Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 73.1% → 83.4%

Time: 21.3s

Alternatives: 29

Speedup: 0.9×

• 58.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+301}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (+ (* x (- (* y z) (* t a))) (* b (- (* a i) (* z c))))
          (* j (- (* t c) (* y i))))))
   (if (<= t_1 5e+301)
     t_1
     (fma
      i
      (fma b a (* y (- j)))
      (fma t (fma j c (* x (- a))) (* z (fma c (- b) (* x y))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) + (b * ((a * i) - (z * c)))) + (j * ((t * c) - (y * i)));
	double tmp;
	if (t_1 <= 5e+301) {
		tmp = t_1;
	} else {
		tmp = fma(i, fma(b, a, (y * -j)), fma(t, fma(j, c, (x * -a)), (z * fma(c, -b, (x * y)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) + Float64(b * Float64(Float64(a * i) - Float64(z * c)))) + Float64(j * Float64(Float64(t * c) - Float64(y * i))))
	tmp = 0.0
	if (t_1 <= 5e+301)
		tmp = t_1;
	else
		tmp = fma(i, fma(b, a, Float64(y * Float64(-j))), fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(z * fma(c, Float64(-b), Float64(x * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+301], t$95$1, N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision] + N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < 5.0000000000000004e301

   1. Initial program 96.6%

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

   if 5.0000000000000004e301 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y))))

   1. Initial program 39.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 67.2%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right) \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ t_2 := \mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), \mathsf{fma}\left(t, t\_1, z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\right)\\ \mathbf{if}\;i \leq -1.6 \cdot 10^{-31}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;i \leq 3.2 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(t, t\_1, \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma j c (* x (- a))))
        (t_2
         (fma
          i
          (fma b a (* y (- j)))
          (fma t t_1 (* z (fma c (- b) (* x y)))))))
   (if (<= i -1.6e-31)
     t_2
     (if (<= i 3.2e+86)
       (fma t t_1 (fma y (fma j (- i) (* x z)) (* b (fma c (- z) (* a i)))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(j, c, (x * -a));
	double t_2 = fma(i, fma(b, a, (y * -j)), fma(t, t_1, (z * fma(c, -b, (x * y)))));
	double tmp;
	if (i <= -1.6e-31) {
		tmp = t_2;
	} else if (i <= 3.2e+86) {
		tmp = fma(t, t_1, fma(y, fma(j, -i, (x * z)), (b * fma(c, -z, (a * i)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(j, c, Float64(x * Float64(-a)))
	t_2 = fma(i, fma(b, a, Float64(y * Float64(-j))), fma(t, t_1, Float64(z * fma(c, Float64(-b), Float64(x * y)))))
	tmp = 0.0
	if (i <= -1.6e-31)
		tmp = t_2;
	elseif (i <= 3.2e+86)
		tmp = fma(t, t_1, fma(y, fma(j, Float64(-i), Float64(x * z)), Float64(b * fma(c, Float64(-z), Float64(a * i)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision] + N[(t * t$95$1 + N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.6e-31], t$95$2, If[LessEqual[i, 3.2e+86], N[(t * t$95$1 + N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.60000000000000009e-31 or 3.2e86 < i

   1. Initial program 65.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 80.9%

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

   if -1.60000000000000009e-31 < i < 3.2e86

   1. Initial program 81.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 88.3%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.6 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\right)\\ \mathbf{elif}\;i \leq 3.2 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ t_2 := \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(t, t\_2, t\_1\right)\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), \mathsf{fma}\left(t, t\_2, z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma c (- z) (* a i)))) (t_2 (fma j c (* x (- a)))))
   (if (<= b -4.2e+115)
     (fma t t_2 t_1)
     (if (<= b 2.6e+207)
       (fma i (fma b a (* y (- j))) (fma t t_2 (* z (fma c (- b) (* x y)))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(c, -z, (a * i));
	double t_2 = fma(j, c, (x * -a));
	double tmp;
	if (b <= -4.2e+115) {
		tmp = fma(t, t_2, t_1);
	} else if (b <= 2.6e+207) {
		tmp = fma(i, fma(b, a, (y * -j)), fma(t, t_2, (z * fma(c, -b, (x * y)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(c, Float64(-z), Float64(a * i)))
	t_2 = fma(j, c, Float64(x * Float64(-a)))
	tmp = 0.0
	if (b <= -4.2e+115)
		tmp = fma(t, t_2, t_1);
	elseif (b <= 2.6e+207)
		tmp = fma(i, fma(b, a, Float64(y * Float64(-j))), fma(t, t_2, Float64(z * fma(c, Float64(-b), Float64(x * y)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.2e+115], N[(t * t$95$2 + t$95$1), $MachinePrecision], If[LessEqual[b, 2.6e+207], N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision] + N[(t * t$95$2 + N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.20000000000000007e115

   1. Initial program 66.5%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

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

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

      18. mul-1-neg N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 70.6%

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

   if -4.20000000000000007e115 < b < 2.5999999999999998e207

   1. Initial program 73.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.1%

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

   if 2.5999999999999998e207 < b

   1. Initial program 78.1%

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

   3. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ t_2 := \mathsf{fma}\left(x, y \cdot z - t \cdot a, i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\right)\\ \mathbf{if}\;i \leq -48000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;i \leq 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(t, t\_1, z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\\ \mathbf{elif}\;i \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(t, t\_1, b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 2.45 \cdot 10^{+238}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma j c (* x (- a))))
        (t_2 (fma x (- (* y z) (* t a)) (* i (fma j (- y) (* a b))))))
   (if (<= i -48000.0)
     t_2
     (if (<= i 1e+38)
       (fma t t_1 (* z (fma c (- b) (* x y))))
       (if (<= i 5e+97)
         (fma t t_1 (* b (fma c (- z) (* a i))))
         (if (<= i 2.45e+238)
           t_2
           (fma i (fma b a (* y (- j))) (* c (fma j t (* z (- b)))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(j, c, (x * -a));
	double t_2 = fma(x, ((y * z) - (t * a)), (i * fma(j, -y, (a * b))));
	double tmp;
	if (i <= -48000.0) {
		tmp = t_2;
	} else if (i <= 1e+38) {
		tmp = fma(t, t_1, (z * fma(c, -b, (x * y))));
	} else if (i <= 5e+97) {
		tmp = fma(t, t_1, (b * fma(c, -z, (a * i))));
	} else if (i <= 2.45e+238) {
		tmp = t_2;
	} else {
		tmp = fma(i, fma(b, a, (y * -j)), (c * fma(j, t, (z * -b))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(j, c, Float64(x * Float64(-a)))
	t_2 = fma(x, Float64(Float64(y * z) - Float64(t * a)), Float64(i * fma(j, Float64(-y), Float64(a * b))))
	tmp = 0.0
	if (i <= -48000.0)
		tmp = t_2;
	elseif (i <= 1e+38)
		tmp = fma(t, t_1, Float64(z * fma(c, Float64(-b), Float64(x * y))));
	elseif (i <= 5e+97)
		tmp = fma(t, t_1, Float64(b * fma(c, Float64(-z), Float64(a * i))));
	elseif (i <= 2.45e+238)
		tmp = t_2;
	else
		tmp = fma(i, fma(b, a, Float64(y * Float64(-j))), Float64(c * fma(j, t, Float64(z * Float64(-b)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -48000.0], t$95$2, If[LessEqual[i, 1e+38], N[(t * t$95$1 + N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5e+97], N[(t * t$95$1 + N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.45e+238], t$95$2, N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision] + N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -48000 or 4.99999999999999999e97 < i < 2.45000000000000013e238

