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

Percentage Accurate: 73.8% → 84.5%

Time: 15.3s

Alternatives: 22

Speedup: 1.0×

• 58.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 84.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.1%

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

   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 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. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 65.1%

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

4. Final simplification 85.6%

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

Alternative 2: 74.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   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 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. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 65.1%

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

4. Final simplification 78.1%

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

Alternative 3: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.02 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-a, x, i \cdot b\right), t, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, j \cdot c\right), a, \mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i 1.02e+92)
   (fma
    (fma (- i) y (* c a))
    j
    (fma (fma (- a) x (* i b)) t (* (fma (- b) c (* y x)) z)))
   (fma (fma (- x) t (* j c)) a (* (fma (- j) y (* b t)) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= 1.02e+92) {
		tmp = fma(fma(-i, y, (c * a)), j, fma(fma(-a, x, (i * b)), t, (fma(-b, c, (y * x)) * z)));
	} else {
		tmp = fma(fma(-x, t, (j * c)), a, (fma(-j, y, (b * t)) * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= 1.02e+92)
		tmp = fma(fma(Float64(-i), y, Float64(c * a)), j, fma(fma(Float64(-a), x, Float64(i * b)), t, Float64(fma(Float64(-b), c, Float64(y * x)) * z)));
	else
		tmp = fma(fma(Float64(-x), t, Float64(j * c)), a, Float64(fma(Float64(-j), y, Float64(b * t)) * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, 1.02e+92], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[((-j) * y + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.02000000000000003e92

   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 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. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 83.6%

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

   if 1.02000000000000003e92 < i

   1. Initial program 56.0%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 79.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.02 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-a, x, i \cdot b\right), t, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, j \cdot c\right), a, \mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, t, j \cdot c\right)\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{if}\;b \leq -100000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq -7.5 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, t\_1 \cdot a\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, a, \mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+159}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, \frac{i}{a}, -x\right) \cdot a\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- x) t (* j c)))
        (t_2 (fma (fma (- z) c (* i t)) b (* (fma (- i) y (* c a)) j))))
   (if (<= b -100000.0)
     t_2
     (if (<= b -7.5e-246)
       (fma (* y x) z (* t_1 a))
       (if (<= b 1.3e+106)
         (fma t_1 a (* (fma (- j) y (* b t)) i))
         (if (<= b 1.1e+159) (* (* (fma b (/ i a) (- x)) a) t) 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, t, (j * c));
	double t_2 = fma(fma(-z, c, (i * t)), b, (fma(-i, y, (c * a)) * j));
	double tmp;
	if (b <= -100000.0) {
		tmp = t_2;
	} else if (b <= -7.5e-246) {
		tmp = fma((y * x), z, (t_1 * a));
	} else if (b <= 1.3e+106) {
		tmp = fma(t_1, a, (fma(-j, y, (b * t)) * i));
	} else if (b <= 1.1e+159) {
		tmp = (fma(b, (i / a), -x) * a) * t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-x), t, Float64(j * c))
	t_2 = fma(fma(Float64(-z), c, Float64(i * t)), b, Float64(fma(Float64(-i), y, Float64(c * a)) * j))
	tmp = 0.0
	if (b <= -100000.0)
		tmp = t_2;
	elseif (b <= -7.5e-246)
		tmp = fma(Float64(y * x), z, Float64(t_1 * a));
	elseif (b <= 1.3e+106)
		tmp = fma(t_1, a, Float64(fma(Float64(-j), y, Float64(b * t)) * i));
	elseif (b <= 1.1e+159)
		tmp = Float64(Float64(fma(b, Float64(i / a), Float64(-x)) * a) * t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b + N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -100000.0], t$95$2, If[LessEqual[b, -7.5e-246], N[(N[(y * x), $MachinePrecision] * z + N[(t$95$1 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3e+106], N[(t$95$1 * a + N[(N[((-j) * y + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e+159], N[(N[(N[(b * N[(i / a), $MachinePrecision] + (-x)), $MachinePrecision] * a), $MachinePrecision] * t), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1e5 or 1.1e159 < b

   1. Initial program 76.0%

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

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.9%

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

   if -1e5 < b < -7.50000000000000049e-246

   1. Initial program 75.2%

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

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 80.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 72.7%

