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

Percentage Accurate: 73.8% → 84.9%

Time: 17.6s

Alternatives: 23

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision] - N[(N[(N[(N[(a * t), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - N[(N[(N[(i * t), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(c * a + N[(N[(N[(N[(b * t), $MachinePrecision] / j), $MachinePrecision] - y), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * j), $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 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i)))) < +inf.0

   1. Initial program 88.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 37.0%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 61.2%

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

4. Final simplification 83.9%

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

Alternative 2: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{if}\;i \leq -1.12 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 2.55 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- y) j (* b t)) i (* (fma (- a) t (* z y)) x))))
   (if (<= i -1.12e+89)
     t_1
     (if (<= i 2.55e+154)
       (fma
        (fma (- i) y (* c a))
        j
        (fma (fma (- c) b (* y x)) z (* (fma (- x) a (* i b)) t)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(fma(-y, j, (b * t)), i, (fma(-a, t, (z * y)) * x));
	double tmp;
	if (i <= -1.12e+89) {
		tmp = t_1;
	} else if (i <= 2.55e+154) {
		tmp = fma(fma(-i, y, (c * a)), j, fma(fma(-c, b, (y * x)), z, (fma(-x, a, (i * b)) * t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-y), j, Float64(b * t)), i, Float64(fma(Float64(-a), t, Float64(z * y)) * x))
	tmp = 0.0
	if (i <= -1.12e+89)
		tmp = t_1;
	elseif (i <= 2.55e+154)
		tmp = fma(fma(Float64(-i), y, Float64(c * a)), j, fma(fma(Float64(-c), b, Float64(y * x)), z, Float64(fma(Float64(-x), a, Float64(i * b)) * t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.12e+89], t$95$1, If[LessEqual[i, 2.55e+154], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.11999999999999995e89 or 2.55e154 < i

   1. Initial program 64.8%

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

   3. Taylor expanded in c around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 80.3%

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

   if -1.11999999999999995e89 < i < 2.55e154

   1. Initial program 76.8%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 84.5%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.12 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{elif}\;i \leq 2.55 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{if}\;z \leq -2.55 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot t, i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right) + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- c) b (* y x)) z (* (fma (- i) y (* c a)) j))))
   (if (<= z -2.55e+135)
     t_1
     (if (<= z 2.3e-25)
       (+
        (fma (* b t) i (* (fma (- a) t (* z y)) x))
        (* (- (* c a) (* i y)) j))
       (if (<= z 1.7e+122)
         (fma (fma (- b) c (* y x)) z (* (fma (- x) a (* i b)) t))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(fma(-c, b, (y * x)), z, (fma(-i, y, (c * a)) * j));
	double tmp;
	if (z <= -2.55e+135) {
		tmp = t_1;
	} else if (z <= 2.3e-25) {
		tmp = fma((b * t), i, (fma(-a, t, (z * y)) * x)) + (((c * a) - (i * y)) * j);
	} else if (z <= 1.7e+122) {
		tmp = fma(fma(-b, c, (y * x)), z, (fma(-x, a, (i * b)) * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-c), b, Float64(y * x)), z, Float64(fma(Float64(-i), y, Float64(c * a)) * j))
	tmp = 0.0
	if (z <= -2.55e+135)
		tmp = t_1;
	elseif (z <= 2.3e-25)
		tmp = Float64(fma(Float64(b * t), i, Float64(fma(Float64(-a), t, Float64(z * y)) * x)) + Float64(Float64(Float64(c * a) - Float64(i * y)) * j));
	elseif (z <= 1.7e+122)
		tmp = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(fma(Float64(-x), a, Float64(i * b)) * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.55e+135], t$95$1, If[LessEqual[z, 2.3e-25], N[(N[(N[(b * t), $MachinePrecision] * i + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.7e+122], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.54999999999999991e135 or 1.7e122 < z

   1. Initial program 67.4%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 87.4%

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

   if -2.54999999999999991e135 < z < 2.2999999999999999e-25

   1. Initial program 77.6%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if 2.2999999999999999e-25 < z < 1.7e122

   1. Initial program 65.2%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 78.3%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot t, i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right) + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-b, c, y \cdot x\right)}{j} \cdot \left(j \cdot z\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(i \cdot t\right) \cdot b\right)\\ \mathbf{elif}\;z \leq 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \left(\left(-a\right) \cdot t\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -1.22e+135)
   (* (/ (fma (- b) c (* y x)) j) (* j z))
   (if (<= z 1.25e-75)
     (fma (fma (- i) y (* c a)) j (* (* i t) b))
     (if (<= z 1e+35)
       (fma (fma (- y) j (* b t)) i (* (* (- a) t) x))
       (* (fma (- c) b (* y x)) z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -1.22e+135) {
		tmp = (fma(-b, c, (y * x)) / j) * (j * z);
	} else if (z <= 1.25e-75) {
		tmp = fma(fma(-i, y, (c * a)), j, ((i * t) * b));
	} else if (z <= 1e+35) {
		tmp = fma(fma(-y, j, (b * t)), i, ((-a * t) * x));
	} else {
		tmp = fma(-c, b, (y * x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -1.22e+135)
		tmp = Float64(Float64(fma(Float64(-b), c, Float64(y * x)) / j) * Float64(j * z));
	elseif (z <= 1.25e-75)
		tmp = fma(fma(Float64(-i), y, Float64(c * a)), j, Float64(Float64(i * t) * b));
	elseif (z <= 1e+35)
		tmp = fma(fma(Float64(-y), j, Float64(b * t)), i, Float64(Float64(Float64(-a) * t) * x));
	else
		tmp = Float64(fma(Float64(-c), b, Float64(y * x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -1.22e+135], N[(N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] / j), $MachinePrecision] * N[(j * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25e-75], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e+35], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-a) * t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.21999999999999996e135