   1. Initial program 64.0%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 76.4%

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

   if -48000 < i < 9.99999999999999977e37

   1. Initial program 82.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 80.7%

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

   if 9.99999999999999977e37 < i < 4.99999999999999999e97

   1. Initial program 83.2%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

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

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

      18. mul-1-neg N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 91.6%

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

   if 2.45000000000000013e238 < i

   1. Initial program 47.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 64.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 76.3%

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

   9. Taylor expanded in c around inf

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

   11. Applied rewrites 76.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -48000:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\right)\\ \mathbf{elif}\;i \leq 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\\ \mathbf{elif}\;i \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{elif}\;i \leq 2.45 \cdot 10^{+238}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ t_2 := \mathsf{fma}\left(c, t\_1, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{if}\;c \leq -4.5 \cdot 10^{-27}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), y \cdot \left(x \cdot z - i \cdot j\right)\right)\\ \mathbf{elif}\;c \leq 9.2 \cdot 10^{+188}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), c \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma j t (* z (- b))))
        (t_2 (fma c t_1 (* x (- (* y z) (* t a))))))
   (if (<= c -4.5e-27)
     t_2
     (if (<= c 7.2e+78)
       (fma t (fma j c (* x (- a))) (* y (- (* x z) (* i j))))
       (if (<= c 9.2e+188) t_2 (fma i (fma b a (* y (- j))) (* c t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(j, t, (z * -b));
	double t_2 = fma(c, t_1, (x * ((y * z) - (t * a))));
	double tmp;
	if (c <= -4.5e-27) {
		tmp = t_2;
	} else if (c <= 7.2e+78) {
		tmp = fma(t, fma(j, c, (x * -a)), (y * ((x * z) - (i * j))));
	} else if (c <= 9.2e+188) {
		tmp = t_2;
	} else {
		tmp = fma(i, fma(b, a, (y * -j)), (c * t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(j, t, Float64(z * Float64(-b)))
	t_2 = fma(c, t_1, Float64(x * Float64(Float64(y * z) - Float64(t * a))))
	tmp = 0.0
	if (c <= -4.5e-27)
		tmp = t_2;
	elseif (c <= 7.2e+78)
		tmp = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(y * Float64(Float64(x * z) - Float64(i * j))));
	elseif (c <= 9.2e+188)
		tmp = t_2;
	else
		tmp = fma(i, fma(b, a, Float64(y * Float64(-j))), Float64(c * t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c * t$95$1 + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4.5e-27], t$95$2, If[LessEqual[c, 7.2e+78], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(y * N[(N[(x * z), $MachinePrecision] - N[(i * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 9.2e+188], t$95$2, N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision] + N[(c * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.5000000000000002e-27 or 7.20000000000000039e78 < c < 9.20000000000000046e188

   1. Initial program 71.3%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 82.6%

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

   if -4.5000000000000002e-27 < c < 7.20000000000000039e78

   1. Initial program 78.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 69.6%

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

   if 9.20000000000000046e188 < c

   1. Initial program 57.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in c around inf

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

   11. Applied rewrites 84.6%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), y \cdot \left(x \cdot z - i \cdot j\right)\right)\\ \mathbf{elif}\;c \leq 9.2 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{if}\;i \leq -14000000000:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;i \leq 1.05 \cdot 10^{-168}:\\ \;\;\;\;\mathsf{fma}\left(t, t\_1, y \cdot \left(x \cdot z\right)\right)\\ \mathbf{elif}\;i \leq 7.6 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(t, t\_1, -c \cdot \left(z \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma j c (* x (- a)))))
   (if (<= i -14000000000.0)
     (+ (* j (- (* t c) (* y i))) (* i (* a b)))
     (if (<= i 1.05e-168)
       (fma t t_1 (* y (* x z)))
       (if (<= i 7.6e+98)
         (fma t t_1 (- (* c (* z b))))
         (* i (fma j (- y) (* a b))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(j, c, (x * -a));
	double tmp;
	if (i <= -14000000000.0) {
		tmp = (j * ((t * c) - (y * i))) + (i * (a * b));
	} else if (i <= 1.05e-168) {
		tmp = fma(t, t_1, (y * (x * z)));
	} else if (i <= 7.6e+98) {
		tmp = fma(t, t_1, -(c * (z * b)));
	} else {
		tmp = i * fma(j, -y, (a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(j, c, Float64(x * Float64(-a)))
	tmp = 0.0
	if (i <= -14000000000.0)
		tmp = Float64(Float64(j * Float64(Float64(t * c) - Float64(y * i))) + Float64(i * Float64(a * b)));
	elseif (i <= 1.05e-168)
		tmp = fma(t, t_1, Float64(y * Float64(x * z)));
	elseif (i <= 7.6e+98)
		tmp = fma(t, t_1, Float64(-Float64(c * Float64(z * b))));
	else
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -14000000000.0], N[(N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.05e-168], N[(t * t$95$1 + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7.6e+98], N[(t * t$95$1 + (-N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -1.4e10

   1. Initial program 65.5%

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

   3. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

   if -1.4e10 < i < 1.04999999999999997e-168

   1. Initial program 83.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 43.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 67.9%

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

   if 1.04999999999999997e-168 < i < 7.5999999999999998e98

   1. Initial program 80.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 54.5%

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

   9. Taylor expanded in c around inf

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

   11. Applied rewrites 74.7%

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

   if 7.5999999999999998e98 < i

   1. Initial program 55.7%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -14000000000:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;i \leq 1.05 \cdot 10^{-168}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), y \cdot \left(x \cdot z\right)\right)\\ \mathbf{elif}\;i \leq 7.6 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), -c \cdot \left(z \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma t (fma j c (* x (- a))) (* z (fma c (- b) (* x y))))))
   (if (<= t -1.6e-31)
     t_1
     (if (<= t 6e-35)
       (fma y (fma j (- i) (* x z)) (* b (fma c (- z) (* a i))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(t, fma(j, c, (x * -a)), (z * fma(c, -b, (x * y))));
	double tmp;
	if (t <= -1.6e-31) {
		tmp = t_1;
	} else if (t <= 6e-35) {
		tmp = fma(y, fma(j, -i, (x * z)), (b * fma(c, -z, (a * i))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(z * fma(c, Float64(-b), Float64(x * y))))
	tmp = 0.0
	if (t <= -1.6e-31)
		tmp = t_1;
	elseif (t <= 6e-35)
		tmp = fma(y, fma(j, Float64(-i), Float64(x * z)), Float64(b * fma(c, Float64(-z), Float64(a * i))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.6e-31], t$95$1, If[LessEqual[t, 6e-35], N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.60000000000000009e-31 or 5.99999999999999978e-35 < t

   1. Initial program 67.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 77.7%

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

   if -1.60000000000000009e-31 < t < 5.99999999999999978e-35

   1. Initial program 81.3%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

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

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. mul-1-neg N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 76.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\right)\\ \mathbf{if}\;i \leq -15500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 1.76 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma i (fma b a (* y (- j))) (* c (fma j t (* z (- b)))))))
   (if (<= i -15500000000.0)
     t_1
     (if (<= i 1.76e+21)
       (fma t (fma j c (* x (- a))) (* z (fma c (- b) (* x y))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(i, fma(b, a, (y * -j)), (c * fma(j, t, (z * -b))));
	double tmp;
	if (i <= -15500000000.0) {
		tmp = t_1;
	} else if (i <= 1.76e+21) {
		tmp = fma(t, fma(j, c, (x * -a)), (z * fma(c, -b, (x * y))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(i, fma(b, a, Float64(y * Float64(-j))), Float64(c * fma(j, t, Float64(z * Float64(-b)))))
	tmp = 0.0
	if (i <= -15500000000.0)
		tmp = t_1;
	elseif (i <= 1.76e+21)
		tmp = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(z * fma(c, Float64(-b), Float64(x * y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision] + N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -15500000000.0], t$95$1, If[LessEqual[i, 1.76e+21], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.55e10 or 1.76e21 < i

