999978e93 < z

   1. Initial program 64.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 73.8

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

   5. Simplified 73.8%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 66.1

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

   8. Simplified 66.1%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 54.0

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

   11. Simplified 54.0%

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

   if -8.0000000000000002e188 < z < -9.50000000000000036e-8

   1. Initial program 59.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 59.7

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

   5. Simplified 59.7%

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

   6. Taylor expanded in c around 0

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

      1. lower-*.f64 41.4

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

   8. Simplified 41.4%

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

   if -9.50000000000000036e-8 < z < 2.05e42

   1. Initial program 82.8%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.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. +-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 52.6

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

   5. Simplified 52.6%

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

   6. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 33.4

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

   8. Simplified 33.4%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 35.3

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

   10. Applied egg-rr 35.3%

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

   if 2.05e42 < z < 2.99999999999999978e93

   1. Initial program 67.1%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 59.5%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-in N/A

         \[\leadsto t \cdot \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto t \cdot \left(a \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto t \cdot \color{blue}{\left(a \cdot \left(-1 \cdot x\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto t \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. lower-neg.f64 59.8

          \[\leadsto t \cdot \left(a \cdot \color{blue}{\left(-x\right)}\right) \]

   8. Simplified 59.8%

      \[\leadsto \color{blue}{t \cdot \left(a \cdot \left(-x\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+188}:\\ \;\;\;\;-c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+42}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+93}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-c \cdot \left(z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)\\ \mathbf{elif}\;c \leq -3 \cdot 10^{-155}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(y, -j, a \cdot b\right)\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, z, i \cdot \left(-j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -2.2e-25)
   (* c (fma j t (* b (- z))))
   (if (<= c -3e-155)
     (* i (fma y (- j) (* a b)))
     (if (<= c 1.4e+77)
       (* y (fma x z (* i (- j))))
       (* c (fma z (- b) (* t j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -2.2e-25) {
		tmp = c * fma(j, t, (b * -z));
	} else if (c <= -3e-155) {
		tmp = i * fma(y, -j, (a * b));
	} else if (c <= 1.4e+77) {
		tmp = y * fma(x, z, (i * -j));
	} else {
		tmp = c * fma(z, -b, (t * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -2.2e-25)
		tmp = Float64(c * fma(j, t, Float64(b * Float64(-z))));
	elseif (c <= -3e-155)
		tmp = Float64(i * fma(y, Float64(-j), Float64(a * b)));
	elseif (c <= 1.4e+77)
		tmp = Float64(y * fma(x, z, Float64(i * Float64(-j))));
	else
		tmp = Float64(c * fma(z, Float64(-b), Float64(t * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -2.2e-25], N[(c * N[(j * t + N[(b * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3e-155], N[(i * N[(y * (-j) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.4e+77], N[(y * N[(x * z + N[(i * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(z * (-b) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.2000000000000002e-25

   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 c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(-1 \cdot z\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      10. lower-neg.f64 70.4

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right) \]

   5. Simplified 70.4%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)} \]

   if -2.2000000000000002e-25 < c < -2.99999999999999984e-155

   1. Initial program 84.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(-1 \cdot j\right) \cdot y} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto i \cdot \left(\color{blue}{y \cdot \left(-1 \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(y \cdot \left(-1 \cdot j\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      6. remove-double-neg N/A

         \[\leadsto i \cdot \left(y \cdot \left(-1 \cdot j\right) + \color{blue}{a \cdot b}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot j, a \cdot b\right)} \]

      8. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(j\right)}, a \cdot b\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto i \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(j\right)}, a \cdot b\right) \]

      10. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(y, \mathsf{neg}\left(j\right), \color{blue}{b \cdot a}\right) \]

      11. lower-*.f64 54.5

          \[\leadsto i \cdot \mathsf{fma}\left(y, -j, \color{blue}{b \cdot a}\right) \]

   5. Simplified 54.5%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(y, -j, b \cdot a\right)} \]

   if -2.99999999999999984e-155 < c < 1.4e77

   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 t around inf

      \[\leadsto \left(\color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + \frac{x \cdot \left(y \cdot z\right)}{t}\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + \frac{x \cdot \left(y \cdot z\right)}{t}\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. +-commutative N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(\frac{x \cdot \left(y \cdot z\right)}{t} + -1 \cdot \left(a \cdot x\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(t \cdot \left(\color{blue}{x \cdot \frac{y \cdot z}{t}} + -1 \cdot \left(a \cdot x\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. *-commutative N/A

         \[\leadsto \left(t \cdot \left(\color{blue}{\frac{y \cdot z}{t} \cdot x} + -1 \cdot \left(a \cdot x\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(t \cdot \left(\frac{y \cdot z}{t} \cdot x + \color{blue}{\left(-1 \cdot a\right) \cdot x}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(x \cdot \left(\frac{y \cdot z}{t} + -1 \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(x \cdot \left(\frac{y \cdot z}{t} + -1 \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. associate-/l* N/A

         \[\leadsto \left(t \cdot \left(x \cdot \left(\color{blue}{y \cdot \frac{z}{t}} + -1 \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(t \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, -1 \cdot a\right)}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{z}{t}}, -1 \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, \color{blue}{\mathsf{neg}\left(a\right)}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. lower-neg.f64 73.6

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, \color{blue}{-a}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 73.6%

      \[\leadsto \left(\color{blue}{t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, -a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   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. lower-*.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. lower-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, z, -1 \cdot \left(i \cdot j\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto y \cdot \mathsf{fma}\left(x, z, \color{blue}{\mathsf{neg}\left(i \cdot j\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(x, z, \mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto y \cdot \mathsf{fma}\left(x, z, \color{blue}{j \cdot \left(\mathsf{neg}\left(i\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto y \cdot \mathsf{fma}\left(x, z, j \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(x, z, \color{blue}{j \cdot \left(-1 \cdot i\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto y \cdot \mathsf{fma}\left(x, z, j \cdot \color{blue}{\left(\mathsf{neg}\left(i\right)\right)}\right) \]