   1. Initial program 66.3%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.0%

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

   if -1.21999999999999996e135 < z < 1.24999999999999995e-75

   1. Initial program 77.6%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 64.8%

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

   if 1.24999999999999995e-75 < z < 9.9999999999999997e34

   1. Initial program 68.4%

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

   3. Taylor expanded in c around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 68.8%

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

   if 9.9999999999999997e34 < z

   1. Initial program 68.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-b, c, y \cdot x\right)}{j} \cdot \left(j \cdot z\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(i \cdot t\right) \cdot b\right)\\ \mathbf{elif}\;z \leq 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \left(\left(-a\right) \cdot t\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(i \cdot t\right) \cdot b\right)\\ \mathbf{elif}\;z \leq 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \left(\left(-a\right) \cdot t\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= z -4.8e+134)
     t_1
     (if (<= z 1.25e-75)
       (fma (fma (- i) y (* c a)) j (* (* i t) b))
       (if (<= z 1e+35)
         (fma (fma (- y) j (* b t)) i (* (* (- a) t) x))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -4.8e+134) {
		tmp = t_1;
	} else if (z <= 1.25e-75) {
		tmp = fma(fma(-i, y, (c * a)), j, ((i * t) * b));
	} else if (z <= 1e+35) {
		tmp = fma(fma(-y, j, (b * t)), i, ((-a * t) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -4.8e+134)
		tmp = t_1;
	elseif (z <= 1.25e-75)
		tmp = fma(fma(Float64(-i), y, Float64(c * a)), j, Float64(Float64(i * t) * b));
	elseif (z <= 1e+35)
		tmp = fma(fma(Float64(-y), j, Float64(b * t)), i, Float64(Float64(Float64(-a) * t) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -4.8e+134], t$95$1, If[LessEqual[z, 1.25e-75], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e+35], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-a) * t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.80000000000000011e134 or 9.9999999999999997e34 < z

   1. Initial program 67.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if -4.80000000000000011e134 < z < 1.24999999999999995e-75

   1. Initial program 77.6%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 64.8%

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

   if 1.24999999999999995e-75 < z < 9.9999999999999997e34

   1. Initial program 68.4%

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

   3. Taylor expanded in c around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 68.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(i \cdot t\right) \cdot b\right)\\ \mathbf{elif}\;z \leq 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \left(\left(-a\right) \cdot t\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, a, \left(\frac{b \cdot t}{j} - y\right) \cdot i\right) \cdot j\\ \mathbf{if}\;j \leq -3.1 \cdot 10^{+66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 2.7 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma c a (* (- (/ (* b t) j) y) i)) j)))
   (if (<= j -3.1e+66)
     t_1
     (if (<= j 2.7e+57)
       (fma (fma (- b) c (* y x)) z (* (fma (- x) a (* i b)) t))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(c, a, ((((b * t) / j) - y) * i)) * j;
	double tmp;
	if (j <= -3.1e+66) {
		tmp = t_1;
	} else if (j <= 2.7e+57) {
		tmp = fma(fma(-b, c, (y * x)), z, (fma(-x, a, (i * b)) * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(c, a, Float64(Float64(Float64(Float64(b * t) / j) - y) * i)) * j)
	tmp = 0.0
	if (j <= -3.1e+66)
		tmp = t_1;
	elseif (j <= 2.7e+57)
		tmp = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(fma(Float64(-x), a, Float64(i * b)) * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(c * a + N[(N[(N[(N[(b * t), $MachinePrecision] / j), $MachinePrecision] - y), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -3.1e+66], t$95$1, If[LessEqual[j, 2.7e+57], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -3.10000000000000019e66 or 2.6999999999999998e57 < j

   1. Initial program 74.6%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 80.7%

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

   if -3.10000000000000019e66 < j < 2.6999999999999998e57

   1. Initial program 71.8%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 71.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -3.1 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(\frac{b \cdot t}{j} - y\right) \cdot i\right) \cdot j\\ \mathbf{elif}\;j \leq 2.7 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(\frac{b \cdot t}{j} - y\right) \cdot i\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ t_2 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+134}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -6.6 \cdot 10^{-145}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 1.01 \cdot 10^{-277}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) a (* i b)) t)) (t_2 (* (fma (- c) b (* y x)) z)))
   (if (<= z -1.25e+134)
     t_2
     (if (<= z -6.6e-145)
       (* (fma (- x) t (* j c)) a)
       (if (<= z 1.01e-277)
         t_1
         (if (<= z 1.4e-120)
           (* (fma c a (* (- i) y)) j)
           (if (<= z 8e+34) 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 = fma(-x, a, (i * b)) * t;
	double t_2 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -1.25e+134) {
		tmp = t_2;
	} else if (z <= -6.6e-145) {
		tmp = fma(-x, t, (j * c)) * a;
	} else if (z <= 1.01e-277) {
		tmp = t_1;
	} else if (z <= 1.4e-120) {
		tmp = fma(c, a, (-i * y)) * j;
	} else if (z <= 8e+34) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-x), a, Float64(i * b)) * t)
	t_2 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -1.25e+134)
		tmp = t_2;
	elseif (z <= -6.6e-145)
		tmp = Float64(fma(Float64(-x), t, Float64(j * c)) * a);
	elseif (z <= 1.01e-277)
		tmp = t_1;
	elseif (z <= 1.4e-120)
		tmp = Float64(fma(c, a, Float64(Float64(-i) * y)) * j);
	elseif (z <= 8e+34)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.25e+134], t$95$2, If[LessEqual[z, -6.6e-145], N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 1.01e-277], t$95$1, If[LessEqual[z, 1.4e-120], N[(N[(c * a + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, 8e+34], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.24999999999999995e134 or 7.99999999999999956e34 < z