   1. Initial program 62.6%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.1%

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

   9. Taylor expanded in c around inf

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

   11. Applied rewrites 68.2%

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

   if -1.55e10 < i < 1.76e21

   1. Initial program 83.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 81.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -15500000000:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\right)\\ \mathbf{elif}\;i \leq 1.76 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ t_2 := \mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), c \cdot t\_1\right)\\ \mathbf{if}\;i \leq -5400000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;i \leq 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(c, t\_1, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma j t (* z (- b))))
        (t_2 (fma i (fma b a (* y (- j))) (* c t_1))))
   (if (<= i -5400000000.0)
     t_2
     (if (<= i 1e+24) (fma c t_1 (* x (- (* y z) (* t a)))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(j, t, (z * -b));
	double t_2 = fma(i, fma(b, a, (y * -j)), (c * t_1));
	double tmp;
	if (i <= -5400000000.0) {
		tmp = t_2;
	} else if (i <= 1e+24) {
		tmp = fma(c, t_1, (x * ((y * z) - (t * a))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(j, t, Float64(z * Float64(-b)))
	t_2 = fma(i, fma(b, a, Float64(y * Float64(-j))), Float64(c * t_1))
	tmp = 0.0
	if (i <= -5400000000.0)
		tmp = t_2;
	elseif (i <= 1e+24)
		tmp = fma(c, t_1, Float64(x * Float64(Float64(y * z) - Float64(t * a))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision] + N[(c * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -5400000000.0], t$95$2, If[LessEqual[i, 1e+24], N[(c * t$95$1 + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if i < -5.4e9 or 9.9999999999999998e23 < i

   1. Initial program 62.6%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.1%

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

   9. Taylor expanded in c around inf

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

   11. Applied rewrites 68.2%

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

   if -5.4e9 < i < 9.9999999999999998e23

   1. Initial program 83.8%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 77.2%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5400000000:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\right)\\ \mathbf{elif}\;i \leq 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, x, i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), y \cdot \left(x \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -2.8e+83)
   (* y (fma z x (* i (- j))))
   (if (<= y 2.65e+157)
     (fma i (fma b a (* y (- j))) (* c (fma j t (* z (- b)))))
     (fma t (fma j c (* x (- a))) (* 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 (y <= -2.8e+83) {
		tmp = y * fma(z, x, (i * -j));
	} else if (y <= 2.65e+157) {
		tmp = fma(i, fma(b, a, (y * -j)), (c * fma(j, t, (z * -b))));
	} else {
		tmp = fma(t, fma(j, c, (x * -a)), (y * (x * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -2.8e+83)
		tmp = Float64(y * fma(z, x, Float64(i * Float64(-j))));
	elseif (y <= 2.65e+157)
		tmp = fma(i, fma(b, a, Float64(y * Float64(-j))), Float64(c * fma(j, t, Float64(z * Float64(-b)))));
	else
		tmp = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(y * Float64(x * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -2.8e+83], N[(y * N[(z * x + N[(i * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.65e+157], N[(i * N[(b * a + N[(y * (-j)), $MachinePrecision]), $MachinePrecision] + N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8e83

   1. Initial program 58.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 30.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 80.4%

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

   12. Taylor expanded in y around inf

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

       3. sub-neg N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

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

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

       9. mul-1-neg N/A

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

       10. lower-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 74.2

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

   14. Applied rewrites 74.2%

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

   if -2.8e83 < y < 2.6499999999999999e157

   1. Initial program 76.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 80.6%

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

   9. Taylor expanded in c around inf

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

   11. Applied rewrites 68.1%

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

   if 2.6499999999999999e157 < y

   1. Initial program 73.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 74.9%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, x, i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(b, a, y \cdot \left(-j\right)\right), c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), y \cdot \left(x \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), y \cdot \left(x \cdot z\right)\right)\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-175}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+38}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma t (fma j c (* x (- a))) (* y (* x z)))))
   (if (<= t -1.55e-75)
     t_1
     (if (<= t -2e-175)
       (* i (fma j (- y) (* a b)))
       (if (<= t 5e+38) (* z (fma y x (* b (- c)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(t, fma(j, c, (x * -a)), (y * (x * z)));
	double tmp;
	if (t <= -1.55e-75) {
		tmp = t_1;
	} else if (t <= -2e-175) {
		tmp = i * fma(j, -y, (a * b));
	} else if (t <= 5e+38) {
		tmp = z * fma(y, x, (b * -c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(y * Float64(x * z)))
	tmp = 0.0
	if (t <= -1.55e-75)
		tmp = t_1;
	elseif (t <= -2e-175)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (t <= 5e+38)
		tmp = Float64(z * fma(y, x, Float64(b * Float64(-c))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.55e-75], t$95$1, If[LessEqual[t, -2e-175], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e+38], N[(z * N[(y * x + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.55000000000000003e-75 or 4.9999999999999997e38 < t

   1. Initial program 68.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 62.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 72.3%

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

   if -1.55000000000000003e-75 < t < -2e-175

   1. Initial program 79.6%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 57.9

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

   5. Applied rewrites 57.9%

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

   if -2e-175 < t < 4.9999999999999997e38

   1. Initial program 79.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

   7. Applied rewrites 56.1%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), y \cdot \left(x \cdot z\right)\right)\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-175}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+38}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), y \cdot \left(x \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, x, i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(t, t\_1, -c \cdot \left(z \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t\_1, y \cdot \left(x \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma j c (* x (- a)))))
   (if (<= y -1.05e+23)
     (* y (fma z x (* i (- j))))
     (if (<= y 1.3e+173)
       (fma t t_1 (- (* c (* z b))))
       (fma t t_1 (* y (* x z)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(j, c, (x * -a));
	double tmp;
	if (y <= -1.05e+23) {
		tmp = y * fma(z, x, (i * -j));
	} else if (y <= 1.3e+173) {
		tmp = fma(t, t_1, -(c * (z * b)));
	} else {
		tmp = fma(t, t_1, (y * (x * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(j, c, Float64(x * Float64(-a)))
	tmp = 0.0
	if (y <= -1.05e+23)
		tmp = Float64(y * fma(z, x, Float64(i * Float64(-j))));
	elseif (y <= 1.3e+173)
		tmp = fma(t, t_1, Float64(-Float64(c * Float64(z * b))));
	else
		tmp = fma(t, t_1, Float64(y * Float64(x * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+23], N[(y * N[(z * x + N[(i * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+173], N[(t * t$95$1 + (-N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(t * t$95$1 + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0500000000000001e23

   1. Initial program 61.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 31.6%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 75.5%