      10. lower-neg.f64 52.2

          \[\leadsto y \cdot \mathsf{fma}\left(x, z, j \cdot \color{blue}{\left(-i\right)}\right) \]

   8. Simplified 52.2%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, z, j \cdot \left(-i\right)\right)} \]

   if 1.4e77 < c

   1. Initial program 58.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \left(\color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + \frac{x \cdot \left(y \cdot z\right)}{t}\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + \frac{x \cdot \left(y \cdot z\right)}{t}\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. +-commutative N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(\frac{x \cdot \left(y \cdot z\right)}{t} + -1 \cdot \left(a \cdot x\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(t \cdot \left(\color{blue}{x \cdot \frac{y \cdot z}{t}} + -1 \cdot \left(a \cdot x\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. *-commutative N/A

         \[\leadsto \left(t \cdot \left(\color{blue}{\frac{y \cdot z}{t} \cdot x} + -1 \cdot \left(a \cdot x\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(t \cdot \left(\frac{y \cdot z}{t} \cdot x + \color{blue}{\left(-1 \cdot a\right) \cdot x}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(x \cdot \left(\frac{y \cdot z}{t} + -1 \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(x \cdot \left(\frac{y \cdot z}{t} + -1 \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. associate-/l* N/A

         \[\leadsto \left(t \cdot \left(x \cdot \left(\color{blue}{y \cdot \frac{z}{t}} + -1 \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(t \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, -1 \cdot a\right)}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{z}{t}}, -1 \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, \color{blue}{\mathsf{neg}\left(a\right)}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. lower-neg.f64 55.1

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, \color{blue}{-a}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 55.1%

      \[\leadsto \left(\color{blue}{t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, -a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. +-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. *-commutative N/A

         \[\leadsto c \cdot \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right) + j \cdot t\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(b\right)\right)} + j \cdot t\right) \]

      8. mul-1-neg N/A

         \[\leadsto c \cdot \left(z \cdot \color{blue}{\left(-1 \cdot b\right)} + j \cdot t\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(z, -1 \cdot b, j \cdot t\right)} \]

      10. mul-1-neg N/A

          \[\leadsto c \cdot \mathsf{fma}\left(z, \color{blue}{\mathsf{neg}\left(b\right)}, j \cdot t\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(z, \color{blue}{\mathsf{neg}\left(b\right)}, j \cdot t\right) \]

      12. *-commutative N/A

          \[\leadsto c \cdot \mathsf{fma}\left(z, \mathsf{neg}\left(b\right), \color{blue}{t \cdot j}\right) \]

      13. lower-*.f64 69.4

          \[\leadsto c \cdot \mathsf{fma}\left(z, -b, \color{blue}{t \cdot j}\right) \]

   8. Simplified 69.4%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)\\ \mathbf{elif}\;c \leq -3 \cdot 10^{-155}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(y, -j, a \cdot b\right)\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, z, i \cdot \left(-j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)\\ \mathbf{elif}\;c \leq -1.35 \cdot 10^{-236}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(y, -j, a \cdot b\right)\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -2.2e-25)
   (* c (fma j t (* b (- z))))
   (if (<= c -1.35e-236)
     (* i (fma y (- j) (* a b)))
     (if (<= c 7.2e+132)
       (* x (- (* y z) (* t a)))
       (* c (fma z (- b) (* t j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -2.2e-25) {
		tmp = c * fma(j, t, (b * -z));
	} else if (c <= -1.35e-236) {
		tmp = i * fma(y, -j, (a * b));
	} else if (c <= 7.2e+132) {
		tmp = x * ((y * z) - (t * a));
	} else {
		tmp = c * fma(z, -b, (t * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -2.2e-25)
		tmp = Float64(c * fma(j, t, Float64(b * Float64(-z))));
	elseif (c <= -1.35e-236)
		tmp = Float64(i * fma(y, Float64(-j), Float64(a * b)));
	elseif (c <= 7.2e+132)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	else
		tmp = Float64(c * fma(z, Float64(-b), Float64(t * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -2.2e-25], N[(c * N[(j * t + N[(b * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.35e-236], N[(i * N[(y * (-j) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.2e+132], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(z * (-b) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.2000000000000002e-25

   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 c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(-1 \cdot z\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      10. lower-neg.f64 70.4

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right) \]

   5. Simplified 70.4%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)} \]

   if -2.2000000000000002e-25 < c < -1.35e-236

   1. Initial program 80.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(-1 \cdot j\right) \cdot y} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto i \cdot \left(\color{blue}{y \cdot \left(-1 \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(y \cdot \left(-1 \cdot j\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      6. remove-double-neg N/A

         \[\leadsto i \cdot \left(y \cdot \left(-1 \cdot j\right) + \color{blue}{a \cdot b}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot j, a \cdot b\right)} \]

      8. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(j\right)}, a \cdot b\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto i \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(j\right)}, a \cdot b\right) \]

      10. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(y, \mathsf{neg}\left(j\right), \color{blue}{b \cdot a}\right) \]

      11. lower-*.f64 54.4

          \[\leadsto i \cdot \mathsf{fma}\left(y, -j, \color{blue}{b \cdot a}\right) \]

   5. Simplified 54.4%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(y, -j, b \cdot a\right)} \]

   if -1.35e-236 < c < 7.20000000000000031e132

   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 x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. lower-*.f64 50.9

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 50.9%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   if 7.20000000000000031e132 < c