   1. Initial program 68.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

   if -1.24999999999999995e134 < z < -6.59999999999999962e-145

   1. Initial program 68.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 60.8

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

   5. Applied rewrites 60.8%

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

   if -6.59999999999999962e-145 < z < 1.01e-277 or 1.39999999999999997e-120 < z < 7.99999999999999956e34

   1. Initial program 76.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   if 1.01e-277 < z < 1.39999999999999997e-120

   1. Initial program 83.3%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 58.8%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -6.6 \cdot 10^{-145}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 1.01 \cdot 10^{-277}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{if}\;j \leq -1.45 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq -6 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{elif}\;j \leq 2.9 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;j \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma c a (* (- i) y)) j)))
   (if (<= j -1.45e+71)
     t_1
     (if (<= j -6e-159)
       (* (fma (- x) a (* i b)) t)
       (if (<= j 2.9e-159)
         (* (fma (- c) b (* y x)) z)
         (if (<= j 1.6e+57) (* (fma (- a) t (* z y)) x) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(c, a, (-i * y)) * j;
	double tmp;
	if (j <= -1.45e+71) {
		tmp = t_1;
	} else if (j <= -6e-159) {
		tmp = fma(-x, a, (i * b)) * t;
	} else if (j <= 2.9e-159) {
		tmp = fma(-c, b, (y * x)) * z;
	} else if (j <= 1.6e+57) {
		tmp = fma(-a, t, (z * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(c, a, Float64(Float64(-i) * y)) * j)
	tmp = 0.0
	if (j <= -1.45e+71)
		tmp = t_1;
	elseif (j <= -6e-159)
		tmp = Float64(fma(Float64(-x), a, Float64(i * b)) * t);
	elseif (j <= 2.9e-159)
		tmp = Float64(fma(Float64(-c), b, Float64(y * x)) * z);
	elseif (j <= 1.6e+57)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(c * a + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -1.45e+71], t$95$1, If[LessEqual[j, -6e-159], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[j, 2.9e-159], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[j, 1.6e+57], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if j < -1.45000000000000004e71 or 1.60000000000000015e57 < j

   1. Initial program 75.0%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 67.9%

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

   if -1.45000000000000004e71 < j < -6.00000000000000018e-159

   1. Initial program 74.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

   if -6.00000000000000018e-159 < j < 2.8999999999999999e-159

   1. Initial program 63.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 60.9

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

   5. Applied rewrites 60.9%

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

   if 2.8999999999999999e-159 < j < 1.60000000000000015e57

   1. Initial program 79.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.45 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{elif}\;j \leq -6 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{elif}\;j \leq 2.9 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;j \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(i \cdot t\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= z -4.8e+134)
     t_1
     (if (<= z 3.1e+33) (fma (fma (- i) y (* c a)) j (* (* i t) b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -4.8e+134) {
		tmp = t_1;
	} else if (z <= 3.1e+33) {
		tmp = fma(fma(-i, y, (c * a)), j, ((i * t) * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -4.8e+134)
		tmp = t_1;
	elseif (z <= 3.1e+33)
		tmp = fma(fma(Float64(-i), y, Float64(c * a)), j, Float64(Float64(i * t) * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -4.8e+134], t$95$1, If[LessEqual[z, 3.1e+33], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.80000000000000011e134 or 3.1e33 < z

   1. Initial program 67.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

   if -4.80000000000000011e134 < z < 3.1e33

   1. Initial program 76.5%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 62.2%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(i \cdot t\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.58 \cdot 10^{-245}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= z -1.25e+134)
     t_1
     (if (<= z -1.58e-245)
       (* (fma (- x) t (* j c)) a)
       (if (<= z 5.8e-10) (* (fma (- y) j (* b t)) 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(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -1.25e+134) {
		tmp = t_1;
	} else if (z <= -1.58e-245) {
		tmp = fma(-x, t, (j * c)) * a;
	} else if (z <= 5.8e-10) {
		tmp = fma(-y, j, (b * t)) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -1.25e+134)
		tmp = t_1;
	elseif (z <= -1.58e-245)
		tmp = Float64(fma(Float64(-x), t, Float64(j * c)) * a);
	elseif (z <= 5.8e-10)
		tmp = Float64(fma(Float64(-y), j, Float64(b * t)) * i);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.25e+134], t$95$1, If[LessEqual[z, -1.58e-245], N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 5.8e-10], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.24999999999999995e134 or 5.79999999999999962e-10 < z

   1. Initial program 67.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   if -1.24999999999999995e134 < z < -1.58e-245