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

   if -7.50000000000000049e-246 < b < 1.3000000000000001e106

   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 c around 0

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 80.2%

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

   if 1.3000000000000001e106 < b < 1.1e159

   1. Initial program 44.3%

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

   3. Taylor expanded in t around 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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 89.9%

      \[\leadsto \left(\mathsf{fma}\left(b, \frac{i}{a}, -x\right) \cdot a\right) \cdot t \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -100000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{elif}\;b \leq -7.5 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, j \cdot c\right), a, \mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+159}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, \frac{i}{a}, -x\right) \cdot a\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, t, j \cdot c\right)\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, t\_1 \cdot a\right)\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, a, \mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- x) t (* j c)))
        (t_2 (fma (fma (- c) b (* y x)) z (* t_1 a))))
   (if (<= z -1.55e-5)
     t_2
     (if (<= z -4.5e-46)
       (fma (fma (- z) c (* i t)) b (* (fma (- i) y (* c a)) j))
       (if (<= z 4.6e-36) (fma t_1 a (* (fma (- j) y (* b t)) i)) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-x, t, (j * c));
	double t_2 = fma(fma(-c, b, (y * x)), z, (t_1 * a));
	double tmp;
	if (z <= -1.55e-5) {
		tmp = t_2;
	} else if (z <= -4.5e-46) {
		tmp = fma(fma(-z, c, (i * t)), b, (fma(-i, y, (c * a)) * j));
	} else if (z <= 4.6e-36) {
		tmp = fma(t_1, a, (fma(-j, y, (b * t)) * i));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-x), t, Float64(j * c))
	t_2 = fma(fma(Float64(-c), b, Float64(y * x)), z, Float64(t_1 * a))
	tmp = 0.0
	if (z <= -1.55e-5)
		tmp = t_2;
	elseif (z <= -4.5e-46)
		tmp = fma(fma(Float64(-z), c, Float64(i * t)), b, Float64(fma(Float64(-i), y, Float64(c * a)) * j));
	elseif (z <= 4.6e-36)
		tmp = fma(t_1, a, Float64(fma(Float64(-j), y, Float64(b * t)) * i));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(t$95$1 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.55e-5], t$95$2, If[LessEqual[z, -4.5e-46], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b + N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.6e-36], N[(t$95$1 * a + N[(N[((-j) * y + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.55000000000000007e-5 or 4.59999999999999993e-36 < z

   1. Initial program 69.2%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 79.2%

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

   if -1.55000000000000007e-5 < z < -4.50000000000000001e-46

   1. Initial program 84.5%

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

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   if -4.50000000000000001e-46 < z < 4.59999999999999993e-36

   1. Initial program 74.3%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.2%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, j \cdot c\right), a, \mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \left(\left(-x\right) \cdot t\right) \cdot a\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -2.4e-39)
   (fma (fma (- c) b (* y x)) z (* (* (- x) t) a))
   (if (<= x 1.2e+56)
     (fma (fma (- z) c (* i t)) b (* (fma (- i) y (* c a)) j))
     (* (fma (- t) a (* 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 (x <= -2.4e-39) {
		tmp = fma(fma(-c, b, (y * x)), z, ((-x * t) * a));
	} else if (x <= 1.2e+56) {
		tmp = fma(fma(-z, c, (i * t)), b, (fma(-i, y, (c * a)) * j));
	} else {
		tmp = fma(-t, a, (z * y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -2.4e-39)
		tmp = fma(fma(Float64(-c), b, Float64(y * x)), z, Float64(Float64(Float64(-x) * t) * a));
	elseif (x <= 1.2e+56)
		tmp = fma(fma(Float64(-z), c, Float64(i * t)), b, Float64(fma(Float64(-i), y, Float64(c * a)) * j));
	else
		tmp = Float64(fma(Float64(-t), a, Float64(z * y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -2.4e-39], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-x) * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e+56], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b + N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.40000000000000016e-39

   1. Initial program 75.9%

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

   3. Taylor expanded in t around 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. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 73.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 70.5%

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

   if -2.40000000000000016e-39 < x < 1.20000000000000007e56

   1. Initial program 73.9%

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

   3. Taylor expanded in t around 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. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.9%