   1. Initial program 55.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \left(\color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + \frac{x \cdot \left(y \cdot z\right)}{t}\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + \frac{x \cdot \left(y \cdot z\right)}{t}\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. +-commutative N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(\frac{x \cdot \left(y \cdot z\right)}{t} + -1 \cdot \left(a \cdot x\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(t \cdot \left(\color{blue}{x \cdot \frac{y \cdot z}{t}} + -1 \cdot \left(a \cdot x\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. *-commutative N/A

         \[\leadsto \left(t \cdot \left(\color{blue}{\frac{y \cdot z}{t} \cdot x} + -1 \cdot \left(a \cdot x\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(t \cdot \left(\frac{y \cdot z}{t} \cdot x + \color{blue}{\left(-1 \cdot a\right) \cdot x}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(x \cdot \left(\frac{y \cdot z}{t} + -1 \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(x \cdot \left(\frac{y \cdot z}{t} + -1 \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. associate-/l* N/A

         \[\leadsto \left(t \cdot \left(x \cdot \left(\color{blue}{y \cdot \frac{z}{t}} + -1 \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(t \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, -1 \cdot a\right)}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{z}{t}}, -1 \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, \color{blue}{\mathsf{neg}\left(a\right)}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. lower-neg.f64 51.1

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, \color{blue}{-a}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 51.1%

      \[\leadsto \left(\color{blue}{t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, -a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. +-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. *-commutative N/A

         \[\leadsto c \cdot \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right) + j \cdot t\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(b\right)\right)} + j \cdot t\right) \]

      8. mul-1-neg N/A

         \[\leadsto c \cdot \left(z \cdot \color{blue}{\left(-1 \cdot b\right)} + j \cdot t\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(z, -1 \cdot b, j \cdot t\right)} \]

      10. mul-1-neg N/A

          \[\leadsto c \cdot \mathsf{fma}\left(z, \color{blue}{\mathsf{neg}\left(b\right)}, j \cdot t\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(z, \color{blue}{\mathsf{neg}\left(b\right)}, j \cdot t\right) \]

      12. *-commutative N/A

          \[\leadsto c \cdot \mathsf{fma}\left(z, \mathsf{neg}\left(b\right), \color{blue}{t \cdot j}\right) \]

      13. lower-*.f64 71.8

          \[\leadsto c \cdot \mathsf{fma}\left(z, -b, \color{blue}{t \cdot j}\right) \]

   8. Simplified 71.8%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)\\ \mathbf{elif}\;c \leq -1.35 \cdot 10^{-236}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(y, -j, a \cdot b\right)\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 53.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-237}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(y, -j, a \cdot b\right)\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -2.2e-25)
   (* c (fma j t (* b (- z))))
   (if (<= c -1.7e-237)
     (* i (fma y (- j) (* a b)))
     (if (<= c 5.6e-11)
       (* a (fma t (- x) (* b i)))
       (* c (fma z (- b) (* t j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -2.2e-25) {
		tmp = c * fma(j, t, (b * -z));
	} else if (c <= -1.7e-237) {
		tmp = i * fma(y, -j, (a * b));
	} else if (c <= 5.6e-11) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = c * fma(z, -b, (t * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -2.2e-25)
		tmp = Float64(c * fma(j, t, Float64(b * Float64(-z))));
	elseif (c <= -1.7e-237)
		tmp = Float64(i * fma(y, Float64(-j), Float64(a * b)));
	elseif (c <= 5.6e-11)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = Float64(c * fma(z, Float64(-b), Float64(t * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -2.2e-25], N[(c * N[(j * t + N[(b * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.7e-237], N[(i * N[(y * (-j) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5.6e-11], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(z * (-b) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.2000000000000002e-25

   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 c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(-1 \cdot z\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      10. lower-neg.f64 70.4

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right) \]

   5. Simplified 70.4%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)} \]

   if -2.2000000000000002e-25 < c < -1.7000000000000001e-237

   1. Initial program 80.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(-1 \cdot j\right) \cdot y} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto i \cdot \left(\color{blue}{y \cdot \left(-1 \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(y \cdot \left(-1 \cdot j\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      6. remove-double-neg N/A

         \[\leadsto i \cdot \left(y \cdot \left(-1 \cdot j\right) + \color{blue}{a \cdot b}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot j, a \cdot b\right)} \]

      8. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(j\right)}, a \cdot b\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto i \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(j\right)}, a \cdot b\right) \]

      10. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(y, \mathsf{neg}\left(j\right), \color{blue}{b \cdot a}\right) \]

      11. lower-*.f64 54.4

          \[\leadsto i \cdot \mathsf{fma}\left(y, -j, \color{blue}{b \cdot a}\right) \]

   5. Simplified 54.4%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(y, -j, b \cdot a\right)} \]

   if -1.7000000000000001e-237 < c < 5.6e-11

   1. Initial program 77.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 45.9

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 45.9%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   if 5.6e-11 < c

   1. Initial program 63.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \left(\color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + \frac{x \cdot \left(y \cdot z\right)}{t}\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + \frac{x \cdot \left(y \cdot z\right)}{t}\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. +-commutative N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(\frac{x \cdot \left(y \cdot z\right)}{t} + -1 \cdot \left(a \cdot x\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(t \cdot \left(\color{blue}{x \cdot \frac{y \cdot z}{t}} + -1 \cdot \left(a \cdot x\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. *-commutative N/A

         \[\leadsto \left(t \cdot \left(\color{blue}{\frac{y \cdot z}{t} \cdot x} + -1 \cdot \left(a \cdot x\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(t \cdot \left(\frac{y \cdot z}{t} \cdot x + \color{blue}{\left(-1 \cdot a\right) \cdot x}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(x \cdot \left(\frac{y \cdot z}{t} + -1 \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(x \cdot \left(\frac{y \cdot z}{t} + -1 \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. associate-/l* N/A

         \[\leadsto \left(t \cdot \left(x \cdot \left(\color{blue}{y \cdot \frac{z}{t}} + -1 \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(t \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, -1 \cdot a\right)}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{z}{t}}, -1 \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, \color{blue}{\mathsf{neg}\left(a\right)}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. lower-neg.f64 59.3