   1. Initial program 77.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 58.3

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

   5. Applied rewrites 58.3%

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

   if -1.58e-245 < z < 5.79999999999999962e-10

   1. Initial program 76.1%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 55.6

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

   5. Applied rewrites 55.6%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -1.58 \cdot 10^{-245}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{if}\;j \leq -0.175:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 3.1 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b\\ \mathbf{elif}\;j \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma c a (* (- i) y)) j)))
   (if (<= j -0.175)
     t_1
     (if (<= j 3.1e-159)
       (* (fma (- c) z (* i t)) b)
       (if (<= j 1.6e+57) (* (fma (- a) t (* z y)) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(c, a, (-i * y)) * j;
	double tmp;
	if (j <= -0.175) {
		tmp = t_1;
	} else if (j <= 3.1e-159) {
		tmp = fma(-c, z, (i * t)) * b;
	} else if (j <= 1.6e+57) {
		tmp = fma(-a, t, (z * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(c, a, Float64(Float64(-i) * y)) * j)
	tmp = 0.0
	if (j <= -0.175)
		tmp = t_1;
	elseif (j <= 3.1e-159)
		tmp = Float64(fma(Float64(-c), z, Float64(i * t)) * b);
	elseif (j <= 1.6e+57)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(c * a + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -0.175], t$95$1, If[LessEqual[j, 3.1e-159], N[(N[((-c) * z + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[j, 1.6e+57], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -0.17499999999999999 or 1.60000000000000015e57 < j

   1. Initial program 75.1%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 63.1%

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

   if -0.17499999999999999 < j < 3.1e-159

   1. Initial program 66.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 54.0

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

   5. Applied rewrites 54.0%

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

   if 3.1e-159 < j < 1.60000000000000015e57

   1. Initial program 79.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -0.175:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{elif}\;j \leq 3.1 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b\\ \mathbf{elif}\;j \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{if}\;j \leq -0.38:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 2.9 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;j \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma c a (* (- i) y)) j)))
   (if (<= j -0.38)
     t_1
     (if (<= j 2.9e-159)
       (* (fma (- c) b (* y x)) z)
       (if (<= j 1.6e+57) (* (fma (- a) t (* z y)) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(c, a, (-i * y)) * j;
	double tmp;
	if (j <= -0.38) {
		tmp = t_1;
	} else if (j <= 2.9e-159) {
		tmp = fma(-c, b, (y * x)) * z;
	} else if (j <= 1.6e+57) {
		tmp = fma(-a, t, (z * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(c, a, Float64(Float64(-i) * y)) * j)
	tmp = 0.0
	if (j <= -0.38)
		tmp = t_1;
	elseif (j <= 2.9e-159)
		tmp = Float64(fma(Float64(-c), b, Float64(y * x)) * z);
	elseif (j <= 1.6e+57)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(c * a + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -0.38], t$95$1, If[LessEqual[j, 2.9e-159], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[j, 1.6e+57], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -0.38 or 1.60000000000000015e57 < j

   1. Initial program 75.1%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 63.1%

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

   if -0.38 < j < 2.8999999999999999e-159

   1. Initial program 66.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 54.0

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

   5. Applied rewrites 54.0%

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

   if 2.8999999999999999e-159 < j < 1.60000000000000015e57

   1. Initial program 79.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -0.38:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{elif}\;j \leq 2.9 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;j \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{if}\;j \leq -0.38:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq -4.2 \cdot 10^{-243}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\\ \mathbf{elif}\;j \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma c a (* (- i) y)) j)))
   (if (<= j -0.38)
     t_1
     (if (<= j -4.2e-243)
       (* (fma (- b) z (* j a)) c)
       (if (<= j 1.6e+57) (* (fma (- a) t (* z y)) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(c, a, (-i * y)) * j;
	double tmp;
	if (j <= -0.38) {
		tmp = t_1;
	} else if (j <= -4.2e-243) {
		tmp = fma(-b, z, (j * a)) * c;
	} else if (j <= 1.6e+57) {
		tmp = fma(-a, t, (z * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(c, a, Float64(Float64(-i) * y)) * j)
	tmp = 0.0
	if (j <= -0.38)
		tmp = t_1;
	elseif (j <= -4.2e-243)
		tmp = Float64(fma(Float64(-b), z, Float64(j * a)) * c);
	elseif (j <= 1.6e+57)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(c * a + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -0.38], t$95$1, If[LessEqual[j, -4.2e-243], N[(N[((-b) * z + N[(j * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[j, 1.6e+57], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -0.38 or 1.60000000000000015e57 < j