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

   12. Taylor expanded in y around inf

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

       3. sub-neg N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

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

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

       9. mul-1-neg N/A

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

       10. lower-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 69.1

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

   14. Applied rewrites 69.1%

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

   if -1.0500000000000001e23 < y < 1.2999999999999999e173

   1. Initial program 76.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 52.5%

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

   9. Taylor expanded in c around inf

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

   11. Applied rewrites 61.0%

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

   if 1.2999999999999999e173 < y

   1. Initial program 78.6%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 41.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 79.8%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, x, i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), -c \cdot \left(z \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), y \cdot \left(x \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{if}\;a \leq -5.2 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -3.8 \cdot 10^{-270}:\\ \;\;\;\;-c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;a \leq 8.6 \cdot 10^{-219}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-21}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (fma t (- x) (* b i)))))
   (if (<= a -5.2e+25)
     t_1
     (if (<= a -3.8e-270)
       (- (* c (* z b)))
       (if (<= a 8.6e-219)
         (* y (* x z))
         (if (<= a 3.6e-21) (* c (* t j)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * fma(t, -x, (b * i));
	double tmp;
	if (a <= -5.2e+25) {
		tmp = t_1;
	} else if (a <= -3.8e-270) {
		tmp = -(c * (z * b));
	} else if (a <= 8.6e-219) {
		tmp = y * (x * z);
	} else if (a <= 3.6e-21) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * fma(t, Float64(-x), Float64(b * i)))
	tmp = 0.0
	if (a <= -5.2e+25)
		tmp = t_1;
	elseif (a <= -3.8e-270)
		tmp = Float64(-Float64(c * Float64(z * b)));
	elseif (a <= 8.6e-219)
		tmp = Float64(y * Float64(x * z));
	elseif (a <= 3.6e-21)
		tmp = Float64(c * Float64(t * j));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5.2e+25], t$95$1, If[LessEqual[a, -3.8e-270], (-N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision]), If[LessEqual[a, 8.6e-219], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.6e-21], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -5.1999999999999997e25 or 3.59999999999999989e-21 < a

   1. Initial program 71.2%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   if -5.1999999999999997e25 < a < -3.80000000000000041e-270

   1. Initial program 69.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 34.6%

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

   10. Applied rewrites 41.1%

       \[\leadsto \left(\left(-b\right) \cdot z\right) \cdot c \]

   if -3.80000000000000041e-270 < a < 8.6000000000000005e-219

   1. Initial program 80.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 24.3%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 43.7%

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

   if 8.6000000000000005e-219 < a < 3.59999999999999989e-21

   1. Initial program 76.7%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 61.4

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 37.8%

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

   10. Applied rewrites 39.4%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+25}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{elif}\;a \leq -3.8 \cdot 10^{-270}:\\ \;\;\;\;-c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;a \leq 8.6 \cdot 10^{-219}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-21}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.05 \cdot 10^{+63}:\\ \;\;\;\;y \cdot \left(i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;i \leq -2.9 \cdot 10^{-180}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;i \leq 6.4 \cdot 10^{-166}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -2.05e+63)
   (* y (* i (- j)))
   (if (<= i -2.9e-180)
     (* j (* t c))
     (if (<= i 6.4e-166)
       (* y (* x z))
       (if (<= i 1.7e+85) (* a (* x (- t))) (* a (* b i)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -2.05e+63) {
		tmp = y * (i * -j);
	} else if (i <= -2.9e-180) {
		tmp = j * (t * c);
	} else if (i <= 6.4e-166) {
		tmp = y * (x * z);
	} else if (i <= 1.7e+85) {
		tmp = a * (x * -t);
	} else {
		tmp = a * (b * i);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (i <= (-2.05d+63)) then
        tmp = y * (i * -j)
    else if (i <= (-2.9d-180)) then
        tmp = j * (t * c)
    else if (i <= 6.4d-166) then
        tmp = y * (x * z)
    else if (i <= 1.7d+85) then
        tmp = a * (x * -t)
    else
        tmp = a * (b * i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -2.05e+63) {
		tmp = y * (i * -j);
	} else if (i <= -2.9e-180) {
		tmp = j * (t * c);
	} else if (i <= 6.4e-166) {
		tmp = y * (x * z);
	} else if (i <= 1.7e+85) {
		tmp = a * (x * -t);
	} else {
		tmp = a * (b * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if i <= -2.05e+63:
		tmp = y * (i * -j)
	elif i <= -2.9e-180:
		tmp = j * (t * c)
	elif i <= 6.4e-166:
		tmp = y * (x * z)
	elif i <= 1.7e+85:
		tmp = a * (x * -t)
	else:
		tmp = a * (b * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -2.05e+63)
		tmp = Float64(y * Float64(i * Float64(-j)));
	elseif (i <= -2.9e-180)
		tmp = Float64(j * Float64(t * c));
	elseif (i <= 6.4e-166)
		tmp = Float64(y * Float64(x * z));
	elseif (i <= 1.7e+85)
		tmp = Float64(a * Float64(x * Float64(-t)));
	else
		tmp = Float64(a * Float64(b * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (i <= -2.05e+63)
		tmp = y * (i * -j);
	elseif (i <= -2.9e-180)
		tmp = j * (t * c);
	elseif (i <= 6.4e-166)
		tmp = y * (x * z);
	elseif (i <= 1.7e+85)
		tmp = a * (x * -t);
	else
		tmp = a * (b * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -2.05e+63], N[(y * N[(i * (-j)), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -2.9e-180], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 6.4e-166], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.7e+85], N[(a * N[(x * (-t)), $MachinePrecision]), $MachinePrecision], N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if i < -2.04999999999999996e63

   1. Initial program 64.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 39.1%

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

   10. Applied rewrites 40.9%

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

   if -2.04999999999999996e63 < i < -2.8999999999999998e-180

   1. Initial program 87.8%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 43.0%

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

   if -2.8999999999999998e-180 < i < 6.40000000000000002e-166

   1. Initial program 76.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 6.3%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 49.8%

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

   if 6.40000000000000002e-166 < i < 1.7000000000000002e85

   1. Initial program 81.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 41.5

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 39.4%

      \[\leadsto a \cdot \left(-t \cdot x\right) \]

   if 1.7000000000000002e85 < i

   1. Initial program 57.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 40.6%

      \[\leadsto a \cdot \left(i \cdot \color{blue}{b}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.05 \cdot 10^{+63}:\\ \;\;\;\;y \cdot \left(i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;i \leq -2.9 \cdot 10^{-180}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;i \leq 6.4 \cdot 10^{-166}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;a \cdot \left(x \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-252}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+92}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (fma y x (* b (- c))))))
   (if (<= z -1.75e+21)
     t_1
     (if (<= z -6.5e-252)
       (* i (fma j (- y) (* a b)))
       (if (<= z 1.05e+92) (* t (fma j c (* x (- a)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * fma(y, x, (b * -c));
	double tmp;
	if (z <= -1.75e+21) {
		tmp = t_1;
	} else if (z <= -6.5e-252) {
		tmp = i * fma(j, -y, (a * b));
	} else if (z <= 1.05e+92) {
		tmp = t * fma(j, c, (x * -a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * fma(y, x, Float64(b * Float64(-c))))
	tmp = 0.0
	if (z <= -1.75e+21)
		tmp = t_1;
	elseif (z <= -6.5e-252)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (z <= 1.05e+92)
		tmp = Float64(t * fma(j, c, Float64(x * Float64(-a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(y * x + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.75e+21], t$95$1, If[LessEqual[z, -6.5e-252], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+92], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.75e21 or 1.04999999999999993e92 < z