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, \color{blue}{-a}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 59.3%

      \[\leadsto \left(\color{blue}{t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, -a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. +-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. *-commutative N/A

         \[\leadsto c \cdot \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right) + j \cdot t\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(b\right)\right)} + j \cdot t\right) \]

      8. mul-1-neg N/A

         \[\leadsto c \cdot \left(z \cdot \color{blue}{\left(-1 \cdot b\right)} + j \cdot t\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(z, -1 \cdot b, j \cdot t\right)} \]

      10. mul-1-neg N/A

          \[\leadsto c \cdot \mathsf{fma}\left(z, \color{blue}{\mathsf{neg}\left(b\right)}, j \cdot t\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(z, \color{blue}{\mathsf{neg}\left(b\right)}, j \cdot t\right) \]

      12. *-commutative N/A

          \[\leadsto c \cdot \mathsf{fma}\left(z, \mathsf{neg}\left(b\right), \color{blue}{t \cdot j}\right) \]

      13. lower-*.f64 60.8

          \[\leadsto c \cdot \mathsf{fma}\left(z, -b, \color{blue}{t \cdot j}\right) \]

   8. Simplified 60.8%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-237}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(y, -j, a \cdot b\right)\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 30.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -c \cdot \left(z \cdot b\right)\\ \mathbf{if}\;z \leq -8 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+69}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (- (* c (* z b)))))
   (if (<= z -8e+188)
     t_1
     (if (<= z -9.5e-8) (* z (* x y)) (if (<= z 6.5e+69) (* 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 = -(c * (z * b));
	double tmp;
	if (z <= -8e+188) {
		tmp = t_1;
	} else if (z <= -9.5e-8) {
		tmp = z * (x * y);
	} else if (z <= 6.5e+69) {
		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 = -(c * (z * b))
    if (z <= (-8d+188)) then
        tmp = t_1
    else if (z <= (-9.5d-8)) then
        tmp = z * (x * y)
    else if (z <= 6.5d+69) 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 = -(c * (z * b));
	double tmp;
	if (z <= -8e+188) {
		tmp = t_1;
	} else if (z <= -9.5e-8) {
		tmp = z * (x * y);
	} else if (z <= 6.5e+69) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = -(c * (z * b))
	tmp = 0
	if z <= -8e+188:
		tmp = t_1
	elif z <= -9.5e-8:
		tmp = z * (x * y)
	elif z <= 6.5e+69:
		tmp = c * (t * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(-Float64(c * Float64(z * b)))
	tmp = 0.0
	if (z <= -8e+188)
		tmp = t_1;
	elseif (z <= -9.5e-8)
		tmp = Float64(z * Float64(x * y));
	elseif (z <= 6.5e+69)
		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 = -(c * (z * b));
	tmp = 0.0;
	if (z <= -8e+188)
		tmp = t_1;
	elseif (z <= -9.5e-8)
		tmp = z * (x * y);
	elseif (z <= 6.5e+69)
		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[(c * N[(z * b), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[z, -8e+188], t$95$1, If[LessEqual[z, -9.5e-8], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.5e+69], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.0000000000000002e188 or 6.5000000000000001e69 < z

   1. Initial program 63.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. mul-1-neg N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\left(\mathsf{neg}\left(\color{blue}{c \cdot b}\right)\right) + x \cdot y\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(b\right)\right)} + x \cdot y\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(b\right), x \cdot y\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 71.0

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 71.0%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \frac{b \cdot \left(c \cdot z\right)}{y} + x \cdot z\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \frac{b \cdot \left(c \cdot z\right)}{y} + x \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z + -1 \cdot \frac{b \cdot \left(c \cdot z\right)}{y}\right)} \]

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(x \cdot z + \color{blue}{\left(\mathsf{neg}\left(\frac{b \cdot \left(c \cdot z\right)}{y}\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z - \frac{b \cdot \left(c \cdot z\right)}{y}\right)} \]

      5. lower--.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z - \frac{b \cdot \left(c \cdot z\right)}{y}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto y \cdot \left(\color{blue}{x \cdot z} - \frac{b \cdot \left(c \cdot z\right)}{y}\right) \]

      7. associate-/l* N/A

         \[\leadsto y \cdot \left(x \cdot z - \color{blue}{b \cdot \frac{c \cdot z}{y}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto y \cdot \left(x \cdot z - \color{blue}{b \cdot \frac{c \cdot z}{y}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto y \cdot \left(x \cdot z - b \cdot \color{blue}{\frac{c \cdot z}{y}}\right) \]

      10. *-commutative N/A

          \[\leadsto y \cdot \left(x \cdot z - b \cdot \frac{\color{blue}{z \cdot c}}{y}\right) \]

      11. lower-*.f64 62.7

          \[\leadsto y \cdot \left(x \cdot z - b \cdot \frac{\color{blue}{z \cdot c}}{y}\right) \]

   8. Simplified 62.7%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot z - b \cdot \frac{z \cdot c}{y}\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot b\right) \cdot \left(c \cdot z\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(-1 \cdot b\right) \cdot \color{blue}{\left(z \cdot c\right)} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(-1 \cdot b\right) \cdot z\right) \cdot c} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right)} \cdot c \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right)\right)} \]

       7. associate-*r* N/A

          \[\leadsto c \cdot \color{blue}{\left(\left(-1 \cdot b\right) \cdot z\right)} \]

       8. *-commutative N/A

          \[\leadsto c \cdot \color{blue}{\left(z \cdot \left(-1 \cdot b\right)\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto c \cdot \color{blue}{\left(z \cdot \left(-1 \cdot b\right)\right)} \]

       10. mul-1-neg N/A

           \[\leadsto c \cdot \left(z \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

       11. lower-neg.f64 51.7

           \[\leadsto c \cdot \left(z \cdot \color{blue}{\left(-b\right)}\right) \]