   1. Initial program 75.1%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 63.1%

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

   if -0.38 < j < -4.2000000000000002e-243

   1. Initial program 71.6%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 47.6

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

   5. Applied rewrites 47.6%

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

   if -4.2000000000000002e-243 < j < 1.60000000000000015e57

   1. Initial program 71.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.3

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

   5. Applied rewrites 52.3%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -0.38:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{elif}\;j \leq -4.2 \cdot 10^{-243}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\\ \mathbf{elif}\;j \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -6.7e+152)
   (* (* z x) y)
   (if (<= z -1.7e-278)
     (* (* (- t) x) a)
     (if (<= z 2.55e-25) (* (* (- i) j) y) (* (* (- c) z) b)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -6.7e+152) {
		tmp = (z * x) * y;
	} else if (z <= -1.7e-278) {
		tmp = (-t * x) * a;
	} else if (z <= 2.55e-25) {
		tmp = (-i * j) * y;
	} else {
		tmp = (-c * z) * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (z <= (-6.7d+152)) then
        tmp = (z * x) * y
    else if (z <= (-1.7d-278)) then
        tmp = (-t * x) * a
    else if (z <= 2.55d-25) then
        tmp = (-i * j) * y
    else
        tmp = (-c * z) * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -6.7e+152) {
		tmp = (z * x) * y;
	} else if (z <= -1.7e-278) {
		tmp = (-t * x) * a;
	} else if (z <= 2.55e-25) {
		tmp = (-i * j) * y;
	} else {
		tmp = (-c * z) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -6.7e+152:
		tmp = (z * x) * y
	elif z <= -1.7e-278:
		tmp = (-t * x) * a
	elif z <= 2.55e-25:
		tmp = (-i * j) * y
	else:
		tmp = (-c * z) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -6.7e+152)
		tmp = Float64(Float64(z * x) * y);
	elseif (z <= -1.7e-278)
		tmp = Float64(Float64(Float64(-t) * x) * a);
	elseif (z <= 2.55e-25)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	else
		tmp = Float64(Float64(Float64(-c) * z) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -6.7e+152)
		tmp = (z * x) * y;
	elseif (z <= -1.7e-278)
		tmp = (-t * x) * a;
	elseif (z <= 2.55e-25)
		tmp = (-i * j) * y;
	else
		tmp = (-c * z) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -6.7e+152], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -1.7e-278], N[(N[((-t) * x), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 2.55e-25], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.69999999999999987e152

   1. Initial program 66.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 57.9%

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

   if -6.69999999999999987e152 < z < -1.7e-278

   1. Initial program 77.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 31.2%

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

   if -1.7e-278 < z < 2.5500000000000001e-25

   1. Initial program 76.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.2%

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

   if 2.5500000000000001e-25 < z

   1. Initial program 67.0%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 58.8

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

   8. Applied rewrites 58.8%

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

   9. Taylor expanded in c around inf

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

   11. Applied rewrites 39.5%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -6.7e+152)
   (* (* z x) y)
   (if (<= z -1.7e-278)
     (* (* (- t) x) a)
     (if (<= z 2.55e-25) (* (* (- i) j) y) (* (* (- b) c) 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 (z <= -6.7e+152) {
		tmp = (z * x) * y;
	} else if (z <= -1.7e-278) {
		tmp = (-t * x) * a;
	} else if (z <= 2.55e-25) {
		tmp = (-i * j) * y;
	} else {
		tmp = (-b * c) * z;
	}
	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 (z <= (-6.7d+152)) then
        tmp = (z * x) * y
    else if (z <= (-1.7d-278)) then
        tmp = (-t * x) * a
    else if (z <= 2.55d-25) then
        tmp = (-i * j) * y
    else
        tmp = (-b * c) * z
    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 (z <= -6.7e+152) {
		tmp = (z * x) * y;
	} else if (z <= -1.7e-278) {
		tmp = (-t * x) * a;
	} else if (z <= 2.55e-25) {
		tmp = (-i * j) * y;
	} else {
		tmp = (-b * c) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -6.7e+152:
		tmp = (z * x) * y
	elif z <= -1.7e-278:
		tmp = (-t * x) * a
	elif z <= 2.55e-25:
		tmp = (-i * j) * y
	else:
		tmp = (-b * c) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -6.7e+152)
		tmp = Float64(Float64(z * x) * y);
	elseif (z <= -1.7e-278)
		tmp = Float64(Float64(Float64(-t) * x) * a);
	elseif (z <= 2.55e-25)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	else
		tmp = Float64(Float64(Float64(-b) * c) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -6.7e+152)
		tmp = (z * x) * y;
	elseif (z <= -1.7e-278)
		tmp = (-t * x) * a;
	elseif (z <= 2.55e-25)
		tmp = (-i * j) * y;
	else
		tmp = (-b * c) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -6.7e+152], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -1.7e-278], N[(N[((-t) * x), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 2.55e-25], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], N[(N[((-b) * c), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.69999999999999987e152

   1. Initial program 66.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 57.9%

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

   if -6.69999999999999987e152 < z < -1.7e-278

   1. Initial program 77.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 31.2%

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

   if -1.7e-278 < z < 2.5500000000000001e-25

   1. Initial program 76.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.2%

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

   if 2.5500000000000001e-25 < z

   1. Initial program 67.0%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 58.8

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

   8. Applied rewrites 58.8%

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

   9. Taylor expanded in c around inf

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

   11. Applied rewrites 39.4%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -6.7e+152)
   (* (* z x) y)
   (if (<= z -1.7e-278)
     (* (* (- t) x) a)
     (if (<= z 2.4e-25) (* (* (- y) j) i) (* (* (- b) c) 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 (z <= -6.7e+152) {
		tmp = (z * x) * y;
	} else if (z <= -1.7e-278) {
		tmp = (-t * x) * a;
	} else if (z <= 2.4e-25) {
		tmp = (-y * j) * i;
	} else {
		tmp = (-b * c) * z;
	}
	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 (z <= (-6.7d+152)) then
        tmp = (z * x) * y
    else if (z <= (-1.7d-278)) then
        tmp = (-t * x) * a
    else if (z <= 2.4d-25) then
        tmp = (-y * j) * i
    else
        tmp = (-b * c) * z
    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 (z <= -6.7e+152) {
		tmp = (z * x) * y;
	} else if (z <= -1.7e-278) {
		tmp = (-t * x) * a;
	} else if (z <= 2.4e-25) {
		tmp = (-y * j) * i;
	} else {
		tmp = (-b * c) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -6.7e+152:
		tmp = (z * x) * y
	elif z <= -1.7e-278:
		tmp = (-t * x) * a
	elif z <= 2.4e-25:
		tmp = (-y * j) * i
	else:
		tmp = (-b * c) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -6.7e+152)
		tmp = Float64(Float64(z * x) * y);
	elseif (z <= -1.7e-278)
		tmp = Float64(Float64(Float64(-t) * x) * a);
	elseif (z <= 2.4e-25)
		tmp = Float64(Float64(Float64(-y) * j) * i);
	else
		tmp = Float64(Float64(Float64(-b) * c) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -6.7e+152)
		tmp = (z * x) * y;
	elseif (z <= -1.7e-278)
		tmp = (-t * x) * a;
	elseif (z <= 2.4e-25)
		tmp = (-y * j) * i;
	else
		tmp = (-b * c) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -6.7e+152], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -1.7e-278], N[(N[((-t) * x), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 2.4e-25], N[(N[((-y) * j), $MachinePrecision] * i), $MachinePrecision], N[(N[((-b) * c), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.69999999999999987e152