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

   if 1.20000000000000007e56 < x

   1. Initial program 58.8%

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

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 72.3

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

   8. Applied rewrites 72.3%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, b, y \cdot x\right), z, \left(\left(-x\right) \cdot t\right) \cdot a\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{+35} \lor \neg \left(b \leq 4.6 \cdot 10^{+141}\right):\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= b -6.8e+35) (not (<= b 4.6e+141)))
   (* (fma (- z) c (* i t)) b)
   (fma (* y x) z (* (fma (- x) t (* j c)) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((b <= -6.8e+35) || !(b <= 4.6e+141)) {
		tmp = fma(-z, c, (i * t)) * b;
	} else {
		tmp = fma((y * x), z, (fma(-x, t, (j * c)) * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((b <= -6.8e+35) || !(b <= 4.6e+141))
		tmp = Float64(fma(Float64(-z), c, Float64(i * t)) * b);
	else
		tmp = fma(Float64(y * x), z, Float64(fma(Float64(-x), t, Float64(j * c)) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[b, -6.8e+35], N[Not[LessEqual[b, 4.6e+141]], $MachinePrecision]], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * z + N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.8000000000000002e35 or 4.6000000000000003e141 < b

   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 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. lower-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   if -6.8000000000000002e35 < b < 4.6000000000000003e141

   1. Initial program 69.6%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 72.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 67.9%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{+35} \lor \neg \left(b \leq 4.6 \cdot 10^{+141}\right):\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-64} \lor \neg \left(t \leq 1.9 \cdot 10^{+116}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(b, \frac{i}{a}, -x\right) \cdot a\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= t -3.3e-64) (not (<= t 1.9e+116)))
   (* (* (fma b (/ i a) (- x)) a) t)
   (fma (* y x) z (* (fma (- b) z (* j a)) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((t <= -3.3e-64) || !(t <= 1.9e+116)) {
		tmp = (fma(b, (i / a), -x) * a) * t;
	} else {
		tmp = fma((y * x), z, (fma(-b, z, (j * a)) * c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((t <= -3.3e-64) || !(t <= 1.9e+116))
		tmp = Float64(Float64(fma(b, Float64(i / a), Float64(-x)) * a) * t);
	else
		tmp = fma(Float64(y * x), z, Float64(fma(Float64(-b), z, Float64(j * a)) * c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[t, -3.3e-64], N[Not[LessEqual[t, 1.9e+116]], $MachinePrecision]], N[(N[(N[(b * N[(i / a), $MachinePrecision] + (-x)), $MachinePrecision] * a), $MachinePrecision] * t), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * z + N[(N[((-b) * z + N[(j * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.2999999999999999e-64 or 1.8999999999999999e116 < t

   1. Initial program 63.8%

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

   3. Taylor expanded in t around 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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 72.4%

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

   if -3.2999999999999999e-64 < t < 1.8999999999999999e116

   1. Initial program 78.6%

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

   3. Taylor expanded in i around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

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

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

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

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 62.4%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-64} \lor \neg \left(t \leq 1.9 \cdot 10^{+116}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(b, \frac{i}{a}, -x\right) \cdot a\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-64} \lor \neg \left(t \leq 1.9 \cdot 10^{+116}\right):\\ \;\;\;\;\mathsf{fma}\left(i, b, \left(-a\right) \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= t -3.3e-64) (not (<= t 1.9e+116)))
   (* (fma i b (* (- a) x)) t)
   (fma (* y x) z (* (fma (- b) z (* j a)) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((t <= -3.3e-64) || !(t <= 1.9e+116)) {
		tmp = fma(i, b, (-a * x)) * t;
	} else {
		tmp = fma((y * x), z, (fma(-b, z, (j * a)) * c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((t <= -3.3e-64) || !(t <= 1.9e+116))
		tmp = Float64(fma(i, b, Float64(Float64(-a) * x)) * t);
	else
		tmp = fma(Float64(y * x), z, Float64(fma(Float64(-b), z, Float64(j * a)) * c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[t, -3.3e-64], N[Not[LessEqual[t, 1.9e+116]], $MachinePrecision]], N[(N[(i * b + N[((-a) * x), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * z + N[(N[((-b) * z + N[(j * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.2999999999999999e-64 or 1.8999999999999999e116 < t

   1. Initial program 63.8%

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

   3. Taylor expanded in t around 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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