   11. Simplified 51.7%

       \[\leadsto \color{blue}{c \cdot \left(z \cdot \left(-b\right)\right)} \]

   if -8.0000000000000002e188 < z < -9.50000000000000036e-8

   1. Initial program 59.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. mul-1-neg N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\left(\mathsf{neg}\left(\color{blue}{c \cdot b}\right)\right) + x \cdot y\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(b\right)\right)} + x \cdot y\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(b\right), x \cdot y\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 59.7

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 59.7%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 41.4

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   8. Simplified 41.4%

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   if -9.50000000000000036e-8 < z < 6.5000000000000001e69

   1. Initial program 82.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. lower-*.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. +-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(\left(\mathsf{neg}\left(i \cdot y\right)\right) + c \cdot t\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(\color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)} + c \cdot t\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(-1 \cdot y\right)} + c \cdot t\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(i, -1 \cdot y, c \cdot t\right)} \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{neg}\left(y\right)}, c \cdot t\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{neg}\left(y\right)}, c \cdot t\right) \]

      9. lower-*.f64 51.5

         \[\leadsto j \cdot \mathsf{fma}\left(i, -y, \color{blue}{c \cdot t}\right) \]

   5. Simplified 51.5%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(i, -y, c \cdot t\right)} \]

   6. Taylor expanded in i around 0

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

      2. lower-*.f64 32.9

         \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

   8. Simplified 32.9%

      \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(j \cdot t\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(j \cdot t\right) \cdot c} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot j\right)} \cdot c \]

      4. lower-*.f64 34.8

         \[\leadsto \color{blue}{\left(t \cdot j\right)} \cdot c \]

   10. Applied egg-rr 34.8%

       \[\leadsto \color{blue}{\left(t \cdot j\right) \cdot c} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+188}:\\ \;\;\;\;-c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+69}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;-c \cdot \left(z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{-95}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -1.1e-95)
   (* c (fma j t (* b (- z))))
   (if (<= c 5.6e-11)
     (* a (fma t (- x) (* b i)))
     (* c (fma z (- b) (* t j))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -1.1e-95) {
		tmp = c * fma(j, t, (b * -z));
	} else if (c <= 5.6e-11) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = c * fma(z, -b, (t * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -1.1e-95)
		tmp = Float64(c * fma(j, t, Float64(b * Float64(-z))));
	elseif (c <= 5.6e-11)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = Float64(c * fma(z, Float64(-b), Float64(t * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -1.1e-95], N[(c * N[(j * t + N[(b * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5.6e-11], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(z * (-b) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.0999999999999999e-95

   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 c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(-1 \cdot z\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      10. lower-neg.f64 61.6

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right) \]

   5. Simplified 61.6%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)} \]

   if -1.0999999999999999e-95 < c < 5.6e-11

   1. Initial program 80.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 44.9

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 44.9%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   if 5.6e-11 < c

   1. Initial program 63.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \left(\color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + \frac{x \cdot \left(y \cdot z\right)}{t}\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + \frac{x \cdot \left(y \cdot z\right)}{t}\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. +-commutative N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(\frac{x \cdot \left(y \cdot z\right)}{t} + -1 \cdot \left(a \cdot x\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(t \cdot \left(\color{blue}{x \cdot \frac{y \cdot z}{t}} + -1 \cdot \left(a \cdot x\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. *-commutative N/A

         \[\leadsto \left(t \cdot \left(\color{blue}{\frac{y \cdot z}{t} \cdot x} + -1 \cdot \left(a \cdot x\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(t \cdot \left(\frac{y \cdot z}{t} \cdot x + \color{blue}{\left(-1 \cdot a\right) \cdot x}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(x \cdot \left(\frac{y \cdot z}{t} + -1 \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(t \cdot \color{blue}{\left(x \cdot \left(\frac{y \cdot z}{t} + -1 \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. associate-/l* N/A

         \[\leadsto \left(t \cdot \left(x \cdot \left(\color{blue}{y \cdot \frac{z}{t}} + -1 \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(t \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, -1 \cdot a\right)}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{z}{t}}, -1 \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. neg-mul-1 N/A

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, \color{blue}{\mathsf{neg}\left(a\right)}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. lower-neg.f64 59.3

          \[\leadsto \left(t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, \color{blue}{-a}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 59.3%

      \[\leadsto \left(\color{blue}{t \cdot \left(x \cdot \mathsf{fma}\left(y, \frac{z}{t}, -a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. +-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. *-commutative N/A

         \[\leadsto c \cdot \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right) + j \cdot t\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(b\right)\right)} + j \cdot t\right) \]

      8. mul-1-neg N/A

         \[\leadsto c \cdot \left(z \cdot \color{blue}{\left(-1 \cdot b\right)} + j \cdot t\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(z, -1 \cdot b, j \cdot t\right)} \]

      10. mul-1-neg N/A

          \[\leadsto c \cdot \mathsf{fma}\left(z, \color{blue}{\mathsf{neg}\left(b\right)}, j \cdot t\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(z, \color{blue}{\mathsf{neg}\left(b\right)}, j \cdot t\right) \]

      12. *-commutative N/A

          \[\leadsto c \cdot \mathsf{fma}\left(z, \mathsf{neg}\left(b\right), \color{blue}{t \cdot j}\right) \]

      13. lower-*.f64 60.8

          \[\leadsto c \cdot \mathsf{fma}\left(z, -b, \color{blue}{t \cdot j}\right) \]

   8. Simplified 60.8%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{-95}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(z, -b, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)\\ \mathbf{if}\;c \leq -1.1 \cdot 10^{-95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (fma j t (* b (- z))))))
   (if (<= c -1.1e-95)
     t_1
     (if (<= c 5.6e-11) (* a (fma t (- x) (* b i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * fma(j, t, (b * -z));
	double tmp;
	if (c <= -1.1e-95) {
		tmp = t_1;
	} else if (c <= 5.6e-11) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * fma(j, t, Float64(b * Float64(-z))))
	tmp = 0.0
	if (c <= -1.1e-95)
		tmp = t_1;
	elseif (c <= 5.6e-11)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(j * t + N[(b * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.1e-95], t$95$1, If[LessEqual[c, 5.6e-11], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.0999999999999999e-95 or 5.6e-11 < c