   1. Initial program 66.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 57.9%

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

   if -6.69999999999999987e152 < z < -1.7e-278

   1. Initial program 77.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 31.2%

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

   if -1.7e-278 < z < 2.40000000000000009e-25

   1. Initial program 76.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.9%

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

   if 2.40000000000000009e-25 < z

   1. Initial program 67.0%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 58.8

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

   8. Applied rewrites 58.8%

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

   9. Taylor expanded in c around inf

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

   11. Applied rewrites 39.4%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 30.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-10}:\\ \;\;\;\;\left(\left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -6.7e+152)
   (* (* z x) y)
   (if (<= z -1.7e-278)
     (* (* (- t) x) a)
     (if (<= z 5.2e-10) (* (* (- y) j) i) (* (* z y) x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -6.7e+152) {
		tmp = (z * x) * y;
	} else if (z <= -1.7e-278) {
		tmp = (-t * x) * a;
	} else if (z <= 5.2e-10) {
		tmp = (-y * j) * i;
	} else {
		tmp = (z * y) * x;
	}
	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 (z <= (-6.7d+152)) then
        tmp = (z * x) * y
    else if (z <= (-1.7d-278)) then
        tmp = (-t * x) * a
    else if (z <= 5.2d-10) then
        tmp = (-y * j) * i
    else
        tmp = (z * y) * x
    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 (z <= -6.7e+152) {
		tmp = (z * x) * y;
	} else if (z <= -1.7e-278) {
		tmp = (-t * x) * a;
	} else if (z <= 5.2e-10) {
		tmp = (-y * j) * i;
	} else {
		tmp = (z * y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -6.7e+152:
		tmp = (z * x) * y
	elif z <= -1.7e-278:
		tmp = (-t * x) * a
	elif z <= 5.2e-10:
		tmp = (-y * j) * i
	else:
		tmp = (z * y) * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -6.7e+152)
		tmp = Float64(Float64(z * x) * y);
	elseif (z <= -1.7e-278)
		tmp = Float64(Float64(Float64(-t) * x) * a);
	elseif (z <= 5.2e-10)
		tmp = Float64(Float64(Float64(-y) * j) * i);
	else
		tmp = Float64(Float64(z * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -6.7e+152)
		tmp = (z * x) * y;
	elseif (z <= -1.7e-278)
		tmp = (-t * x) * a;
	elseif (z <= 5.2e-10)
		tmp = (-y * j) * i;
	else
		tmp = (z * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -6.7e+152], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -1.7e-278], N[(N[((-t) * x), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 5.2e-10], N[(N[((-y) * j), $MachinePrecision] * i), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.69999999999999987e152

   1. Initial program 66.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 57.9%

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

   if -6.69999999999999987e152 < z < -1.7e-278

   1. Initial program 77.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 31.2%

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

   if -1.7e-278 < z < 5.19999999999999962e-10

   1. Initial program 74.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 40.6

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 34.0%

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

   if 5.19999999999999962e-10 < z

   1. Initial program 68.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 45.0

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 40.2%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-10}:\\ \;\;\;\;\left(\left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 53.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{if}\;j \leq -7.8 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma c a (* (- i) y)) j)))
   (if (<= j -7.8e+68)
     t_1
     (if (<= j 1.6e+57) (* (fma (- a) t (* z y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(c, a, (-i * y)) * j;
	double tmp;
	if (j <= -7.8e+68) {
		tmp = t_1;
	} else if (j <= 1.6e+57) {
		tmp = fma(-a, t, (z * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(c, a, Float64(Float64(-i) * y)) * j)
	tmp = 0.0
	if (j <= -7.8e+68)
		tmp = t_1;
	elseif (j <= 1.6e+57)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(c * a + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -7.8e+68], t$95$1, If[LessEqual[j, 1.6e+57], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -7.80000000000000037e68 or 1.60000000000000015e57 < j

   1. Initial program 75.3%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in j around inf

      \[\leadsto \mathsf{fma}\left(c, a, -1 \cdot \left(i \cdot y\right)\right) \cdot j \]
   7. Step-by-step derivation