   1. Initial program 60.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 69.2

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

   5. Applied rewrites 69.2%

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

   7. Applied rewrites 70.2%

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

   if -1.75e21 < z < -6.4999999999999998e-252

   1. Initial program 80.7%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 51.6

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

   5. Applied rewrites 51.6%

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

   if -6.4999999999999998e-252 < z < 1.04999999999999993e92

   1. Initial program 83.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 57.8

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

   5. Applied rewrites 57.8%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+21}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-252}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+92}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{-101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-252}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+92}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (fma c (- b) (* x y)))))
   (if (<= z -1.05e-101)
     t_1
     (if (<= z -6.5e-252)
       (* i (fma j (- y) (* a b)))
       (if (<= z 1.05e+92) (* t (fma j c (* x (- a)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * fma(c, -b, (x * y));
	double tmp;
	if (z <= -1.05e-101) {
		tmp = t_1;
	} else if (z <= -6.5e-252) {
		tmp = i * fma(j, -y, (a * b));
	} else if (z <= 1.05e+92) {
		tmp = t * fma(j, c, (x * -a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * fma(c, Float64(-b), Float64(x * y)))
	tmp = 0.0
	if (z <= -1.05e-101)
		tmp = t_1;
	elseif (z <= -6.5e-252)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (z <= 1.05e+92)
		tmp = Float64(t * fma(j, c, Float64(x * Float64(-a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.05e-101], t$95$1, If[LessEqual[z, -6.5e-252], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+92], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.05000000000000008e-101 or 1.04999999999999993e92 < z

   1. Initial program 66.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   if -1.05000000000000008e-101 < z < -6.4999999999999998e-252

   1. Initial program 75.5%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   if -6.4999999999999998e-252 < z < 1.04999999999999993e92

   1. Initial program 83.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 57.8

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

   5. Applied rewrites 57.8%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-101}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-252}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+92}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{-29}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq -1.4 \cdot 10^{-236}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, y, t \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -3.2e-29)
   (* c (fma j t (* z (- b))))
   (if (<= c -1.4e-236)
     (* i (fma j (- y) (* a b)))
     (if (<= c 7.2e+132)
       (* x (fma z y (* t (- a))))
       (* c (fma b (- z) (* t j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -3.2e-29) {
		tmp = c * fma(j, t, (z * -b));
	} else if (c <= -1.4e-236) {
		tmp = i * fma(j, -y, (a * b));
	} else if (c <= 7.2e+132) {
		tmp = x * fma(z, y, (t * -a));
	} else {
		tmp = c * fma(b, -z, (t * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -3.2e-29)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-b))));
	elseif (c <= -1.4e-236)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (c <= 7.2e+132)
		tmp = Float64(x * fma(z, y, Float64(t * Float64(-a))));
	else
		tmp = Float64(c * fma(b, Float64(-z), Float64(t * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -3.2e-29], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.4e-236], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.2e+132], N[(x * N[(z * y + N[(t * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(b * (-z) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.2e-29

   1. Initial program 73.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -3.2e-29 < c < -1.39999999999999993e-236

   1. Initial program 80.9%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

   if -1.39999999999999993e-236 < c < 7.20000000000000031e132

   1. Initial program 76.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 48.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 76.9%

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

   12. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. *-commutative N/A

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

       4. mul-1-neg N/A

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

       5. lower-fma.f64 N/A

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

       6. mul-1-neg N/A

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

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

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

       8. mul-1-neg N/A

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

       9. lower-*.f64 N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 52.9

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

   14. Applied rewrites 52.9%

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

   if 7.20000000000000031e132 < c

   1. Initial program 55.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 53.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 53.5%

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

   12. Taylor expanded in c around inf

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

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

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

       7. mul-1-neg N/A

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

       8. lower-fma.f64 N/A

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

       9. mul-1-neg N/A

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

       10. lower-neg.f64 N/A

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

       11. lower-*.f64 71.8

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

   14. Applied rewrites 71.8%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{-29}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq -1.4 \cdot 10^{-236}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, y, t \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 51.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{-29}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{-236}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -3.2e-29)
   (* c (fma j t (* z (- b))))
   (if (<= c -1.1e-236)
     (* i (fma j (- y) (* a b)))
     (if (<= c 7.2e+132)
       (* x (- (* y z) (* t a)))
       (* c (fma b (- z) (* t j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -3.2e-29) {
		tmp = c * fma(j, t, (z * -b));
	} else if (c <= -1.1e-236) {
		tmp = i * fma(j, -y, (a * b));
	} else if (c <= 7.2e+132) {
		tmp = x * ((y * z) - (t * a));
	} else {
		tmp = c * fma(b, -z, (t * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -3.2e-29)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-b))));
	elseif (c <= -1.1e-236)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (c <= 7.2e+132)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	else
		tmp = Float64(c * fma(b, Float64(-z), Float64(t * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -3.2e-29], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.1e-236], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.2e+132], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(b * (-z) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.2e-29

   1. Initial program 73.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -3.2e-29 < c < -1.09999999999999996e-236

   1. Initial program 80.9%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

   if -1.09999999999999996e-236 < c < 7.20000000000000031e132

   1. Initial program 76.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 50.9

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

   5. Applied rewrites 50.9%

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

   if 7.20000000000000031e132 < c

   1. Initial program 55.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 53.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 53.5%

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

   12. Taylor expanded in c around inf

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

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

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

       7. mul-1-neg N/A

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

       8. lower-fma.f64 N/A

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

       9. mul-1-neg N/A

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

       10. lower-neg.f64 N/A

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

       11. lower-*.f64 71.8

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

   14. Applied rewrites 71.8%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{-29}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{-236}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 53.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{-29}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-236}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-10}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -3.2e-29)
   (* c (fma j t (* z (- b))))
   (if (<= c -1.25e-236)
     (* i (fma j (- y) (* a b)))
     (if (<= c 6e-10)
       (* a (fma t (- x) (* b i)))
       (* c (fma b (- z) (* t j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -3.2e-29) {
		tmp = c * fma(j, t, (z * -b));
	} else if (c <= -1.25e-236) {
		tmp = i * fma(j, -y, (a * b));
	} else if (c <= 6e-10) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = c * fma(b, -z, (t * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -3.2e-29)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-b))));
	elseif (c <= -1.25e-236)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (c <= 6e-10)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = Float64(c * fma(b, Float64(-z), Float64(t * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -3.2e-29], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.25e-236], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6e-10], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(b * (-z) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.2e-29

   1. Initial program 73.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -3.2e-29 < c < -1.2499999999999999e-236

   1. Initial program 80.9%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

   if -1.2499999999999999e-236 < c < 6e-10

   1. Initial program 77.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 45.9

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

   5. Applied rewrites 45.9%

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

   if 6e-10 < c

   1. Initial program 63.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, \color{blue}{a \cdot \left(-x\right)}\right), \left(b \cdot a\right) \cdot i\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, \color{blue}{a \cdot \left(\mathsf{neg}\left(x\right)\right)}\right), y \cdot \left(-1 \cdot \left(i \cdot j\right) + \left(x \cdot z + \frac{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)}{y}\right)\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 65.0%

       \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, \color{blue}{a \cdot \left(-x\right)}\right), y \cdot \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(c, -z, i \cdot a\right)}{y}, z \cdot x - j \cdot i\right)\right) \]

   12. Taylor expanded in c around inf

       \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

       2. sub-neg N/A

          \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

       4. +-commutative N/A

          \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

       5. mul-1-neg N/A

          \[\leadsto c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} + j \cdot t\right) \]

       6. distribute-rgt-neg-in N/A

          \[\leadsto c \cdot \left(\color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + j \cdot t\right) \]

       7. mul-1-neg N/A

          \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)} + j \cdot t\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, j \cdot t\right)} \]

       9. mul-1-neg N/A

          \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

       10. lower-neg.f64 N/A

           \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

       11. lower-*.f64 60.8

           \[\leadsto c \cdot \mathsf{fma}\left(b, -z, \color{blue}{j \cdot t}\right) \]