   1. Initial program 68.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(-1 \cdot z\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      10. lower-neg.f64 60.5

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right) \]

   5. Simplified 60.5%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)} \]

   if -1.0999999999999999e-95 < c < 5.6e-11

   1. Initial program 80.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 44.9

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 44.9%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{-95}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+194}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (* x y))))
   (if (<= y -2.7e+126)
     t_1
     (if (<= y 1.25e+194) (* b (fma c (- 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 = z * (x * y);
	double tmp;
	if (y <= -2.7e+126) {
		tmp = t_1;
	} else if (y <= 1.25e+194) {
		tmp = b * fma(c, -z, (a * 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 (y <= -2.7e+126)
		tmp = t_1;
	elseif (y <= 1.25e+194)
		tmp = Float64(b * fma(c, Float64(-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[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.7e+126], t$95$1, If[LessEqual[y, 1.25e+194], N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.70000000000000002e126 or 1.24999999999999997e194 < y

   1. Initial program 63.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. mul-1-neg N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\left(\mathsf{neg}\left(\color{blue}{c \cdot b}\right)\right) + x \cdot y\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(b\right)\right)} + x \cdot y\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(b\right), x \cdot y\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 65.0

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 65.0%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 60.2

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   8. Simplified 60.2%

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   if -2.70000000000000002e126 < y < 1.24999999999999997e194

   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 b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right)} \]

      2. remove-double-neg N/A

         \[\leadsto b \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)} + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(a \cdot i\right)\right) + c \cdot z\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \]

      6. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \]

      10. sub-neg N/A

          \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      12. remove-double-neg N/A

          \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{a \cdot i}\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)} + a \cdot i\right) \]

      14. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot i\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot z, a \cdot i\right)} \]

      16. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      17. lower-neg.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      18. *-commutative N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(z\right), \color{blue}{i \cdot a}\right) \]

      19. lower-*.f64 42.5

          \[\leadsto b \cdot \mathsf{fma}\left(c, -z, \color{blue}{i \cdot a}\right) \]

   5. Simplified 42.5%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+126}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+194}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 30.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(t \cdot j\right)\\ \mathbf{if}\;j \leq -2.45 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 1.12 \cdot 10^{+73}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* t j))))
   (if (<= j -2.45e+17) t_1 (if (<= j 1.12e+73) (* z (* x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2.45e+17) {
		tmp = t_1;
	} else if (j <= 1.12e+73) {
		tmp = z * (x * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (t * j)
    if (j <= (-2.45d+17)) then
        tmp = t_1
    else if (j <= 1.12d+73) then
        tmp = z * (x * y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2.45e+17) {
		tmp = t_1;
	} else if (j <= 1.12e+73) {
		tmp = z * (x * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (t * j)
	tmp = 0
	if j <= -2.45e+17:
		tmp = t_1
	elif j <= 1.12e+73:
		tmp = z * (x * y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(t * j))
	tmp = 0.0
	if (j <= -2.45e+17)
		tmp = t_1;
	elseif (j <= 1.12e+73)
		tmp = Float64(z * Float64(x * y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (t * j);
	tmp = 0.0;
	if (j <= -2.45e+17)
		tmp = t_1;
	elseif (j <= 1.12e+73)
		tmp = z * (x * y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -2.45e+17], t$95$1, If[LessEqual[j, 1.12e+73], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -2.45e17 or 1.12e73 < j

   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 j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-*.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. +-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(\left(\mathsf{neg}\left(i \cdot y\right)\right) + c \cdot t\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(\color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)} + c \cdot t\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(-1 \cdot y\right)} + c \cdot t\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(i, -1 \cdot y, c \cdot t\right)} \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{neg}\left(y\right)}, c \cdot t\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{neg}\left(y\right)}, c \cdot t\right) \]

      9. lower-*.f64 60.3

         \[\leadsto j \cdot \mathsf{fma}\left(i, -y, \color{blue}{c \cdot t}\right) \]

   5. Simplified 60.3%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(i, -y, c \cdot t\right)} \]

   6. Taylor expanded in i around 0

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

      2. lower-*.f64 40.5

         \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

   8. Simplified 40.5%

      \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(j \cdot t\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(j \cdot t\right) \cdot c} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot j\right)} \cdot c \]

      4. lower-*.f64 42.9

         \[\leadsto \color{blue}{\left(t \cdot j\right)} \cdot c \]

   10. Applied egg-rr 42.9%

       \[\leadsto \color{blue}{\left(t \cdot j\right) \cdot c} \]

   if -2.45e17 < j < 1.12e73

   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 z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. mul-1-neg N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\left(\mathsf{neg}\left(\color{blue}{c \cdot b}\right)\right) + x \cdot y\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(b\right)\right)} + x \cdot y\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(b\right), x \cdot y\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 54.5

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 54.5%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 34.0

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   8. Simplified 34.0%

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.45 \cdot 10^{+17}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;j \leq 1.12 \cdot 10^{+73}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 28.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+241}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (* x y))))
   (if (<= y -3.8e-11) t_1 (if (<= y 2.35e+241) (* j (* t c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (y <= -3.8e-11) {
		tmp = t_1;
	} else if (y <= 2.35e+241) {
		tmp = j * (t * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (x * y)
    if (y <= (-3.8d-11)) then
        tmp = t_1
    else if (y <= 2.35d+241) then
        tmp = j * (t * c)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (y <= -3.8e-11) {
		tmp = t_1;
	} else if (y <= 2.35e+241) {
		tmp = j * (t * c);
	} 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 y <= -3.8e-11:
		tmp = t_1
	elif y <= 2.35e+241:
		tmp = j * (t * c)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * Float64(x * y))
	tmp = 0.0
	if (y <= -3.8e-11)
		tmp = t_1;
	elseif (y <= 2.35e+241)
		tmp = Float64(j * Float64(t * c));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = z * (x * y);
	tmp = 0.0;
	if (y <= -3.8e-11)
		tmp = t_1;
	elseif (y <= 2.35e+241)
		tmp = j * (t * c);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.8e-11], t$95$1, If[LessEqual[y, 2.35e+241], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7999999999999998e-11 or 2.34999999999999991e241 < y