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

   if -3.2999999999999999e-64 < t < 1.8999999999999999e116

   1. Initial program 78.6%

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

   3. Taylor expanded in i around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

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

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

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

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 62.4%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-64} \lor \neg \left(t \leq 1.9 \cdot 10^{+116}\right):\\ \;\;\;\;\mathsf{fma}\left(i, b, \left(-a\right) \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+171}:\\ \;\;\;\;\mathsf{fma}\left(i, b, \left(-a\right) \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -4.5e+35)
   (* (fma (- z) c (* i t)) b)
   (if (<= b 6.5e+87)
     (* (fma (- x) t (* j c)) a)
     (if (<= b 7.5e+171)
       (* (fma i b (* (- a) x)) t)
       (* (fma (- b) c (* 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 (b <= -4.5e+35) {
		tmp = fma(-z, c, (i * t)) * b;
	} else if (b <= 6.5e+87) {
		tmp = fma(-x, t, (j * c)) * a;
	} else if (b <= 7.5e+171) {
		tmp = fma(i, b, (-a * x)) * t;
	} else {
		tmp = fma(-b, c, (y * x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -4.5e+35)
		tmp = Float64(fma(Float64(-z), c, Float64(i * t)) * b);
	elseif (b <= 6.5e+87)
		tmp = Float64(fma(Float64(-x), t, Float64(j * c)) * a);
	elseif (b <= 7.5e+171)
		tmp = Float64(fma(i, b, Float64(Float64(-a) * x)) * t);
	else
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -4.5e+35], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[b, 6.5e+87], N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[b, 7.5e+171], N[(N[(i * b + N[((-a) * x), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.4999999999999997e35

   1. Initial program 73.4%

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

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

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if -4.4999999999999997e35 < b < 6.5000000000000002e87

   1. Initial program 70.1%

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

   3. Taylor expanded in 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. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

   if 6.5000000000000002e87 < b < 7.4999999999999998e171

   1. Initial program 57.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 86.3%

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

   if 7.4999999999999998e171 < b

   1. Initial program 81.4%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+171}:\\ \;\;\;\;\mathsf{fma}\left(i, b, \left(-a\right) \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(i, b, \left(-a\right) \cdot x\right) \cdot t\\ \mathbf{elif}\;t \leq -9.8 \cdot 10^{-207}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -6.6e-55)
   (* (fma i b (* (- a) x)) t)
   (if (<= t -9.8e-207)
     (* (fma (- i) j (* z x)) y)
     (if (<= t 5.5e-24)
       (* (fma (- b) c (* y x)) z)
       (* (fma (- a) x (* i b)) t)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= -6.6e-55) {
		tmp = fma(i, b, (-a * x)) * t;
	} else if (t <= -9.8e-207) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (t <= 5.5e-24) {
		tmp = fma(-b, c, (y * x)) * z;
	} else {
		tmp = fma(-a, x, (i * b)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -6.6e-55)
		tmp = Float64(fma(i, b, Float64(Float64(-a) * x)) * t);
	elseif (t <= -9.8e-207)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (t <= 5.5e-24)
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	else
		tmp = Float64(fma(Float64(-a), x, Float64(i * b)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -6.6e-55], N[(N[(i * b + N[((-a) * x), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t, -9.8e-207], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t, 5.5e-24], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -6.5999999999999999e-55

   1. Initial program 61.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 66.9

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.9%

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

   if -6.5999999999999999e-55 < t < -9.7999999999999999e-207

   1. Initial program 84.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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. mul-1-neg N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 55.8

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

   5. Applied rewrites 55.8%

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

   if -9.7999999999999999e-207 < t < 5.4999999999999999e-24

   1. Initial program 78.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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. cancel-sign-sub-inv N/A

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

   if 5.4999999999999999e-24 < t

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

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(i, b, \left(-a\right) \cdot x\right) \cdot t\\ \mathbf{elif}\;t \leq -9.8 \cdot 10^{-207}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-27} \lor \neg \left(a \leq 4.2 \cdot 10^{+35}\right):\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= a -5.3e-27) (not (<= a 4.2e+35)))
   (* (fma (- x) t (* j c)) a)
   (* (fma (- b) c (* 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 ((a <= -5.3e-27) || !(a <= 4.2e+35)) {
		tmp = fma(-x, t, (j * c)) * a;
	} else {
		tmp = fma(-b, c, (y * x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((a <= -5.3e-27) || !(a <= 4.2e+35))
		tmp = Float64(fma(Float64(-x), t, Float64(j * c)) * a);
	else
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[a, -5.3e-27], N[Not[LessEqual[a, 4.2e+35]], $MachinePrecision]], N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.30000000000000006e-27 or 4.1999999999999998e35 < a

   1. Initial program 65.6%

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

   3. Taylor expanded in 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. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   if -5.30000000000000006e-27 < a < 4.1999999999999998e35