   14. Applied rewrites 60.8%

       \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(b, -z, j \cdot t\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{-29}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-236}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-10}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 30.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -c \cdot \left(z \cdot b\right)\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+189}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+68}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (- (* c (* z b)))))
   (if (<= z -6.2e+189)
     t_1
     (if (<= z -1.35e-6)
       (* z (* x y))
       (if (<= z 3.1e+68) (* c (* t j)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = -(c * (z * b));
	double tmp;
	if (z <= -6.2e+189) {
		tmp = t_1;
	} else if (z <= -1.35e-6) {
		tmp = z * (x * y);
	} else if (z <= 3.1e+68) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -(c * (z * b))
    if (z <= (-6.2d+189)) then
        tmp = t_1
    else if (z <= (-1.35d-6)) then
        tmp = z * (x * y)
    else if (z <= 3.1d+68) then
        tmp = c * (t * j)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = -(c * (z * b));
	double tmp;
	if (z <= -6.2e+189) {
		tmp = t_1;
	} else if (z <= -1.35e-6) {
		tmp = z * (x * y);
	} else if (z <= 3.1e+68) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = -(c * (z * b))
	tmp = 0
	if z <= -6.2e+189:
		tmp = t_1
	elif z <= -1.35e-6:
		tmp = z * (x * y)
	elif z <= 3.1e+68:
		tmp = c * (t * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(-Float64(c * Float64(z * b)))
	tmp = 0.0
	if (z <= -6.2e+189)
		tmp = t_1;
	elseif (z <= -1.35e-6)
		tmp = Float64(z * Float64(x * y));
	elseif (z <= 3.1e+68)
		tmp = Float64(c * Float64(t * j));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = -(c * (z * b));
	tmp = 0.0;
	if (z <= -6.2e+189)
		tmp = t_1;
	elseif (z <= -1.35e-6)
		tmp = z * (x * y);
	elseif (z <= 3.1e+68)
		tmp = c * (t * j);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = (-N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[z, -6.2e+189], t$95$1, If[LessEqual[z, -1.35e-6], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.1e+68], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.1999999999999999e189 or 3.0999999999999998e68 < z

   1. Initial program 63.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 71.0

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.9%

      \[\leadsto -b \cdot \left(c \cdot z\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 51.7%

       \[\leadsto \left(\left(-b\right) \cdot z\right) \cdot c \]

   if -6.1999999999999999e189 < z < -1.34999999999999999e-6

   1. Initial program 59.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 59.7

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Applied rewrites 59.7%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto z \cdot \left(x \cdot \color{blue}{y}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 41.4%

      \[\leadsto z \cdot \left(y \cdot \color{blue}{x}\right) \]

   if -1.34999999999999999e-6 < z < 3.0999999999999998e68

   1. Initial program 82.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \mathsf{neg}\left(\color{blue}{y \cdot i}\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(\mathsf{neg}\left(i\right)\right)}\right) \]

      6. neg-mul-1 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(-1 \cdot i\right)}\right) \]

      8. neg-mul-1 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(\mathsf{neg}\left(i\right)\right)}\right) \]

      9. lower-neg.f64 50.8

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-i\right)}\right) \]

   5. Applied rewrites 50.8%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 32.1%

      \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{t} \]
   9. Step-by-step derivation

   10. Applied rewrites 34.8%

       \[\leadsto \left(t \cdot j\right) \cdot c \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+189}:\\ \;\;\;\;-c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+68}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;-c \cdot \left(z \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 52.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{-95}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-10}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -1.1e-95)
   (* c (fma j t (* z (- b))))
   (if (<= c 6e-10) (* a (fma t (- x) (* b i))) (* c (fma b (- z) (* t j))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -1.1e-95) {
		tmp = c * fma(j, t, (z * -b));
	} else if (c <= 6e-10) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = c * fma(b, -z, (t * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -1.1e-95)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-b))));
	elseif (c <= 6e-10)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = Float64(c * fma(b, Float64(-z), Float64(t * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -1.1e-95], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6e-10], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(b * (-z) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.0999999999999999e-95

   1. Initial program 73.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(-1 \cdot z\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      10. lower-neg.f64 61.6

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right) \]

   5. Applied rewrites 61.6%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)} \]

   if -1.0999999999999999e-95 < c < 6e-10

   1. Initial program 80.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 44.9

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 44.9%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   if 6e-10 < c

   1. Initial program 63.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(c \cdot \left(j \cdot t\right) + y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)\right)\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, \color{blue}{a \cdot \left(\mathsf{neg}\left(x\right)\right)}\right), a \cdot \left(b \cdot i\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, \color{blue}{a \cdot \left(-x\right)}\right), \left(b \cdot a\right) \cdot i\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, \color{blue}{a \cdot \left(\mathsf{neg}\left(x\right)\right)}\right), y \cdot \left(-1 \cdot \left(i \cdot j\right) + \left(x \cdot z + \frac{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)}{y}\right)\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 65.0%

       \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, \color{blue}{a \cdot \left(-x\right)}\right), y \cdot \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(c, -z, i \cdot a\right)}{y}, z \cdot x - j \cdot i\right)\right) \]

   12. Taylor expanded in c around inf

       \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

       2. sub-neg N/A

          \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

       4. +-commutative N/A

          \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

       5. mul-1-neg N/A

          \[\leadsto c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} + j \cdot t\right) \]

       6. distribute-rgt-neg-in N/A

          \[\leadsto c \cdot \left(\color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + j \cdot t\right) \]

       7. mul-1-neg N/A

          \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)} + j \cdot t\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, j \cdot t\right)} \]

       9. mul-1-neg N/A

          \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

       10. lower-neg.f64 N/A

           \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

       11. lower-*.f64 60.8

           \[\leadsto c \cdot \mathsf{fma}\left(b, -z, \color{blue}{j \cdot t}\right) \]

   14. Applied rewrites 60.8%

       \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(b, -z, j \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{-95}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-10}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 53.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \mathbf{if}\;c \leq -1950000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-10}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (fma b (- z) (* t j)))))
   (if (<= c -1950000000.0)
     t_1
     (if (<= c 6e-10) (* a (fma t (- x) (* b i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * fma(b, -z, (t * j));
	double tmp;
	if (c <= -1950000000.0) {
		tmp = t_1;
	} else if (c <= 6e-10) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * fma(b, Float64(-z), Float64(t * j)))
	tmp = 0.0
	if (c <= -1950000000.0)
		tmp = t_1;
	elseif (c <= 6e-10)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(b * (-z) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1950000000.0], t$95$1, If[LessEqual[c, 6e-10], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.95e9 or 6e-10 < c

   1. Initial program 67.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(c \cdot \left(j \cdot t\right) + y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]

   5. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)\right)\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, \color{blue}{a \cdot \left(\mathsf{neg}\left(x\right)\right)}\right), a \cdot \left(b \cdot i\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 52.0%

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, \color{blue}{a \cdot \left(-x\right)}\right), \left(b \cdot a\right) \cdot i\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, \color{blue}{a \cdot \left(\mathsf{neg}\left(x\right)\right)}\right), y \cdot \left(-1 \cdot \left(i \cdot j\right) + \left(x \cdot z + \frac{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)}{y}\right)\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 62.9%

       \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, \color{blue}{a \cdot \left(-x\right)}\right), y \cdot \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(c, -z, i \cdot a\right)}{y}, z \cdot x - j \cdot i\right)\right) \]

   12. Taylor expanded in c around inf

       \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

       2. sub-neg N/A

          \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

       3. mul-1-neg N/A

          \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

       4. +-commutative N/A

          \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

       5. mul-1-neg N/A

          \[\leadsto c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} + j \cdot t\right) \]