   1. Initial program 63.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. mul-1-neg N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\left(\mathsf{neg}\left(\color{blue}{c \cdot b}\right)\right) + x \cdot y\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(b\right)\right)} + x \cdot y\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(b\right), x \cdot y\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 60.3

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 60.3%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 52.1

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   8. Simplified 52.1%

      \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

   if -3.7999999999999998e-11 < y < 2.34999999999999991e241

   1. Initial program 77.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-*.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. +-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(\left(\mathsf{neg}\left(i \cdot y\right)\right) + c \cdot t\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(\color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)} + c \cdot t\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(-1 \cdot y\right)} + c \cdot t\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(i, -1 \cdot y, c \cdot t\right)} \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{neg}\left(y\right)}, c \cdot t\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{neg}\left(y\right)}, c \cdot t\right) \]

      9. lower-*.f64 42.2

         \[\leadsto j \cdot \mathsf{fma}\left(i, -y, \color{blue}{c \cdot t}\right) \]

   5. Simplified 42.2%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(i, -y, c \cdot t\right)} \]

   6. Taylor expanded in i around 0

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

      2. lower-*.f64 30.1

         \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

   8. Simplified 30.1%

      \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 30.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(t \cdot c\right)\\ \mathbf{if}\;c \leq -2 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-49}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (* t c))))
   (if (<= c -2e+107) t_1 (if (<= c 7e-49) (* y (* x z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * (t * c);
	double tmp;
	if (c <= -2e+107) {
		tmp = t_1;
	} else if (c <= 7e-49) {
		tmp = y * (x * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = j * (t * c)
    if (c <= (-2d+107)) then
        tmp = t_1
    else if (c <= 7d-49) then
        tmp = y * (x * z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * (t * c);
	double tmp;
	if (c <= -2e+107) {
		tmp = t_1;
	} else if (c <= 7e-49) {
		tmp = y * (x * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * (t * c)
	tmp = 0
	if c <= -2e+107:
		tmp = t_1
	elif c <= 7e-49:
		tmp = y * (x * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(t * c))
	tmp = 0.0
	if (c <= -2e+107)
		tmp = t_1;
	elseif (c <= 7e-49)
		tmp = Float64(y * Float64(x * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * (t * c);
	tmp = 0.0;
	if (c <= -2e+107)
		tmp = t_1;
	elseif (c <= 7e-49)
		tmp = y * (x * z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2e+107], t$95$1, If[LessEqual[c, 7e-49], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.9999999999999999e107 or 7.00000000000000012e-49 < c

   1. Initial program 66.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. lower-*.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. +-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(\left(\mathsf{neg}\left(i \cdot y\right)\right) + c \cdot t\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(\color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)} + c \cdot t\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(-1 \cdot y\right)} + c \cdot t\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(i, -1 \cdot y, c \cdot t\right)} \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{neg}\left(y\right)}, c \cdot t\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{neg}\left(y\right)}, c \cdot t\right) \]

      9. lower-*.f64 52.1

         \[\leadsto j \cdot \mathsf{fma}\left(i, -y, \color{blue}{c \cdot t}\right) \]

   5. Simplified 52.1%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(i, -y, c \cdot t\right)} \]

   6. Taylor expanded in i around 0

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

      2. lower-*.f64 42.2

         \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

   8. Simplified 42.2%

      \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

   if -1.9999999999999999e107 < c < 7.00000000000000012e-49

   1. Initial program 79.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. mul-1-neg N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\left(\mathsf{neg}\left(\color{blue}{c \cdot b}\right)\right) + x \cdot y\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(b\right)\right)} + x \cdot y\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(b\right), x \cdot y\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 43.4

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 43.4%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \frac{b \cdot \left(c \cdot z\right)}{y} + x \cdot z\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \frac{b \cdot \left(c \cdot z\right)}{y} + x \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z + -1 \cdot \frac{b \cdot \left(c \cdot z\right)}{y}\right)} \]

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(x \cdot z + \color{blue}{\left(\mathsf{neg}\left(\frac{b \cdot \left(c \cdot z\right)}{y}\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z - \frac{b \cdot \left(c \cdot z\right)}{y}\right)} \]

      5. lower--.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z - \frac{b \cdot \left(c \cdot z\right)}{y}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto y \cdot \left(\color{blue}{x \cdot z} - \frac{b \cdot \left(c \cdot z\right)}{y}\right) \]

      7. associate-/l* N/A

         \[\leadsto y \cdot \left(x \cdot z - \color{blue}{b \cdot \frac{c \cdot z}{y}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto y \cdot \left(x \cdot z - \color{blue}{b \cdot \frac{c \cdot z}{y}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto y \cdot \left(x \cdot z - b \cdot \color{blue}{\frac{c \cdot z}{y}}\right) \]

      10. *-commutative N/A

          \[\leadsto y \cdot \left(x \cdot z - b \cdot \frac{\color{blue}{z \cdot c}}{y}\right) \]

      11. lower-*.f64 41.1

          \[\leadsto y \cdot \left(x \cdot z - b \cdot \frac{\color{blue}{z \cdot c}}{y}\right) \]

   8. Simplified 41.1%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot z - b \cdot \frac{z \cdot c}{y}\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto y \cdot \color{blue}{\left(x \cdot z\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 32.0

          \[\leadsto y \cdot \color{blue}{\left(x \cdot z\right)} \]