   8. Applied rewrites 68.2%

      \[\leadsto \mathsf{fma}\left(c, a, \left(-y\right) \cdot i\right) \cdot j \]

   if -7.80000000000000037e68 < j < 1.60000000000000015e57

   1. Initial program 71.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 45.5

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 45.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -7.8 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{elif}\;j \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 30.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-142}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+41}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -6.7e+152)
   (* (* z x) y)
   (if (<= z -2.5e-142)
     (* (* (- t) x) a)
     (if (<= z 4.8e+41) (* (* i t) b) (* (* z y) x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -6.7e+152) {
		tmp = (z * x) * y;
	} else if (z <= -2.5e-142) {
		tmp = (-t * x) * a;
	} else if (z <= 4.8e+41) {
		tmp = (i * t) * b;
	} else {
		tmp = (z * y) * x;
	}
	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 (z <= (-6.7d+152)) then
        tmp = (z * x) * y
    else if (z <= (-2.5d-142)) then
        tmp = (-t * x) * a
    else if (z <= 4.8d+41) then
        tmp = (i * t) * b
    else
        tmp = (z * y) * x
    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 (z <= -6.7e+152) {
		tmp = (z * x) * y;
	} else if (z <= -2.5e-142) {
		tmp = (-t * x) * a;
	} else if (z <= 4.8e+41) {
		tmp = (i * t) * b;
	} else {
		tmp = (z * y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -6.7e+152:
		tmp = (z * x) * y
	elif z <= -2.5e-142:
		tmp = (-t * x) * a
	elif z <= 4.8e+41:
		tmp = (i * t) * b
	else:
		tmp = (z * y) * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -6.7e+152)
		tmp = Float64(Float64(z * x) * y);
	elseif (z <= -2.5e-142)
		tmp = Float64(Float64(Float64(-t) * x) * a);
	elseif (z <= 4.8e+41)
		tmp = Float64(Float64(i * t) * b);
	else
		tmp = Float64(Float64(z * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -6.7e+152)
		tmp = (z * x) * y;
	elseif (z <= -2.5e-142)
		tmp = (-t * x) * a;
	elseif (z <= 4.8e+41)
		tmp = (i * t) * b;
	else
		tmp = (z * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -6.7e+152], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -2.5e-142], N[(N[((-t) * x), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 4.8e+41], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.69999999999999987e152

   1. Initial program 66.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(j \cdot i\right)} + x \cdot z\right) \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right) \cdot i} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 70.4

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 57.9%

      \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -6.69999999999999987e152 < z < -2.5000000000000001e-142

   1. Initial program 70.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 37.5

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around inf

      \[\leadsto -1 \cdot \color{blue}{\left(a \cdot \left(t \cdot x\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.9%

      \[\leadsto \left(t \cdot x\right) \cdot \color{blue}{\left(-a\right)} \]

   if -2.5000000000000001e-142 < z < 4.8000000000000003e41

   1. Initial program 78.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j + \color{blue}{\left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j + \color{blue}{\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \color{blue}{\left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)} \]

   5. Applied rewrites 81.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-c, b, x \cdot y\right), z, \mathsf{fma}\left(-x, a, b \cdot i\right) \cdot t\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot t - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. mul-1-neg N/A

         \[\leadsto \left(i \cdot t + \color{blue}{-1 \cdot \left(c \cdot z\right)}\right) \cdot b \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right) \cdot b} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)} + i \cdot t\right) \cdot b \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + i \cdot t\right) \cdot b \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + i \cdot t\right) \cdot b \]

      9. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + i \cdot t\right) \cdot b \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, i \cdot t\right)} \cdot b \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, i \cdot t\right) \cdot b \]

      12. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, i \cdot t\right) \cdot b \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{t \cdot i}\right) \cdot b \]

      14. lower-*.f64 35.7

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{t \cdot i}\right) \cdot b \]

   8. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, t \cdot i\right) \cdot b} \]

   9. Taylor expanded in c around 0

      \[\leadsto \left(i \cdot t\right) \cdot b \]
   10. Step-by-step derivation

   11. Applied rewrites 30.5%

       \[\leadsto \left(t \cdot i\right) \cdot b \]

   if 4.8000000000000003e41 < z

   1. Initial program 68.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 47.3

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 45.4%

      \[\leadsto \left(z \cdot y\right) \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-142}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+41}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 40.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x 8.5e+75) (* (fma c a (* (- i) y)) j) (* (* z x) y)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= 8.5e+75) {
		tmp = fma(c, a, (-i * y)) * j;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= 8.5e+75)
		tmp = Float64(fma(c, a, Float64(Float64(-i) * y)) * j);
	else
		tmp = Float64(Float64(z * x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, 8.5e+75], N[(N[(c * a + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.4999999999999993e75

   1. Initial program 72.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(j \cdot \left(-1 \cdot \left(a \cdot c - i \cdot y\right) + -1 \cdot \frac{x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(c \cdot z - i \cdot t\right)}{j}\right)\right)} \]

   5. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, a, \mathsf{fma}\left(-i, y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-c, z, t \cdot i\right), b, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)}{j}\right)\right) \cdot j} \]

   6. Taylor expanded in j around inf

      \[\leadsto \mathsf{fma}\left(c, a, -1 \cdot \left(i \cdot y\right)\right) \cdot j \]
   7. Step-by-step derivation

   8. Applied rewrites 44.7%

      \[\leadsto \mathsf{fma}\left(c, a, \left(-y\right) \cdot i\right) \cdot j \]

   if 8.4999999999999993e75 < x

   1. Initial program 73.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(j \cdot i\right)} + x \cdot z\right) \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right) \cdot i} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 56.7