   1. Initial program 78.3%

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

   3. Taylor expanded in 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. cancel-sign-sub-inv N/A

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-27} \lor \neg \left(a \leq 4.2 \cdot 10^{+35}\right):\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-70} \lor \neg \left(t \leq 2.45 \cdot 10^{-118}\right):\\ \;\;\;\;\mathsf{fma}\left(i, b, \left(-a\right) \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= t -1.55e-70) (not (<= t 2.45e-118)))
   (* (fma i b (* (- a) x)) t)
   (* (fma c a (* (- i) y)) j)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((t <= -1.55e-70) || !(t <= 2.45e-118)) {
		tmp = fma(i, b, (-a * x)) * t;
	} else {
		tmp = fma(c, a, (-i * y)) * j;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((t <= -1.55e-70) || !(t <= 2.45e-118))
		tmp = Float64(fma(i, b, Float64(Float64(-a) * x)) * t);
	else
		tmp = Float64(fma(c, a, Float64(Float64(-i) * y)) * j);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[t, -1.55e-70], N[Not[LessEqual[t, 2.45e-118]], $MachinePrecision]], N[(N[(i * b + N[((-a) * x), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(c * a + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.55e-70 or 2.4499999999999999e-118 < t

   1. Initial program 66.4%

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

   3. Taylor expanded in t around 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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 64.3

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

   5. Applied rewrites 64.3%

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

   7. Applied rewrites 64.3%

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

   if -1.55e-70 < t < 2.4499999999999999e-118

   1. Initial program 82.8%

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

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 49.3%

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

   10. Applied rewrites 49.3%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-70} \lor \neg \left(t \leq 2.45 \cdot 10^{-118}\right):\\ \;\;\;\;\mathsf{fma}\left(i, b, \left(-a\right) \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 41.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{+232}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (- a) (* t x))))
   (if (<= t -2.5e+232)
     t_1
     (if (<= t 2.6e-23)
       (* (fma c a (* (- i) y)) j)
       (if (<= t 1.8e+25) t_1 (* (* b t) i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = -a * (t * x);
	double tmp;
	if (t <= -2.5e+232) {
		tmp = t_1;
	} else if (t <= 2.6e-23) {
		tmp = fma(c, a, (-i * y)) * j;
	} else if (t <= 1.8e+25) {
		tmp = t_1;
	} else {
		tmp = (b * t) * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(-a) * Float64(t * x))
	tmp = 0.0
	if (t <= -2.5e+232)
		tmp = t_1;
	elseif (t <= 2.6e-23)
		tmp = Float64(fma(c, a, Float64(Float64(-i) * y)) * j);
	elseif (t <= 1.8e+25)
		tmp = t_1;
	else
		tmp = Float64(Float64(b * t) * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.5e+232], t$95$1, If[LessEqual[t, 2.6e-23], N[(N[(c * a + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[t, 1.8e+25], t$95$1, N[(N[(b * t), $MachinePrecision] * i), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.49999999999999993e232 or 2.6e-23 < t < 1.80000000000000008e25

   1. Initial program 65.3%

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

   3. Taylor expanded in t around 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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.3%

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

   if -2.49999999999999993e232 < t < 2.6e-23

   1. Initial program 74.5%

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

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 42.1%

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

   10. Applied rewrites 42.1%

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

   if 1.80000000000000008e25 < t

   1. Initial program 69.2%

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

   3. Taylor expanded in 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. fp-cancel-sub-sign-inv N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(b \cdot t\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 47.1%