   11. Simplified 32.0%

       \[\leadsto y \cdot \color{blue}{\left(x \cdot z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 30.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+142}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (* y z))))
   (if (<= y -3.8e-11) t_1 (if (<= y 2e+142) (* j (* t c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (y <= -3.8e-11) {
		tmp = t_1;
	} else if (y <= 2e+142) {
		tmp = j * (t * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y * z)
    if (y <= (-3.8d-11)) then
        tmp = t_1
    else if (y <= 2d+142) then
        tmp = j * (t * c)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (y <= -3.8e-11) {
		tmp = t_1;
	} else if (y <= 2e+142) {
		tmp = j * (t * c);
	} 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 y <= -3.8e-11:
		tmp = t_1
	elif y <= 2e+142:
		tmp = j * (t * c)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(x * Float64(y * z))
	tmp = 0.0
	if (y <= -3.8e-11)
		tmp = t_1;
	elseif (y <= 2e+142)
		tmp = Float64(j * Float64(t * c));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = x * (y * z);
	tmp = 0.0;
	if (y <= -3.8e-11)
		tmp = t_1;
	elseif (y <= 2e+142)
		tmp = j * (t * c);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.8e-11], t$95$1, If[LessEqual[y, 2e+142], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7999999999999998e-11 or 2.0000000000000001e142 < y

   1. Initial program 65.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. mul-1-neg N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\left(\mathsf{neg}\left(\color{blue}{c \cdot b}\right)\right) + x \cdot y\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(b\right)\right)} + x \cdot y\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(b\right), x \cdot y\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 59.2

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]

      2. lower-*.f64 46.6

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   8. Simplified 46.6%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]

   if -3.7999999999999998e-11 < y < 2.0000000000000001e142

   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 j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-*.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. +-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(\left(\mathsf{neg}\left(i \cdot y\right)\right) + c \cdot t\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(\color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)} + c \cdot t\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(-1 \cdot y\right)} + c \cdot t\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(i, -1 \cdot y, c \cdot t\right)} \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{neg}\left(y\right)}, c \cdot t\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{neg}\left(y\right)}, c \cdot t\right) \]

      9. lower-*.f64 41.5

         \[\leadsto j \cdot \mathsf{fma}\left(i, -y, \color{blue}{c \cdot t}\right) \]

   5. Simplified 41.5%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(i, -y, c \cdot t\right)} \]

   6. Taylor expanded in i around 0

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

      2. lower-*.f64 29.9

         \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

   8. Simplified 29.9%

      \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 30.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot i\right)\\ \mathbf{if}\;i \leq -1.15 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 1.15 \cdot 10^{+38}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (* b i))))
   (if (<= i -1.15e+110) t_1 (if (<= i 1.15e+38) (* j (* t c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (b * i);
	double tmp;
	if (i <= -1.15e+110) {
		tmp = t_1;
	} else if (i <= 1.15e+38) {
		tmp = j * (t * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * (b * i)
    if (i <= (-1.15d+110)) then
        tmp = t_1
    else if (i <= 1.15d+38) then
        tmp = j * (t * c)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (b * i);
	double tmp;
	if (i <= -1.15e+110) {
		tmp = t_1;
	} else if (i <= 1.15e+38) {
		tmp = j * (t * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = a * (b * i)
	tmp = 0
	if i <= -1.15e+110:
		tmp = t_1
	elif i <= 1.15e+38:
		tmp = j * (t * c)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * Float64(b * i))
	tmp = 0.0
	if (i <= -1.15e+110)
		tmp = t_1;
	elseif (i <= 1.15e+38)
		tmp = Float64(j * Float64(t * c));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = a * (b * i);
	tmp = 0.0;
	if (i <= -1.15e+110)
		tmp = t_1;
	elseif (i <= 1.15e+38)
		tmp = j * (t * c);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.15e+110], t$95$1, If[LessEqual[i, 1.15e+38], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.15e110 or 1.1500000000000001e38 < i

   1. Initial program 62.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 40.4

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 40.4%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]

      2. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

      3. lower-*.f64 33.8

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   8. Simplified 33.8%

      \[\leadsto \color{blue}{a \cdot \left(i \cdot b\right)} \]

   if -1.15e110 < i < 1.1500000000000001e38

   1. Initial program 80.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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. lower-*.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. +-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(\left(\mathsf{neg}\left(i \cdot y\right)\right) + c \cdot t\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(\color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)} + c \cdot t\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \left(i \cdot \color{blue}{\left(-1 \cdot y\right)} + c \cdot t\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(i, -1 \cdot y, c \cdot t\right)} \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{neg}\left(y\right)}, c \cdot t\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{neg}\left(y\right)}, c \cdot t\right) \]

      9. lower-*.f64 41.8

         \[\leadsto j \cdot \mathsf{fma}\left(i, -y, \color{blue}{c \cdot t}\right) \]

   5. Simplified 41.8%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(i, -y, c \cdot t\right)} \]

   6. Taylor expanded in i around 0

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

      2. lower-*.f64 33.2

         \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

   8. Simplified 33.2%

      \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.15 \cdot 10^{+110}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;i \leq 1.15 \cdot 10^{+38}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 22.5% 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 73.3%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

   3. mul-1-neg N/A

      \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

   7. remove-double-neg N/A

      \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

   9. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

   10. lower-neg.f64 N/A

       \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

   11. *-commutative N/A

       \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

   12. lower-*.f64 32.4

       \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

5. Simplified 32.4%

   \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]

   2. *-commutative N/A

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   3. lower-*.f64 18.9

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

8. Simplified 18.9%

   \[\leadsto \color{blue}{a \cdot \left(i \cdot b\right)} \]

9. Final simplification 18.9%

   \[\leadsto a \cdot \left(b \cdot i\right) \]
10. Add Preprocessing

Developer Target 1: 68.1% 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 2024219 
(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)))))