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 52.2%

      \[\leadsto \left(z \cdot x\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 30.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+134}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+38}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -4.8e+134)
   (* (* z x) y)
   (if (<= z 6.5e+38) (* (* b t) i) (* (* z y) x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -4.8e+134) {
		tmp = (z * x) * y;
	} else if (z <= 6.5e+38) {
		tmp = (b * t) * i;
	} else {
		tmp = (z * y) * x;
	}
	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 (z <= (-4.8d+134)) then
        tmp = (z * x) * y
    else if (z <= 6.5d+38) then
        tmp = (b * t) * i
    else
        tmp = (z * y) * x
    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 (z <= -4.8e+134) {
		tmp = (z * x) * y;
	} else if (z <= 6.5e+38) {
		tmp = (b * t) * i;
	} else {
		tmp = (z * y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -4.8e+134:
		tmp = (z * x) * y
	elif z <= 6.5e+38:
		tmp = (b * t) * i
	else:
		tmp = (z * y) * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -4.8e+134)
		tmp = Float64(Float64(z * x) * y);
	elseif (z <= 6.5e+38)
		tmp = Float64(Float64(b * t) * i);
	else
		tmp = Float64(Float64(z * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -4.8e+134)
		tmp = (z * x) * y;
	elseif (z <= 6.5e+38)
		tmp = (b * t) * i;
	else
		tmp = (z * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -4.8e+134], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 6.5e+38], N[(N[(b * t), $MachinePrecision] * i), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.80000000000000011e134

   1. Initial program 66.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(j \cdot i\right)} + x \cdot z\right) \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right) \cdot i} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 61.1

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 61.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 51.1%

      \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -4.80000000000000011e134 < z < 6.5e38

   1. Initial program 76.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right)} \cdot i \]

      4. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \cdot i \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{y \cdot j}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \cdot i \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \cdot i \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right)} \cdot j + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot t\right)\right)\right)\right) \cdot i \]

      8. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot t\right)\right)}\right)\right)\right) \cdot i \]

      9. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{b \cdot t}\right) \cdot i \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, b \cdot t\right)} \cdot i \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, b \cdot t\right) \cdot i \]

      12. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, b \cdot t\right) \cdot i \]

      13. lower-*.f64 46.2

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot t}\right) \cdot i \]

   5. Applied rewrites 46.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i} \]

   6. Taylor expanded in b around inf

      \[\leadsto \left(b \cdot t\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 27.6%

      \[\leadsto \left(b \cdot t\right) \cdot i \]

   if 6.5e38 < z

   1. Initial program 68.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 47.3

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 45.4%

      \[\leadsto \left(z \cdot y\right) \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 23.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+123}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x 2.1e+123) (* (* z y) x) (* (* z x) y)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= 2.1e+123) {
		tmp = (z * y) * x;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (x <= 2.1d+123) then
        tmp = (z * y) * x
    else
        tmp = (z * x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= 2.1e+123) {
		tmp = (z * y) * x;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if x <= 2.1e+123:
		tmp = (z * y) * x
	else:
		tmp = (z * x) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= 2.1e+123)
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = Float64(Float64(z * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (x <= 2.1e+123)
		tmp = (z * y) * x;
	else
		tmp = (z * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, 2.1e+123], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.09999999999999994e123

   1. Initial program 73.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 29.9

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 17.6%

      \[\leadsto \left(z \cdot y\right) \cdot x \]

   if 2.09999999999999994e123 < x

   1. Initial program 70.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(j \cdot i\right)} + x \cdot z\right) \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right) \cdot i} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 60.0

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 56.6%

      \[\leadsto \left(z \cdot x\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 23.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot y\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* z y) x))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * y) * x;
}

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 = (z * y) * x
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 (z * y) * x;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (z * y) * x

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(z * y) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (z * y) * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

   5. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

   6. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

   9. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   11. lower-*.f64 37.9

       \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

5. Applied rewrites 37.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

6. Taylor expanded in a around 0

   \[\leadsto \left(y \cdot z\right) \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 22.9%

   \[\leadsto \left(z \cdot y\right) \cdot x \]
9. Add Preprocessing

Developer Target 1: 59.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_2 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + t\_1\\ \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* c a) (* y i))))
        (t_2
         (+
          (-
           (* x (- (* y z) (* t a)))
           (/
            (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0)))
            (+ (* c z) (* t i))))
          t_1)))
   (if (< x -1.469694296777705e-64)
     t_2
     (if (< x 3.2113527362226803e-147)
       (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) t_1))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (pow((c * z), 2.0) - pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = j * ((c * a) - (y * i))
    t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ** 2.0d0) - ((t * i) ** 2.0d0))) / ((c * z) + (t * i)))) + t_1
    if (x < (-1.469694296777705d-64)) then
        tmp = t_2
    else if (x < 3.2113527362226803d-147) then
        tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (Math.pow((c * z), 2.0) - Math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((c * a) - (y * i))
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (math.pow((c * z), 2.0) - math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1
	tmp = 0
	if x < -1.469694296777705e-64:
		tmp = t_2
	elif x < 3.2113527362226803e-147:
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_2 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(b * Float64((Float64(c * z) ^ 2.0) - (Float64(t * i) ^ 2.0))) / Float64(Float64(c * z) + Float64(t * i)))) + t_1)
	tmp = 0.0
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = Float64(Float64(Float64(Float64(b * i) - Float64(x * a)) * t) - Float64(Float64(z * Float64(c * b)) - t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * ((c * a) - (y * i));
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ^ 2.0) - ((t * i) ^ 2.0))) / ((c * z) + (t * i)))) + t_1;
	tmp = 0.0;
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[Power[N[(c * z), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(t * i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Less[x, -1.469694296777705e-64], t$95$2, If[Less[x, 3.2113527362226803e-147], N[(N[(N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] - N[(N[(z * N[(c * b), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024268 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -293938859355541/2000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 32113527362226803/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))