      \[\leadsto \left(b \cdot t\right) \cdot i \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+232}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(c, a, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+25}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b \cdot t\right) \cdot i\\ \mathbf{if}\;b \leq -3.6 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+143}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{+198}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot z\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* b t) i)))
   (if (<= b -3.6e+36)
     t_1
     (if (<= b 3.4e+143)
       (* (- a) (* t x))
       (if (<= b 2.05e+198) t_1 (* (* (- b) z) c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (b * t) * i;
	double tmp;
	if (b <= -3.6e+36) {
		tmp = t_1;
	} else if (b <= 3.4e+143) {
		tmp = -a * (t * x);
	} else if (b <= 2.05e+198) {
		tmp = t_1;
	} else {
		tmp = (-b * z) * c;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (b * t) * i
    if (b <= (-3.6d+36)) then
        tmp = t_1
    else if (b <= 3.4d+143) then
        tmp = -a * (t * x)
    else if (b <= 2.05d+198) then
        tmp = t_1
    else
        tmp = (-b * z) * c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (b * t) * i;
	double tmp;
	if (b <= -3.6e+36) {
		tmp = t_1;
	} else if (b <= 3.4e+143) {
		tmp = -a * (t * x);
	} else if (b <= 2.05e+198) {
		tmp = t_1;
	} else {
		tmp = (-b * z) * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (b * t) * i
	tmp = 0
	if b <= -3.6e+36:
		tmp = t_1
	elif b <= 3.4e+143:
		tmp = -a * (t * x)
	elif b <= 2.05e+198:
		tmp = t_1
	else:
		tmp = (-b * z) * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(b * t) * i)
	tmp = 0.0
	if (b <= -3.6e+36)
		tmp = t_1;
	elseif (b <= 3.4e+143)
		tmp = Float64(Float64(-a) * Float64(t * x));
	elseif (b <= 2.05e+198)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-b) * z) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (b * t) * i;
	tmp = 0.0;
	if (b <= -3.6e+36)
		tmp = t_1;
	elseif (b <= 3.4e+143)
		tmp = -a * (t * x);
	elseif (b <= 2.05e+198)
		tmp = t_1;
	else
		tmp = (-b * z) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(b * t), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[b, -3.6e+36], t$95$1, If[LessEqual[b, 3.4e+143], N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.05e+198], t$95$1, N[(N[((-b) * z), $MachinePrecision] * c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5999999999999997e36 or 3.39999999999999982e143 < b < 2.0500000000000001e198

   1. Initial program 72.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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. fp-cancel-sub-sign-inv N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(b \cdot t\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 50.9%

      \[\leadsto \left(b \cdot t\right) \cdot i \]

   if -3.5999999999999997e36 < b < 3.39999999999999982e143

   1. Initial program 69.6%

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

   3. Taylor expanded in 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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 43.2

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 38.2%

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

   if 2.0500000000000001e198 < b

   1. Initial program 82.5%

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

   3. Taylor expanded in 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. fp-cancel-sub-sign-inv N/A

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 49.2%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+36}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+143}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{+198}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot z\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b \cdot t\right) \cdot i\\ \mathbf{if}\;b \leq -3.6 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+143}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+248}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(c \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* b t) i)))
   (if (<= b -3.6e+36)
     t_1
     (if (<= b 3.4e+143)
       (* (- a) (* t x))
       (if (<= b 7e+248) t_1 (* (- 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 t_1 = (b * t) * i;
	double tmp;
	if (b <= -3.6e+36) {
		tmp = t_1;
	} else if (b <= 3.4e+143) {
		tmp = -a * (t * x);
	} else if (b <= 7e+248) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (b * t) * i
    if (b <= (-3.6d+36)) then
        tmp = t_1
    else if (b <= 3.4d+143) then
        tmp = -a * (t * x)
    else if (b <= 7d+248) then
        tmp = t_1
    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 t_1 = (b * t) * i;
	double tmp;
	if (b <= -3.6e+36) {
		tmp = t_1;
	} else if (b <= 3.4e+143) {
		tmp = -a * (t * x);
	} else if (b <= 7e+248) {
		tmp = t_1;
	} else {
		tmp = -b * (c * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (b * t) * i
	tmp = 0
	if b <= -3.6e+36:
		tmp = t_1
	elif b <= 3.4e+143:
		tmp = -a * (t * x)
	elif b <= 7e+248:
		tmp = t_1
	else:
		tmp = -b * (c * z)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(b * t) * i)
	tmp = 0.0
	if (b <= -3.6e+36)
		tmp = t_1;
	elseif (b <= 3.4e+143)
		tmp = Float64(Float64(-a) * Float64(t * x));
	elseif (b <= 7e+248)
		tmp = t_1;
	else
		tmp = Float64(Float64(-b) * Float64(c * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (b * t) * i;
	tmp = 0.0;
	if (b <= -3.6e+36)
		tmp = t_1;
	elseif (b <= 3.4e+143)
		tmp = -a * (t * x);
	elseif (b <= 7e+248)
		tmp = t_1;
	else
		tmp = -b * (c * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(b * t), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[b, -3.6e+36], t$95$1, If[LessEqual[b, 3.4e+143], N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e+248], t$95$1, N[((-b) * N[(c * z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5999999999999997e36 or 3.39999999999999982e143 < b < 7.00000000000000044e248

   1. Initial program 72.8%

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

   3. Taylor expanded in 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. fp-cancel-sub-sign-inv N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(b \cdot t\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 47.5%

      \[\leadsto \left(b \cdot t\right) \cdot i \]

   if -3.5999999999999997e36 < b < 3.39999999999999982e143

   1. Initial program 69.6%

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

   3. Taylor expanded in 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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 43.2