       6. distribute-rgt-neg-in N/A

          \[\leadsto c \cdot \left(\color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + j \cdot t\right) \]

       7. mul-1-neg N/A

          \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)} + j \cdot t\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, j \cdot t\right)} \]

       9. mul-1-neg N/A

          \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

       10. lower-neg.f64 N/A

           \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

       11. lower-*.f64 65.8

           \[\leadsto c \cdot \mathsf{fma}\left(b, -z, \color{blue}{j \cdot t}\right) \]

   14. Applied rewrites 65.8%

       \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(b, -z, j \cdot t\right)} \]

   if -1.95e9 < c < 6e-10

   1. Initial program 78.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 42.6

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 42.6%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1950000000:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-10}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 43.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+178}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (* x y))))
   (if (<= y -2.2e+124)
     t_1
     (if (<= y 1.05e+178) (* b (fma c (- z) (* a i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (y <= -2.2e+124) {
		tmp = t_1;
	} else if (y <= 1.05e+178) {
		tmp = b * fma(c, -z, (a * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * Float64(x * y))
	tmp = 0.0
	if (y <= -2.2e+124)
		tmp = t_1;
	elseif (y <= 1.05e+178)
		tmp = Float64(b * fma(c, Float64(-z), Float64(a * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e+124], t$95$1, If[LessEqual[y, 1.05e+178], N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2000000000000001e124 or 1.0499999999999999e178 < y

   1. Initial program 63.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 65.0

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Applied rewrites 65.0%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto z \cdot \left(x \cdot \color{blue}{y}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 60.2%

      \[\leadsto z \cdot \left(y \cdot \color{blue}{x}\right) \]

   if -2.2000000000000001e124 < y < 1.0499999999999999e178

   1. Initial program 76.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \]

      3. remove-double-neg N/A

         \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \]

      9. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \color{blue}{a \cdot i}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot z, a \cdot i\right)} \]

      15. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      16. lower-neg.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(z\right), \color{blue}{i \cdot a}\right) \]

      18. lower-*.f64 42.5

          \[\leadsto b \cdot \mathsf{fma}\left(c, -z, \color{blue}{i \cdot a}\right) \]

   5. Applied rewrites 42.5%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+124}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+178}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 30.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(t \cdot j\right)\\ \mathbf{if}\;j \leq -3.8 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq -7 \cdot 10^{+18}:\\ \;\;\;\;-b \cdot \left(z \cdot c\right)\\ \mathbf{elif}\;j \leq 5.5 \cdot 10^{+70}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* t j))))
   (if (<= j -3.8e+92)
     t_1
     (if (<= j -7e+18)
       (- (* b (* z c)))
       (if (<= j 5.5e+70) (* z (* x y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -3.8e+92) {
		tmp = t_1;
	} else if (j <= -7e+18) {
		tmp = -(b * (z * c));
	} else if (j <= 5.5e+70) {
		tmp = z * (x * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (t * j)
    if (j <= (-3.8d+92)) then
        tmp = t_1
    else if (j <= (-7d+18)) then
        tmp = -(b * (z * c))
    else if (j <= 5.5d+70) then
        tmp = z * (x * y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -3.8e+92) {
		tmp = t_1;
	} else if (j <= -7e+18) {
		tmp = -(b * (z * c));
	} else if (j <= 5.5e+70) {
		tmp = z * (x * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (t * j)
	tmp = 0
	if j <= -3.8e+92:
		tmp = t_1
	elif j <= -7e+18:
		tmp = -(b * (z * c))
	elif j <= 5.5e+70:
		tmp = z * (x * y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(t * j))
	tmp = 0.0
	if (j <= -3.8e+92)
		tmp = t_1;
	elseif (j <= -7e+18)
		tmp = Float64(-Float64(b * Float64(z * c)));
	elseif (j <= 5.5e+70)
		tmp = Float64(z * Float64(x * y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (t * j);
	tmp = 0.0;
	if (j <= -3.8e+92)
		tmp = t_1;
	elseif (j <= -7e+18)
		tmp = -(b * (z * c));
	elseif (j <= 5.5e+70)
		tmp = z * (x * y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -3.8e+92], t$95$1, If[LessEqual[j, -7e+18], (-N[(b * N[(z * c), $MachinePrecision]), $MachinePrecision]), If[LessEqual[j, 5.5e+70], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -3.8e92 or 5.49999999999999986e70 < j

   1. Initial program 74.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \mathsf{neg}\left(\color{blue}{y \cdot i}\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(\mathsf{neg}\left(i\right)\right)}\right) \]

      6. neg-mul-1 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(-1 \cdot i\right)}\right) \]

      8. neg-mul-1 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(\mathsf{neg}\left(i\right)\right)}\right) \]

      9. lower-neg.f64 61.5

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-i\right)}\right) \]

   5. Applied rewrites 61.5%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.1%

      \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{t} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.8%

       \[\leadsto \left(t \cdot j\right) \cdot c \]

   if -3.8e92 < j < -7e18

   1. Initial program 70.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 53.6

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.9%

      \[\leadsto -b \cdot \left(c \cdot z\right) \]

   if -7e18 < j < 5.49999999999999986e70

   1. Initial program 72.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 54.5

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Applied rewrites 54.5%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto z \cdot \left(x \cdot \color{blue}{y}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 34.0%

      \[\leadsto z \cdot \left(y \cdot \color{blue}{x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -3.8 \cdot 10^{+92}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;j \leq -7 \cdot 10^{+18}:\\ \;\;\;\;-b \cdot \left(z \cdot c\right)\\ \mathbf{elif}\;j \leq 5.5 \cdot 10^{+70}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 30.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(t \cdot j\right)\\ \mathbf{if}\;j \leq -1.25 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 5.5 \cdot 10^{+70}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* t j))))
   (if (<= j -1.25e+19) t_1 (if (<= j 5.5e+70) (* z (* x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -1.25e+19) {
		tmp = t_1;
	} else if (j <= 5.5e+70) {
		tmp = z * (x * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (t * j)
    if (j <= (-1.25d+19)) then
        tmp = t_1
    else if (j <= 5.5d+70) then
        tmp = z * (x * y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -1.25e+19) {
		tmp = t_1;
	} else if (j <= 5.5e+70) {
		tmp = z * (x * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (t * j)
	tmp = 0
	if j <= -1.25e+19:
		tmp = t_1
	elif j <= 5.5e+70:
		tmp = z * (x * y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(t * j))
	tmp = 0.0
	if (j <= -1.25e+19)
		tmp = t_1;
	elseif (j <= 5.5e+70)
		tmp = Float64(z * Float64(x * y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (t * j);
	tmp = 0.0;
	if (j <= -1.25e+19)
		tmp = t_1;
	elseif (j <= 5.5e+70)
		tmp = z * (x * y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -1.25e+19], t$95$1, If[LessEqual[j, 5.5e+70], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -1.25e19 or 5.49999999999999986e70 < j

   1. Initial program 74.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \mathsf{neg}\left(\color{blue}{y \cdot i}\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(\mathsf{neg}\left(i\right)\right)}\right) \]

      6. neg-mul-1 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(-1 \cdot i\right)}\right) \]

      8. neg-mul-1 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(\mathsf{neg}\left(i\right)\right)}\right) \]

      9. lower-neg.f64 60.3

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-i\right)}\right) \]