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 38.2%

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

   if 7.00000000000000044e248 < b

   1. Initial program 91.4%

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

   3. Taylor expanded in c around inf

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 66.9%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+36}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+143}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+248}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(c \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(i, b, \left(-a\right) \cdot x\right) \cdot t\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -3.3e-64)
   (* (fma i b (* (- a) x)) t)
   (if (<= t 5.5e-24)
     (* (fma (- b) c (* y x)) z)
     (* (fma (- a) x (* i b)) t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= -3.3e-64) {
		tmp = fma(i, b, (-a * x)) * t;
	} else if (t <= 5.5e-24) {
		tmp = fma(-b, c, (y * x)) * z;
	} else {
		tmp = fma(-a, x, (i * b)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -3.3e-64)
		tmp = Float64(fma(i, b, Float64(Float64(-a) * x)) * t);
	elseif (t <= 5.5e-24)
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	else
		tmp = Float64(fma(Float64(-a), x, Float64(i * b)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -3.3e-64], N[(N[(i * b + N[((-a) * x), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t, 5.5e-24], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.2999999999999999e-64

   1. Initial program 61.8%

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

   3. Taylor expanded in t around 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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   7. Applied rewrites 66.1%

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

   if -3.2999999999999999e-64 < t < 5.4999999999999999e-24

   1. Initial program 80.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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. cancel-sign-sub-inv N/A

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 50.1

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

   5. Applied rewrites 50.1%

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

   if 5.4999999999999999e-24 < t

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

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(i, b, \left(-a\right) \cdot x\right) \cdot t\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+36}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{elif}\;b \leq 10^{+144}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -3.6e+36)
   (* (* b t) i)
   (if (<= b 1e+144) (* (- a) (* t x)) (* (* i b) t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -3.6e+36) {
		tmp = (b * t) * i;
	} else if (b <= 1e+144) {
		tmp = -a * (t * x);
	} else {
		tmp = (i * b) * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (b <= (-3.6d+36)) then
        tmp = (b * t) * i
    else if (b <= 1d+144) then
        tmp = -a * (t * x)
    else
        tmp = (i * b) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -3.6e+36) {
		tmp = (b * t) * i;
	} else if (b <= 1e+144) {
		tmp = -a * (t * x);
	} else {
		tmp = (i * b) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -3.6e+36:
		tmp = (b * t) * i
	elif b <= 1e+144:
		tmp = -a * (t * x)
	else:
		tmp = (i * b) * t
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -3.6e+36)
		tmp = Float64(Float64(b * t) * i);
	elseif (b <= 1e+144)
		tmp = Float64(Float64(-a) * Float64(t * x));
	else
		tmp = Float64(Float64(i * b) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -3.6e+36)
		tmp = (b * t) * i;
	elseif (b <= 1e+144)
		tmp = -a * (t * x);
	else
		tmp = (i * b) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -3.6e+36], N[(N[(b * t), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[b, 1e+144], N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5999999999999997e36

   1. Initial program 73.4%

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

   3. Taylor expanded in i around inf

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(b \cdot t\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 45.8%

      \[\leadsto \left(b \cdot t\right) \cdot i \]

   if -3.5999999999999997e36 < b < 1.00000000000000002e144

   1. Initial program 69.6%

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

   3. Taylor expanded in 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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 43.2

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 38.2%

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

   if 1.00000000000000002e144 < b

   1. Initial program 77.1%

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

   3. Taylor expanded in t around inf

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

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

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 55.4

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 44.5%

      \[\leadsto \left(i \cdot b\right) \cdot t \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+36}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{elif}\;b \leq 10^{+144}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 30.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-6} \lor \neg \left(z \leq 2.1 \cdot 10^{-54}\right):\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -4.2e-6) (not (<= z 2.1e-54))) (* (* z y) x) (* (* i b) t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((z <= -4.2e-6) || !(z <= 2.1e-54)) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * b) * t;
	}
	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 <= (-4.2d-6)) .or. (.not. (z <= 2.1d-54))) then
        tmp = (z * y) * x
    else
        tmp = (i * b) * t
    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 <= -4.2e-6) || !(z <= 2.1e-54)) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * b) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (z <= -4.2e-6) or not (z <= 2.1e-54):
		tmp = (z * y) * x
	else:
		tmp = (i * b) * t
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -4.2e-6) || !(z <= 2.1e-54))
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = Float64(Float64(i * b) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((z <= -4.2e-6) || ~((z <= 2.1e-54)))
		tmp = (z * y) * x;
	else
		tmp = (i * b) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -4.2e-6], N[Not[LessEqual[z, 2.1e-54]], $MachinePrecision]], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.1999999999999996e-6 or 2.1e-54 < z