   5. Applied rewrites 60.3%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.7%

      \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{t} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.9%

       \[\leadsto \left(t \cdot j\right) \cdot c \]

   if -1.25e19 < j < 5.49999999999999986e70

   1. Initial program 72.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 54.5

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Applied rewrites 54.5%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto z \cdot \left(x \cdot \color{blue}{y}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 34.0%

      \[\leadsto z \cdot \left(y \cdot \color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.25 \cdot 10^{+19}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;j \leq 5.5 \cdot 10^{+70}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 30.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(t \cdot j\right)\\ \mathbf{if}\;j \leq -2.25 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 3.8 \cdot 10^{+76}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* t j))))
   (if (<= j -2.25e+18) t_1 (if (<= j 3.8e+76) (* y (* x z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2.25e+18) {
		tmp = t_1;
	} else if (j <= 3.8e+76) {
		tmp = y * (x * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (t * j)
    if (j <= (-2.25d+18)) then
        tmp = t_1
    else if (j <= 3.8d+76) then
        tmp = y * (x * z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2.25e+18) {
		tmp = t_1;
	} else if (j <= 3.8e+76) {
		tmp = y * (x * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (t * j)
	tmp = 0
	if j <= -2.25e+18:
		tmp = t_1
	elif j <= 3.8e+76:
		tmp = y * (x * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(t * j))
	tmp = 0.0
	if (j <= -2.25e+18)
		tmp = t_1;
	elseif (j <= 3.8e+76)
		tmp = Float64(y * Float64(x * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (t * j);
	tmp = 0.0;
	if (j <= -2.25e+18)
		tmp = t_1;
	elseif (j <= 3.8e+76)
		tmp = y * (x * z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -2.25e+18], t$95$1, If[LessEqual[j, 3.8e+76], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -2.25e18 or 3.80000000000000024e76 < j

   1. Initial program 74.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \mathsf{neg}\left(\color{blue}{y \cdot i}\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(\mathsf{neg}\left(i\right)\right)}\right) \]

      6. neg-mul-1 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(-1 \cdot i\right)}\right) \]

      8. neg-mul-1 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(\mathsf{neg}\left(i\right)\right)}\right) \]

      9. lower-neg.f64 60.3

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-i\right)}\right) \]

   5. Applied rewrites 60.3%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.7%

      \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{t} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.9%

       \[\leadsto \left(t \cdot j\right) \cdot c \]

   if -2.25e18 < j < 3.80000000000000024e76

   1. Initial program 72.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \color{blue}{\left(\left(j \cdot y\right) \cdot i\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

      10. cancel-sign-sub N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(a\right)\right) \cdot b\right)} \]

      11. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(-1 \cdot a\right)} \cdot b\right) \]

      12. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{-1 \cdot \left(a \cdot b\right)}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - a \cdot t, i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)\right)} \]

   5. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - t \cdot a, i \cdot \mathsf{fma}\left(j, -y, b \cdot a\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto -1 \cdot \color{blue}{\left(i \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.3%

      \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(-y\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 33.3%

       \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.25 \cdot 10^{+18}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;j \leq 3.8 \cdot 10^{+76}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 29.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+142}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (* y z))))
   (if (<= y -4.5e-15) t_1 (if (<= y 3.9e+142) (* c (* t j)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (y <= -4.5e-15) {
		tmp = t_1;
	} else if (y <= 3.9e+142) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y * z)
    if (y <= (-4.5d-15)) then
        tmp = t_1
    else if (y <= 3.9d+142) then
        tmp = c * (t * j)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (y <= -4.5e-15) {
		tmp = t_1;
	} else if (y <= 3.9e+142) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = x * (y * z)
	tmp = 0
	if y <= -4.5e-15:
		tmp = t_1
	elif y <= 3.9e+142:
		tmp = c * (t * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(x * Float64(y * z))
	tmp = 0.0
	if (y <= -4.5e-15)
		tmp = t_1;
	elseif (y <= 3.9e+142)
		tmp = Float64(c * Float64(t * j));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = x * (y * z);
	tmp = 0.0;
	if (y <= -4.5e-15)
		tmp = t_1;
	elseif (y <= 3.9e+142)
		tmp = c * (t * j);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.5e-15], t$95$1, If[LessEqual[y, 3.9e+142], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4999999999999998e-15 or 3.9e142 < y

   1. Initial program 65.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 59.2

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Applied rewrites 59.2%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.0%

      \[\leadsto -b \cdot \left(c \cdot z\right) \]

   9. Taylor expanded in c around 0

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 46.6%

       \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

   if -4.4999999999999998e-15 < y < 3.9e142

   1. Initial program 77.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \mathsf{neg}\left(\color{blue}{y \cdot i}\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(\mathsf{neg}\left(i\right)\right)}\right) \]

      6. neg-mul-1 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(-1 \cdot i\right)}\right) \]

      8. neg-mul-1 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(\mathsf{neg}\left(i\right)\right)}\right) \]

      9. lower-neg.f64 40.9

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-i\right)}\right) \]

   5. Applied rewrites 40.9%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 28.0%

      \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{t} \]
   9. Step-by-step derivation

   10. Applied rewrites 31.0%

       \[\leadsto \left(t \cdot j\right) \cdot c \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+142}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 23.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(t \cdot j\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* c (* t j)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return c * (t * j);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = c * (t * j)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return c * (t * j);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return c * (t * j)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(c * Float64(t * j))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = c * (t * j);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.3%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in j around inf

   \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

   2. sub-neg N/A

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto j \cdot \mathsf{fma}\left(c, t, \mathsf{neg}\left(\color{blue}{y \cdot i}\right)\right) \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(\mathsf{neg}\left(i\right)\right)}\right) \]

   6. neg-mul-1 N/A

      \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(-1 \cdot i\right)}\right) \]

   8. neg-mul-1 N/A

      \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(\mathsf{neg}\left(i\right)\right)}\right) \]

   9. lower-neg.f64 40.5

      \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-i\right)}\right) \]

5. Applied rewrites 40.5%

   \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right)} \]

6. Taylor expanded in c around inf

   \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
7. Step-by-step derivation

8. Applied rewrites 24.6%

   \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{t} \]
9. Step-by-step derivation

10. Applied rewrites 26.1%

    \[\leadsto \left(t \cdot j\right) \cdot c \]

11. Final simplification 26.1%

    \[\leadsto c \cdot \left(t \cdot j\right) \]
12. Add Preprocessing

Alternative 29: 23.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ t \cdot \left(c \cdot j\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* t (* c j)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return t * (c * j);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = t * (c * j)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return t * (c * j);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return t * (c * j)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(t * Float64(c * j))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = t * (c * j);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(t * N[(c * j), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.3%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in j around inf

   \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

   2. sub-neg N/A

      \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto j \cdot \mathsf{fma}\left(c, t, \mathsf{neg}\left(\color{blue}{y \cdot i}\right)\right) \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(\mathsf{neg}\left(i\right)\right)}\right) \]

   6. neg-mul-1 N/A

      \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(-1 \cdot i\right)}\right) \]

   8. neg-mul-1 N/A

      \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(\mathsf{neg}\left(i\right)\right)}\right) \]

   9. lower-neg.f64 40.5

      \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-i\right)}\right) \]

5. Applied rewrites 40.5%

   \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right)} \]

6. Taylor expanded in c around inf

   \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
7. Step-by-step derivation

8. Applied rewrites 24.6%

   \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{t} \]

9. Final simplification 24.6%

   \[\leadsto t \cdot \left(c \cdot j\right) \]
10. Add Preprocessing

Developer Target 1: 68.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1015122364899489/125000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -942510763643697/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -238547917063487/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 10535888557455487/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))