   1. Initial program 69.4%

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

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 80.3%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 79.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 35.2%

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

   13. Applied rewrites 36.5%

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

   if -4.1999999999999996e-6 < z < 2.1e-54

   1. Initial program 75.6%

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

   3. Taylor expanded in t around 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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 40.6%

      \[\leadsto \left(i \cdot b\right) \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-6} \lor \neg \left(z \leq 2.1 \cdot 10^{-54}\right):\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 30.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-7} \lor \neg \left(z \leq 2.1 \cdot 10^{-54}\right):\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -9e-7) (not (<= z 2.1e-54))) (* (* z y) x) (* (* i t) b)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((z <= -9e-7) || !(z <= 2.1e-54)) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * t) * 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 <= (-9d-7)) .or. (.not. (z <= 2.1d-54))) then
        tmp = (z * y) * x
    else
        tmp = (i * t) * 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 <= -9e-7) || !(z <= 2.1e-54)) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * t) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (z <= -9e-7) or not (z <= 2.1e-54):
		tmp = (z * y) * x
	else:
		tmp = (i * t) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -9e-7) || !(z <= 2.1e-54))
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = Float64(Float64(i * t) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((z <= -9e-7) || ~((z <= 2.1e-54)))
		tmp = (z * y) * x;
	else
		tmp = (i * t) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -9e-7], N[Not[LessEqual[z, 2.1e-54]], $MachinePrecision]], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999959e-7 or 2.1e-54 < z

   1. Initial program 69.4%

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

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 80.3%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 79.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 35.2%

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

   13. Applied rewrites 36.5%

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

   if -8.99999999999999959e-7 < z < 2.1e-54

   1. Initial program 75.6%

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

   3. Taylor expanded in t around 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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 38.3%

      \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-7} \lor \neg \left(z \leq 2.1 \cdot 10^{-54}\right):\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 30.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-7} \lor \neg \left(z \leq 1.46 \cdot 10^{-55}\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -9e-7) (not (<= z 1.46e-55))) (* (* y x) z) (* (* i t) b)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((z <= -9e-7) || !(z <= 1.46e-55)) {
		tmp = (y * x) * z;
	} else {
		tmp = (i * t) * 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 <= (-9d-7)) .or. (.not. (z <= 1.46d-55))) then
        tmp = (y * x) * z
    else
        tmp = (i * t) * 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 <= -9e-7) || !(z <= 1.46e-55)) {
		tmp = (y * x) * z;
	} else {
		tmp = (i * t) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (z <= -9e-7) or not (z <= 1.46e-55):
		tmp = (y * x) * z
	else:
		tmp = (i * t) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -9e-7) || !(z <= 1.46e-55))
		tmp = Float64(Float64(y * x) * z);
	else
		tmp = Float64(Float64(i * t) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((z <= -9e-7) || ~((z <= 1.46e-55)))
		tmp = (y * x) * z;
	else
		tmp = (i * t) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -9e-7], N[Not[LessEqual[z, 1.46e-55]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999959e-7 or 1.46000000000000009e-55 < z

   1. Initial program 69.4%

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

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

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

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

   5. Applied rewrites 80.3%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 79.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 35.2%

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

   if -8.99999999999999959e-7 < z < 1.46000000000000009e-55

   1. Initial program 75.6%

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

   3. Taylor expanded in t around 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. fp-cancel-sub-sign-inv N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 38.3%

      \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-7} \lor \neg \left(z \leq 1.46 \cdot 10^{-55}\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 22.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(i \cdot t\right) \cdot b \end{array} \]

FPCore

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

C

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

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 = (i * t) * b
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 (i * t) * b;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]

Derivation

1. Initial program 71.9%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
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. fp-cancel-sub-sign-inv N/A

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

   4. associate-*r* N/A

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

   5. mul-1-neg N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 47.9

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

5. Applied rewrites 47.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 24.3%

   \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]

9. Final simplification 24.3%

   \[\leadsto \left(i \cdot t\right) \cdot b \]
10. 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 2024326 
(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)))))